ISSN2188－1200

Proceedings of Forum "Math-for-Industry" 2021
-Mathematics for Digital EconomyChief Editors: Osamu Saeki, Ho Tu Bao

Editors: Shizuo Kaji, Kenji Kajiwara,Nguyen Ha Nam, Ta Hai Tung,

Melanie Roberts, Masato Wakayama, Le Minh Ha,
Philip Broadbridge
九州大学マス・フォア・インダストリ研究所

MI Lecture Note Vol.87 : Kyushu University

Proceedings of Forum “Math-for-Industry” 2021
-Mathematics for Digital Economy-■Chief Editors:
Osamu Saeki (Kyushu University, Japan)
Ho Tu Bao (Vietnam Institute for Advanced Study in Mathematics, Vietnam)

Editors:
Shizuo Kaji (Kyushu University, Japan)
Kenji Kajiwara (Kyushu University, Japan)
Nguyen Ha Nam (Vietnam Institute for Advanced Study in Mathematics,
Vietnam)
Ta Hai Tung (Hanoi University of Science and Technology, Vietnam)
Melanie Roberts (Griffith University in Brisbane, Australia)
Masato Wakayama (Institute for Fundamental Mathematics, NTT, Japan)
Le Minh Ha (Vietnam Institute for Advanced Study in Mathematics, Vietnam)
Philip Broadbridge (La Trobe University, Australia)

AAbout MI Lecture Note Series
The Math-for-Industry (MI) Lecture Note Series is the successor to the COE Lecture
Notes, which were published for the 21st COE Program “Development of Dynamic
Mathematics with High Functionality,” sponsored by Japan’s Ministry of Education,
Culture, Sports, Science and Technology (MEXT) from 2003 to 2007. The MI Lecture Note Series has published the notes of lectures organized under the following two
programs: “Training Program for Ph.D. and New Master’s Degree in Mathematics as
Required by Industry,” adopted as a Support Program for Improving Graduate School
Education by MEXT from 2007 to 2009; and “Education-and-Research Hub for
Mathematics-for-Industry,” adopted as a Global COE Program by MEXT from 2008 to2012.
In accordance with the establishment of the Institute of Mathematics for Industry (IMI)in April 2011 and the authorization of IMI’s Joint Research Center for Advanced andFundamental Mathematics-for-Industry as a MEXT Joint Usage / Research Center in
April 2013, hereafter the MI Lecture Notes Series will publish lecture notes and proceedings by worldwide researchers of MI to contribute to the development of MI.
October 2018
Osamu Saeki
Director
Institute of Mathematics for Industry

Proceedings of Forum “Math-for-Industry” 2021
-Mathematics for Digital Economy-

MI Lecture Note Vol.87, Institute of Mathematics for Industry, Kyushu University
ISSN 2188-1200
Date of issue: March 28, 2022
Chief Editors: O samu Saeki, Ho Tu Bao hizuo Kaji, Kenji Kajiwara, Nguyen Ha Nam, Ta Hai Tung,
Editors: S
Melanie Roberts, Masato Wakayama, Le Minh Ha, Philip Broadbridge
Publisher:
Institute of Mathematics for Industry, Kyushu University
Graduate School of Mathematics, Kyushu University
Motooka 744, Nishi-ku, Fukuoka, 819-0395, JAPANTel +81-(0)92-802-4402, Fax +81-(0)92-802-4405
URL https://www.imi.kyushu-u.ac.jp/

BPreface
The FMfI2021 has been organized by the Vietnam Institute for Advanced Study in Mathematics, theInstitute of Mathematics for Industry, and the Asia Pacific Consortium of Mathematics for Industry.
This is the first time that the Forum was held in Hanoi, Vietnam - December 13-16, 2021.
The dramatic acceleration of digital transformation and the increasing role of applied mathematics
across the world inspired us to run the FMfI2021 theme “Mathematics for Digital Economy”. It
converged about 30 prominent scholars and industry experts to deliver excellent lectures in acomprehensive program.
A special session named ‘Mathematics of Covid-19’ was also included in response to the pandemic,
with various modelling insights into the fight against it. There were further 28 posters selected topresent at the Forum, covering a wide range of topics in many branches of mathematics. Their
presentations, though short, were delivered in an unexpectedly interactive and interesting manner.
FMfI2021 was attended by more than 200 participants both in person and online, many attending it for
the first time. I do believe the participants have gained fruitful and unforgettable experience at the
FMfI2021.
I would like to acknowledge each and every person in the organizing staff, the organizing committee,
invited speakers committee, and poster prize committee for their cooperative spirit and tremendous
support. They worked very hard to make FMfI2021 a continued success in the history of the 11-year
Institute. FMfI2021 indeed met its intended goals and reached broad participation. I would also like toexpress my gratitude to the FMfI2021 speakers for contributing their research results to the conference.
Without their commitment and dedication, the proceedings could not have been produced.
I once again sincerely thank you all for making it all happen. It was with great pleasure that mycolleagues and I had this opportunity to host a conference of this magnitude.
Le Minh Ha
Managing Director
Vietnam Institute for Advanced Study in Mathematics (VIASM)

I

0

II

The Asia-Pacific Consortium of Mathematics for Industry(APCMfI)
Mathematics for Industry (MfI) aims at the development of mathematics and its
applications to enhance the quality of life on the planet by creating newtechnologies, improve industrial mathematical research and stimulate the twoway interaction between mathematics and industry. In Industrial Mathematics,
it is the questions spawned by real world applications that drive the resulting
two-way interaction between a particular application and the associatedmathematics that is utilized and developed, and that sometimes involves, quite
unexpectedly, deeper aspects and new areas of mathematics than initiallyanticipated.
Though its significance has often been overlooked, industrial mathematics has
always been an essential aspect of the history, culture, traditions anddevelopment of mathematics, including much of modern theoretical mathematics.
Directly and indirectly, developments in mathematics can be traced to the initialattempts to answer quite practical questions. The development of Galileo's
telescope and the design of clocks represent early stimuli. Harmonic analysis andFourier analysis have their origins in the study of heat transfer in metals. The
conservation and minimization of energy engendered in the study ofthermodynamics and fluid motion underlie much of the foundations of modern
theoretical mathematics, as well as applied and industrial mathematics.
The increasing sophistication of modern industry, reflected in, for example,
medical measurements, game theory applications in economics, psychology,
behavioral science and biology, computer-controlled instrumentation, theefficient development of geothermal energy, the microbial treatment of waste
water, Ito calculus in finance, etc., has generated a need and demand formathematical expertise to stimulate, foster and implement the associatedinnovations. Even the theoretical areas of algebraic geometry, abstract algebra,
topology, differential geometry and group theory are playing an increasinglyIIIimportant role in industrial endeavors connected with entertainment (such asgames and movies), architecture, analysis of protein structure and errorcorrecting codes.
There is general agreement and support in the Asia-Pacific region to have regularindustrial mathematics exchanges, conferences, internships, etc., which build on
the activities already occurring. In fact, over the years since the concept of an
Asian Consortium of Mathematics for Industry was first proposed and more
recently when planning to formalize possibilities, there has been strong supportand encouragement from colleagues in China, Hawaii, Korea, Malaysia andSingapore as well as Australia, New Zealand and Japan.
A small group, with the encouragement of various colleagues throughout the
Asia-Pacific region, met in Canberra, March 31 to April 2, 2014, to do the initialplanning for the formation and launch of APCMfI, with the emphasis being
fundamentally Mathematics-for-Industry. Those directly involved in thediscussions in Canberra were Bob Anderssen (Australia), Zainal Aziz (Malaysia),
Frank de Hoog (Australia), Yasuhide Fukumoto (Japan), Alexandra Hogan(Australia), Geoff Mercer (Australia), Masato Wakayama (Japan) and Graeme
Wake (New Zealand).
In any endeavours that involve the initiation and implementation of a newopportunity, the situation is similar to planting and nurturing a seed which willgrow into a strong and robust tree. The meeting and deliberations of this grouprepresented the preparation of the ground for the planting of the seed. The
subsequent planting and nurturing involves the wide distribution of thisinitiative throughout the Asia-Pacific region; the seeking of seed funding fromvarious mathematics departments, societies, agencies and industry; theestablishment of a website; the launch of APCMfI under the MfI banner.
In 2021 the APCMfI turned into its second generation and reorganized theadministration; the Council is chaired by Zainal Aziz (President) and driven by
the Steering Committee, Philip Broadbridge(Australia, Vice President), Kenji
Kajiwara (Japan, Secretary), Shizuo Kaji (Japan, Treasurer), and MelanieIVRoberts (Australia, Communications). Other eight Council Members are from
Australia, China, Korea, New Zealand, and Thailand. Among thirteen Council
Members there are five female Members. The APCMfI will expand its activities
to form a platform of collaborations of industrial and applied mathematics in the
Asia Pacific region.

Planned Activities for APCMfI
An important component of the plans for APCMfI is a number of activities
through which it interacts directly with the Asia-Pacific MfI communities andindirectly with the various international industrial mathematics consortia,
organizations and individuals.
The underling goal is to stimulate the development of mathematics and its
applications to enhance the quality of life on the planet by creating newtechnologies, improve industrial mathematical research and stimulate the twoway interaction between mathematics and industry.
The planned activities include:a. facilitating the creation of internships for graduate students to work on
industrial and governmental research projects in the Asia Pacific region; inprinciple, interns will spend several months working at their home institution
and several months working with an industrial partner.
b. the promotion of regular Mathematics-for-Industry Study Groups (MfISG)
having a strong Asia Pacific component with respect to both the problems to bestudied and participation, taking advantage of study groups already operating inAustralia, New Zealand, Japan and Malaysia,
c. the development within APCMfI of similar events e.g. “year projects” to theregular Mathematics-for-Industry Forums and Workshops, building on the
successful annual Forums organized by the Institute of Mathematics for Industry(“IMI”) at Kyushu University,Vd. the utilization of APCMfI for the exchange of information and publicity
materials about industrial mathematics activities in the Asia Pacific region, suchas electronic newsletters, publications, websites, etc.,
e. the organization of lectures and programs, either live or by video conference,that foster student participation by taking advantage of the similar time zones
in the Asia Pacific region,f. the fostering of a strong two-way interaction between (i) individuals andinstitutions engaged in mathematical and statistical research, and (ii) the needs
and opportunities of industrial mathematics,
g. the development of synergetic links with other similar or relevantorganizations, andh. the identification of an international project that several governments mightvalue and support.

History of the Forums "Math-for-Industry"
The Forums now have a decade-long history. Initiated by the Institute ofMathematics for Industry (IMI) at Kyushu University in Japan in 2010, theForums have provided a meeting place for mathematical minds, and also toprovide insights that enable the endeavors of industry-focused researchers to beshared within the region.
2010
Oct 21–23

Fukuoka, Japan2011
Oct 24–28

Honolulu, US
TSUNAMI - Mathematical Modelling Using Mathematics for
Natural Disaster: Prediction, Recovery and Provision for the
Future2012
Oct 22–26

Fukuoka, Japan

Information Security, Visualization, and Inverse Problems, on thebasis of Optimization Techniques

VI

Information Recovery and Discovery
2013
Nov 4–8

Fukuoka, JapanThe Impact of Applications on Mathematics

2014
Oct 27–31

Fukuoka, Japan
Applications + Practical Conceptualization + Mathematics =Fruitful Innovation

In 2014, the Asia-Pacific Consortium of Mathematics for Industry (APCMfI) was
formed, and the forums started to move around the Consortium's member
countries, with themes that reflected each country's interests.
2015
Fukuoka, Japan
Oct. 26–30 The Role and Importance of Mathematics in Innovation
2016
Nov. 21–
23

Brisbane, AU
Agriculture as a Metaphor for Creativity in all Human Endeavors

2017
Honolulu, US
Oct. 23–26 Responding to the Challenges of Climate Change: Exploiting,Harnessing and Enhancing the Opportunities of Clean Energy
2018
Shanghai, PRC
Nov. 17-21 Big Data Analysis, AI, Fintech, Math in Finance and Economics
2019
Nov. 18–
21

Auckland, NZ
Mathematics for the Primary Industries and the Environment2021
Dec 13-16

Hanoi, Vietnam
Mathematics for Digital Economy

FMfI2022 will be held in late November or mid-December in Melbourne,
Australia, hosted by La Trobe University. FMfI2023 will be hosted by the IMI,
Kyushu University, Japan. It is planned to be a satellite meeting of theInternational Congress of Industrial and Applied Mathematics in Tokyo

VII

(ICIAM2023). It will be held one week prior to or after ICIAM2023 which isscheduled during 20-26 August 2023.
It is clear that the Forums traverse a wide range of topics, and that the abilities
of mathematicians to address these affirm the importance of such specialists in
the increasingly-complex ways in which society operates. The value thatquantitative scientists and engineers provide to all communities cannot be underestimated. While most people appreciate effective and efficiently-operating
systems, they often do not realize how these come about, and who is providing
the sophisticated processes that underlie their efficiency.
While the speakers are experienced in their fields, the students who present
posters and give talks about their work are the future leaders in APCMfI; they
are valuable members of the "Math-for-Industry" community, and areparticularly welcome at this Forum.
*Information about the APCMfI and FMfI is extracted from the APCMfI website andthe FMfI2019*

VIII

The Vietnam Institute for Advanced Study in Mathematics
The Vietnam Institute for Advanced Study in Mathematics (VIASM) was established in late2010 and officially came into operation on June 1st, 2011. The scientific director of the Instituteis Professor Ngo Bao Chau, the 2010 Field Medalist.
Since then, our institute has become the meeting point for international and Vietnamesemathematicians, exchanging ideas, initiating new research projects, collaborating andconnecting with young Vietnamese researchers and students.
We aim to promote and initiate basic research activities in mathematics and mathematicaleducation in Vietnam, collaborating with other academic and research institutes around theworld to strengthen the research and education ecosystem.
The main activity of the Institute is organizing research groups to conduct research programs
and projects of high quality. Scientists in the same field will gather and work together at theInstitute on a short-term basis. It aims to attract Vietnamese mathematicians from abroad andinternational mathematicians to Vietnam to participate in research and training together with
their colleagues in Vietnam. This activity will strengthen the research branches which havetaken root in Vietnam, and will incubate the formation of new branches of Mathematics.
Every year, the institute offers up to 5 Postdoctoral fellowships, and organizes conferences,
workshops, seminars on topics associated with research groups working at the Institute in orderto implement their research projects as well as attract new students to do research.
Our institute is also responsible for the implementation of the National Program for theDevelopment of Mathematics in Vietnam, now in the second phase from 2021 to 2030. Under
this program, we help organize many teacher training seminars, and outreach activities toencourage young students to learn mathematics, improve the quality of teaching and learning
mathematics, as well as disseminate scientific knowledge to the public.

IX

FMfI 2021

Forum "Math-for-Industry" 2021

ProgramPosterRegistration

AccessAccommodation

Organising Committee

Invited Speakers Committee

Le Minh Ha (Chair), Vietnam Institute for

Ho Tu Bao (Chair), Vietnam Institute forAdvanced Study in Mathematics

Advanced Study in Mathematics

Trinh Thi Thuy Giang, Vietnam Institute for

Alona Ben-Tal, Massey University, New Zealand

Advanced Study in Mathematics

Philip Broadbridge, La Trobe University,

Start 08:30, ﬁnish 15:30 each day (Hanoi time)

Shizuo Kaji, Kyushu University, Japan

Australia

Nguyen Ha Nam, Vietnam Institute for Advanced

Jin Cheng, Fudan University, China

12/12 Sunday morning from 8am: IMI-IAB,

Study in Mathematics

Yasuhide Fukumoto, Kyushu University, JapanTa Hai Tung, Hanoi University of Science andNguyen Xuan Hung, Ho Chi Minh City

Technology, Vietnam

University of Technology, Vietnam

Melanie Roberts, Griﬃth University in Brisbane,

Kenji Kajiwara, Kyushu University, Japan

Australia

Vu Hoang Linh, Vietnam National University,

Osamu Saeki, Kyushu University, Japan

Hanoi

Masato Wakayama, Tokyo University of Science,

Vu Ha Van, Yale University, USA and VingroupJapan

Big Data Institute, Vietnam-Mathematics for Digital EconomyDate: December 13-16, 2021.
Venue: Vietnam Institute for Advanced Study inMathematics

APCMfI (Council and AGM), Journal Board
meetings
14/12 Tuesday afternoon: Poster Session andShort Communication
14/12 Tuesday evening: Banquet (depending on
the covid-19 situation in Hanoi, Vietnam)
15/12 Wednesday morning: Special Session:
Mathematics of Covid-19

Theme may include:

FMfI2021 will include:

Digital economy can be understood as an* Invited talks

economy in the digital environment where much

* Emerging Researcher talks

more data than ever and connection of* Student posters

everything. It consists of three groups:* Special Session: Mathematics of Covid-19- ICT sector* Short Communication- Industries in which business models are closely* Meetings of APCMfI + IMI-IAB + Journal

related to digital technology (e.g. Finance)

Editorial Boards- Traditional industries trying to supplement theirpractices with digital technology
Math for digital economy may include or be
related to:- Machine Learning/Data Science- Optimization in Industry- Information Security- Blockchain- Math Modelling- Big Data
- Business Analytics- and others

X

XI

XII

XIII

Local news about FMfI2021
I am very pleased to share that the FMfI2021 has been covered in leading Vietnamese magazines andnewspapers. This shows the Forum as well as Math for Industry have gained positive local recognition,and the collective effort of the FMfI2021 team did not go unnoticed.
1. Local News about FMfI2021(from Vietnam)
https://apcmfi.org/event/view/179
2. Các nhà khoa học Việt ứng dụng Toán học trong ứng phó với Covid-19
https://dantri.com.vn/giao-duc-huong-nghiep/cac-nha-khoa-hoc-viet-ung-dung-toan-hoctrong-ung-pho-voi-covid1920211213172455652.htm?zarsrc=31&utm_source=zalo&utm_medium=zalo&utm_campaign=zalo
3. Ứng dụng Toán học trong các hoạt động phòng chống Covid-19
https://vov2.vov.vn/giao-duc-dao-tao/ung-dung-toan-hoc-trong-cac-hoat-dong-phong-chongcovid-19-31383.vov2
4. Ứng dụng Toán học trong các hoạt động phòng chống Covid-19
https://vietnamnet.vn/vn/giao-duc/toan-hoc-co-nhieu-ung-dung-trong-viec-chong-lai-covid19-800897.html?fbclid=IwAR0O4EiHbcPL39XxO4wNDXgWkM6MxUc3uVtp76HVpLVo5P-5Dg6j1gldJE

5. VIASM chủ trì hội nghị quốc tế về ứng dụng toán học cho nền kinh tế sốhttps://khoahocphattrien.vn/thoi-su-trong-nuoc/viasm-chu-tri-hoi-nghi-quoc-te-ve-ung-dungtoan-hoc-cho-nen-kinh-te-so/20211213105218775p882c918.htm

6. Ứng dụng toán học cho nền kinh tế số như thế nào?
https://dantri.com.vn/giao-duc-huong-nghiep/ung-dung-toan-hoc-cho-nen-kinh-te-so-nhu-thenao-20211212113916798.htm
7. Giáo sư Katsuki Fujisawa: "Không thể tạo ra tiến bộ nếu không có Toán học"https://dantri.com.vn/giao-duc-huong-nghiep/giao-su-katsuki-fujisawa-khong-the-tao-ra-tienbo-neu-khong-co-toan-hoc-20211225093121436.htm

XIV

XV

Join the FMfI2021:

Washio Takashi
Japan

Ngo Duc Thanh
VietnamGraham Williams
Australia

Lunch

Alexander Lipton
IsraelBreak

Julian Jang-Jaccard
New Zealand

Volkan Cevher
Switzerland

10.30-11.10

11.10-11.50

11.50-12.30

12.30-13.30

13.30-14.10

14.10-14.20

14.20-15.00

15.00-15.40

11.00. Journal Boards

Refreshment break

Nathan Kutz
USA

10.10-10.30

09.30-10.10

9.00. APCMfI Board

Welcome and Opening

Registration

10.00. APCMfI AGM

09.00-09.30

08.30–09.00

Monday 13 Dec

8.00. IMI IAB

Sunday 12 Dec○ Passcode: 107102○ Meeting ID: 939 1135 9124

14.00-16.00

13.30-14.00

12.00-13.30

11.20-12.00

10.40-11.20

10.00-10.40

09.50-10.00

09.10-09.50

08.30-09.10

15.00-15.40

14.20-15.00

Jessada Tanthanuch
Thailand
Poster Session

14.10–14.20

Amir Mosavic
Hungary

Vincent Y. F. Tan
Singapore

Break

Mai Anh Tien
Singapore

13.30–14.10

17.00-20.00

15.00-15.20

14.10–14.20

BanquetClosing
Osamu Saeki

Break

Jin Cheng
China

Lunch

12.00-13.30
Lunch

12.00-13.30

13.30–14.10

Nguyen Dinh Hoa
Japan

Kohei HatanoJapan

Xiaoping Lu
Australia

10.00-10.40
10.40-11.20

Break

Caleb Moses
New Zealand

Yaniv Gal
New Zealand

09.50-10.00

09.10-09.50

08.30-09.10

11.20-12.00

Nguyen Ngoc Doanh
Vietnam

Emily Harvey
New Zealand

Break

Michael Lydeamore
Australia

Stefan Canzar
Chile

Thursday 16 Dec

Shingo Iwami
Japan

11.20-12.00

10.40-11.20

10.00-10.40

09.50-10.00

09.10-09.50

08.30-09.10

Mathematics of Covid-19

Wednesday 15 Dec

Shortcommunications
Aung Zaw Myint
Myanmar

Lunch

Takashi TsuchiyaJapan

Yu Jiang
China

Hien Nguyen
Australia

Break

Lim Ee-Peng
Singapore

Kazue SakoJapan

Tuesday 14 Dec

https://zoom.us/j/93911359124?pwd=R0lmZmYycEpJT1BVbGRPZnRLM2tpUT09

THEME: Mathematics for Digital Economy

December 12-16

Program FMfI2021 Vietnam

Kenji Kajiwara
Le Minh Ha

13.30-14.10

Melanie Roberts

Jin Cheng

Alona Ben-tal

Philip Broadbridge

Shizuo Kaji

Hung Nguyen Xuan

Hoang Linh Vu

Zainal Aziz

Ho Tu Bao

Yasuhide Fukumoto

Osamu Saeki

10.00-12.00

Thu 8.30-9.50

13.30-15.40

10.00-12.00

Wed 8.30-9.50

14.00-16.00

13.30-14.00

10.00-12.00

Tue 8.30-9.50

13.30-15.40

10.30-12.30

Mon 9.30-10.10

Session Chairs

XVI

2.00

3.00

4.00

5.00

23.00

0.00

1.00

0.00

20.00

22.00

23.00

19.00

1.00

22.00

18.00

21.00

Chile

USA (EST)World Time (December 2021)

9.00

8.00

7.00

6.00

5.00

4.00

3.00

2.00

Europe

14.30

13.30

12.30

11.30

10.30

9.30

8.30

7.30

Myanmar

15.00

14.00

13.00

12.00

11.00

10.00

9.00

8.00

Vietnam

16.00

15.00

14.00

13.00

12.00

11.00

10.00

9.00

Singapore/China

17.00

16.00

15.00

14.00

13.00

12.00

11.00

10.00

Japan

19.00

18.00

17.00

16.00

15.00

14.00

13.00

12.00

Australia(AEDT)

21.00

20.00

19.00

18.00

17.00

16.00

15.00

14.00

New Zealand

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Towards Minimax Optimal Best Arm Identification In Linear Bandits
Vincent Y. F. Tan and Junwen Yang
Global and Local Prediction Methods of COVID-19 Time Series with Machine Learning
Amir Mosavi, Sina Ardebili, Annamaria R and Varkonyi-Koczy
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Yaniv Gal

Language models in industry and around the world ��������������������������������������������������������������� 75
Caleb Moses

Option pricing with transaction costs –mathematical modelling in new digital economy ������������� 76
Xiaoping Lu

Blackwell game and its applications in online prediction tasks������������������������������������������������� 83
Kohei Hatano

Mutuality between AI and Optimization������������������������������������������������������������������������������ 84
Nguyen Dinh Hoa

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Jin Cheng

XIXForum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Learning Dynamical Systems Models from Data
J. Nathan Kutz

Applied Mathematics, University of Washington, USA

A major challenge in the study of dynamical systems is that of model discovery:
turning data into reduced order models that are not just predictive, but provide insight
into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamicalsystems and their embeddings from data. We consider two canonical cases: (i) systems
for which we have full measurements of the governing variables, and (ii) systems forwhich we have incomplete measurements. For systems with full state measurements,
we show that the recent sparse identiﬁcation of nonlinear dynamical systems (SINDy)
method can discover governing equations with relatively little data and introduce asampling method that allows SINDy to scale eﬃciently to problems with multiple timescales, noise and parametric dependencies. For systems with incomplete observations,
we show that the Hankel alternative view of Koopman (HAVOK) method, based ontime-delay embedding coordinates and the dynamic mode decomposition, can be used
to obtain a linear model and Koopman invariant measurement systems that nearly
perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks
are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to
discover and model nonlinear multiscale systems.

1Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Rare Event Search and Fast Data Assimilation
for Industry in the Digital Twin Era
Takashi WASHIOISIR, Osaka University, and
AIRC, The National Institute of Advanced Industrial Science and Technology, Japan

Modern society has now entered the digital twin era, where simulation models of
many systems are constructed, and highly reliable and eﬃcient designs and operationsof the systems are expected to be carried out using simulations. Under this movement,
enormous research activities on developing simulation techniques and models are currently underway in various ﬁelds. However, generic techniques to eﬃciently constructhigh quality designs and operation plans of the systems using the simulations have notbeen suﬃciently studied. Such techniques must be developed by fusing mathematicaloptimization and simulation approaches in elaborating manners.
In this talk, ﬁrst, we show techniques to eﬃciently discover rare events, which occur
under very special conditions with extremely low probabilities, using simulations guided
by mathematical search principles [1]. We demonstrate an eﬃcient scheme to design
highly reliable products in industry using the techniques. Second, we show techniques
for data assimilation which automatically and eﬃciently tune the simulation model
parameters to reﬂect real system dynamics [2]. Particularly, our techniques enable toﬁnd the accurate parameter values using only a few observations of the real system.
We demonstrate quick monitoring of dynamics changes of an industrial factory and its
prompt operation alteration to maintain the productivity.
This talk suggests an important R&D direction of applied mathematics for futureindustry.
References[1] Keiichi Kisamori, Takashi Washio, Yoshio Kameda and Ryohei Fujimaki, A Rare and Critical Condition Search Technique and its Application to Telescope Stray Light Analysis, Proc.
the 2018 SIAM International Conference on Data Mining (SDM2018), 2018. Read More:https://epubs.siam.org/doi/pdf/10.1137/1.9781611975321.64
https://archive.siam.org/meetings/sdm18/[2] Keiichi Kisamori, Motonobu Kanagawa and Keisuke Yamazaki, Simulator Calibration
under Covariate Shift with Kernels, Proc. the 23rdInternational Conference on Artiﬁcial Intelligence and Statistics (AISTATS2020), PMLR: Vol.108, 2020. Read More:
http://proceedings.mlr.press/v108/kisamori20a/kisamori20a.pdf
https://aistats.org/aistats2020/2

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Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Climate change modelling in Southeast Asia andfuture climate information for the society
Thanh NGO-DUC

Department of Space and Applications, University of Science and Technology of
Hanoi, Vietnam(joint work with CORDEX-SEA’s team & Quentin DESMET, LEGOS, France)

Today, 8.6% of the world population is living in Southeast Asia (SEA). Any change
in the climate system can have unequivocal impacts on the region’s socio-economic
structures and living conditions. Given the high exposure and vulnerability of theregion to extreme events, countries in SEA need to implement adaptation measures
to lower their risk. Detailed information on future climate scenarios is thus needed.
However, such information is still lacking in the region or generally based on global
climate models (GCMs) that may have large uncertainties in a complex region such as
SEA. In order to ﬁll the gap, the Coordinated Regional Climate Downscaling EXperiment - Southeast Asia (CORDEX-SEA) project was established and had successfully
gathered members from several countries to carry out a high resolution multi-model
regional climate downscaling experiment.
In this presentation, an overview of climate change modeling activities in Southeast
Asia and the recent ﬁndings of the CORDEX-SEA downscaling activities with the
Coupled Model Intercomparison Project Phase 5 (CMIP5) are ﬁrst introduced. We
address how simulation of present-day extremes is inﬂuenced by the choices of various
physical parameterizations to determine which schemes are well suited to simulate theclimate extremes over the region. Future projected rainfall, extremes, and surface windin association with tropical cyclone activities in SEA are subsequently analyzed. Lastly,
we focus on a regional evaluation of 26 CMIP6 GCMs over SEA by introducing a novel
ranking method based on temperature, rainfall, and wind distributions. The evaluation
provides the CORDEX-SEA community with a reduced number of CMIP6 models
with better performance over the region, which can be used in a further downscaling
experiment.

8Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Simply Deploying AI and ML
Graham Williams

Software Innovation Institute, Australian National University, Australia

With the extraordinary growth in research outputs in artiﬁcial intelligence, machinelearning, and data science, industry struggles to keep pace. Developers in industry
generally have limited time to explore and experiment with new algorithms coming
out of our research labs at their current pace. Trialling a new technique can takeconsiderable eﬀort, even when the developers in industry have solid experience anddata at the ready. The MLHub.ai initiative is a fully open source framework that aims
to facilitate the exploration of new algorithms with minimal initial overhead. This
presentation will set the scene and introduce a framework for easing our access to thelatest research, illustrating its utility with industry collaborators.

9Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Forex Trading Utilizing Consensus as a Service on
Blockchains
Alexander Lipton

Jerusalem Business School, Hebrew University of Jerusalem, Israel(joint work with Artur Sepp, Sygnum Bank, Zurich, Switzerland)

We present an automated market-making (AMM) cross-settlement mechanism for
digital assets on interoperable blockchains, focusing on central bank digital currencies(CBDCs) and stable coins. We develop an innovative approach for generating fairexchange rates for on-chain assets consistent with traditional oﬀ-chain markets. We
illustrate the eﬃcacy of our approach on realized FX rates for G-10 currencies.

10Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Artiﬁcial Intelligence (AI) for Intrusion Detection
and Math
Julian Jang-Jaccard

Massey University, New Zealand

Cybersecurity Lab at Massey University, founded in 2016, has been one of thefastest-growing research labs in NZ dedicated to providing cutting-edge research theory, tools, and methodologies to improve the cybersecurity posture. With generous
funds awarded from the NZ government, the lab has been dedicated to developing a
set of novel cyber-resilient systems using the advancement of the latest AI techniques,both including machine and deep learnings, that can rapidly detect and classify various intrusions including malware. In this presentation, I will present a set of AI-based
techniques (e.g., Autoencoder, Multi-Layer Perceptron, Deep Q-learning based Reinforcement Learning, Generative Adversarial Network) we have developed in the last
few years and discuss the type of math skills demanded in these areas.

11Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Optimization challenges in adversarial machinelearning
Volkan Cevher

EPFL - Swiss Federal Institute of Technology Lausanne, Switzerland(joint work with Panayotis Mertikopoulos, Thomas Pethick, Ya-Ping Hsieh, Nadav
Hallak, Ali Kavis)
Thanks to neural networks (NNs), faster computation, and massive datasets, machine learning (ML) is under increasing pressure to provide automated solutions toeven harder real-world tasks beyond human performance with ever faster response
times due to potentially huge technological and societal beneﬁts. Unsurprisingly, theNN learning formulations present a fundamental challenge to the back-end learningalgorithms despite their scalability, in particular due to the existence of traps in the
non-convex optimization landscape, such as saddle points, that can prevent algorithms
from obtaining “good” solutions.
In this talk, we describe our recent research that has demonstrated that the nonconvex optimization dogma is false by showing that scalable stochastic optimization
algorithms can avoid traps and rapidly obtain locally optimal solutions. Coupled withthe progress in representation learning, such as over-parameterized neural networks,
such local solutions can be globally optimal.
Unfortunately, this talk will also demonstrate that the central min-max optimization
problems in ML, such as generative adversarial networks (GANs), robust reinforcement
learning (RL), and distributionally robust ML, contain spurious attractors that do notinclude any stationary points of the original learning formulation. Indeed, we will
describe how algorithms are subject to a grander challenge, including unavoidable
convergence failures, which could explain the stagnation in their progress despite theimpressive earlier demonstrations. We will conclude with promising new preliminary
results from our recent progress on some of these diﬃcult challenges.

12Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Cryptography and Transparency
Kazue Sako

Waseda University, Japan

In a digitalized society, we everyday use computers to receive messages from ourfriends, buy tickets online, and receive personal ads for attractive products on sale.
However, as these are represented as digital data, it is diﬃcult to verify whether thesedata sent from other computers are trustworthy. In this talk, we will discuss sometools using cryptography that makes the procedures occurring on the other computertransparent, thus increasing trustworthiness.
Keywords: veriﬁability, digital signature, zero-knowledge proofs, blockchain.

13

Forum:We will be using Green and Red Buttons

Maths for Industry
2021.12.14

Cryptography and Transparency
Kazue Sako
Dept. Computer Science and Engineering
Waseda University:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

Zoom Poll 1

2

Cryptographic Foundations I

Do you know there are public-key encryptions and secret-key
encryptions
Yes
I have heard of it, but not sure how they’re different
No
Don’t want to answer

Secret-key Encryption

Public-key Encryption

a.k.a Symmetric key encryption

a.k.a Asymmetric key encryption

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34RSA Cryptosystem(Rivest-Shamir-Adleman)1978

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Prof. Kazue Sako

• 㻹㼍㼖㼛㼞㼑㼐 㼙㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟 㼕㼚 㻷㼥㼛㼠㼛 㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥• 㻿㼛㼛㼚㻌㼍㼒㼠㼑㼞㻌㼖㼛㼕㼚㼕㼚㼓㻌㻺㻱㻯㻘㻌㼙㼥㻌㼎㼛㼟㼟㻌㼓㼍㼢㼑㻌㼙㼑㻌㼍㼚㻌㼍㼞㼠㼕㼏㼘㼑㻌㼛㼚㻌㻾㻿㻭㻌㼏㼞㼥㼜㼠㼛㼟㼥㼟㼠㼑㼙(QFU\SWLRQe

C = M (mod N)$Q\RQH FDQ HQFU\SW'HFU\SWLRQd

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5

14

6

RSA Cryptosystem(Rivest-Shamir-Adleman)1978

N is a productof 2 primes p,q.H\ 6HWXS)LQGHG DQG1VW IRUDOO0ed00 PRG1
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Factorization ofN(2048bits)
being difficult, dis kept secure!

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C = M (mod N)'HFU\SWLRQd

M = C (mod N)

Self Introduction
Prof. Kazue Sako

ed=1 mod Φ(N)ΦʁEuler Function
Fermat’s little theorem$Q\RQH FDQ HQFU\SW• 㻹㼍㼖㼛㼞㼑㼐 㼙㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟 㼕㼚 㻷㼥㼛㼠㼛 㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥• 㻿㼛㼛㼚㻌㼍㼒㼠㼑㼞㻌㼖㼛㼕㼚㼕㼚㼓㻌㻺㻱㻯㻘㻌㼙㼥㻌㼎㼛㼟㼟㻌㼓㼍㼢㼑㻌㼙㼑㻌㼍㼚㻌㼍㼞㼠㼕㼏㼘㼑㻌㼛㼚㻌㻾㻿㻭㻌㼏㼞㼥㼜㼠㼛㼟㼥㼟㼠㼑㼙• 㻵㻌㼒㼑㼘㼘㻌㼕㼚㻌㼘㼛㼢㼑㻌㼣㼕㼠㼔㻌㼏㼞㼥㼜㼠㼛㼓㼞㼍㼜㼔㼥㻚㻌㼀㼔㼑㼥㻌㼍㼞㼑㻌㼠㼔㼑㻌㼠㼛㼛㼘㼟㻌㼠㼛㻌㼙㼍㼗㼑㻌㼛㼡㼞㻌㼟㼛㼏㼕㼑㼠㼥㻌㼙㼛㼞㼑㻌㼟㼑㼏㼡㼞㼑㻘㻌㼙㼛㼞㼑㻌㼜㼞㼕㼢㼍㼏㼥㻙㼒㼞㼕㼑㼚㼐㼘㼥㻌㼍㼚㼐㻌㼙㼛㼞㼑㻌㼒㼍㼕㼞㻚• 㻵㻌㼖㼛㼕㼚㼑㼐㻌㼃㼍㼟㼑㼐㼍 㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼒㼞㼛㼙㻌㻞㻜㻞㻜㻘㻌㼐㼑㼠㼑㼞㼙㼕㼚㼑㼐㻌㼠㼔㼍㼠㻌㼕㼠䇻㼟㻌㼥㼛㼡㼚㼓㻌㼟㼠㼡㼐㼑㼚㼠㼟㻌㼣㼔㼛㻌㼣㼕㼘㼘㻌㼡㼟㼑㻌㼠㼔㼑㼟㼑㻌㼠㼛㼛㼘㼟㻌㼠㼛㻌㼙㼍㼗㼑㻌㼛㼡㼞㻌㼟㼛㼏㼕㼑㼠㼥㻌㼎㼑㼠㼠㼑㼞㻚

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7

Self Introduction

8

Today’s talkProf. Kazue Sako

Cryptography and Transparency• 㻹㼍㼖㼛㼞㼑㼐 㼙㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟 㼕㼚 㻷㼥㼛㼠㼛 㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥In a digitalized society, we everyday use computers toreceive messages from our friends, buy tickets online, andreceive personal ads for attractive products on sale.
However, as these are represented as digital data, it isdifficult to verify whether these data sent from othercomputers are trustworthy. In this talk, we will discuss
some tools using cryptography that makes the procedures
occurring on the other computer transparent, thusincreasing trustworthiness.

• 㻿㼛㼛㼚㻌㼍㼒㼠㼑㼞㻌㼖㼛㼕㼚㼕㼚㼓㻌㻺㻱㻯㻘㻌㼙㼥㻌㼎㼛㼟㼟㻌㼓㼍㼢㼑㻌㼙㼑㻌㼍㼚㻌㼍㼞㼠㼕㼏㼘㼑㻌㼛㼚㻌㻾㻿㻭㻌㼏㼞㼥㼜㼠㼛㼟㼥㼟㼠㼑㼙• 㻵㻌㼒㼑㼘㼘㻌㼕㼚㻌㼘㼛㼢㼑㻌㼣㼕㼠㼔㻌㼏㼞㼥㼜㼠㼛㼓㼞㼍㼜㼔㼥㻚㻌㼀㼔㼑㼥㻌㼍㼞㼑㻌㼠㼔㼑㻌㼠㼛㼛㼘㼟㻌㼠㼛㻌㼙㼍㼗㼑㻌㼛㼡㼞㻌㼟㼛㼏㼕㼑㼠㼥㻌㼙㼛㼞㼑㻌㼟㼑㼏㼡㼞㼑㻘㻌㼙㼛㼞㼑㻌㼜㼞㼕㼢㼍㼏㼥㻙㼒㼞㼕㼑㼚㼐㼘㼥㻌㼍㼚㼐㻌㼙㼛㼞㼑㻌㼒㼍㼕㼞㻚• 㻵㻌㼖㼛㼕㼚㼑㼐㻌㼃㼍㼟㼑㼐㼍 㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼒㼞㼛㼙㻌㻞㻜㻞㻜㻘㻌㼐㼑㼠㼑㼞㼙㼕㼚㼑㼐㻌㼠㼔㼍㼠㻌㼕㼠䇻㼟㻌㼥㼛㼡㼚㼓㻌㼟㼠㼡㼐㼑㼚㼠㼟㻌㼣㼔㼛㻌㼣㼕㼘㼘㻌㼡㼟㼑㻌㼠㼔㼑㼟㼑㻌㼠㼛㼛㼘㼟㻌㼠㼛㻌㼙㼍㼗㼑㻌㼛㼡㼞㻌㼟㼛㼏㼕㼑㼠㼥㻌㼎㼑㼠㼠㼑㼞㻚• 㻲㼛㼞㼙㼑㼞㻌㻼㼞㼑㼟㼕㼐㼑㼚㼠㻌㼛㼒㻌㻶㼍㼜㼍㼚㻌㻿㼛㼏㼕㼑㼠㼥㻌㼛㼒㻌㻵㼚㼐㼡㼟㼠㼞㼕㼍㼘㻌㼍㼚㼐㻌㻭㼜㼜㼘㼕㼑㼐㻌㻹㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟㻌㻔㻶㻿㻵㻭㻹㻕

• 㻹㼑㼙㼎㼑㼞㻌㼛㼒㻌㻿㼏㼕㼑㼚㼏㼑㻌㻯㼛㼡㼚㼏㼕㼘㻌㼛㼒㻌㻶㼍㼜㼍㼚• 㼂㼕㼏㼑㻌㻯㼔㼍㼕㼞㻌㼛㼒㻌㻹㼥㻰㼍㼠㼍㻶㼍㼜㼍㼚:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

9Today’s title:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

10:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

12Zoom Poll 2

Do you know digital signature schemes?

Cryptography and TransparencyYes
I have heard of it, but not sure what they areNo
Don’t want to answer

If you used encryption,
wouldn’t everything is
obscured and nottransparent?

:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

1115

Cryptographic Foundations II

Zoom Poll 3

Public-key encryption

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Yes
No
Not true. Encryption is indeed used in Bitcoin
Don’t want to answer

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Did you know that in cryptocurrency Bitcoin, encryptionfunction is not used?

Digital Signature scheme

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Bitcoin and Digital Signature schemes

Today’s title

Public key is the account number(address) !
ࢴ

A

14

Cryptography and Transparency

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Cryptography/ Digital signatures clarifies who said what.
Cryptography ensures that one can not cheat in the system.
Cryptography ensures that if anyone cheated, it can bedetected.

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Cryptography provides transparency to operations in IT
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15

Today’s title
Cryptography and Transparency

Cryptography provides transparency to operations in IT
systems, even the software running on the other computeris not visible.
Ex: votingᵟᶊᶇᶁᶃᴾ

Example: secret e-votingᵴᶍᶒᶇᶌᶅ
ᵟᶓᶒᶆᶍᶐᶇᶒᶗᴾ

How can we design a system to avoid cheating?

Will he count myvote correctly?
Will my votesecrecy
respected?

Is she a voter,and haven’t vote
before?

:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

17

16

16

What kind of cheating do we want to prevent in e-voting?

What kind of cheating do we want to prevent in e-voting?
Only legitimate voters vote, but once

Tally must be correct

But vote secrecy must be maintained!

:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

How do we do in paper-based voting?

ENC Data

ENC Data

BallotBallotAlice KE9SLIWELBob SJAJIWE54S
Chris GKX3RPB9U
Eva QKS769WML
Dave GR83F80BUY

DEC

BallotInput

DEC

Shuffle Enc-Data
And then decrypt
Prove inZero-Knowledge
That process
is correct:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

21

How to shuffle digital data?

Encrypt Ballot
And Sign

SIG

DEC

ENC Data

openSupervisor
ChecksThe process:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

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open

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open

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Ballot in
doubleenvelopes

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Mixnet based voting protocol

VOTE

Ballot in
doubleenvelopes:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

19

22

Probabilistic Encryption

ENC
C+
Randomm Value

After Shuffle???
SJAJIWE54S
QKS769WML
GR83F80BUY
GKX3RPB9U
KE9SLIWEL

Ciphertext SpaceDEC
KE9SLIWEL

message1

IE8XJFN39
P849XKJSN

message1䞉䞉䞉(DV\WRWUDFHEDFN:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

PQJ0D4NXH

23

17:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

24

Probabilistic Encryption

Re-encryption

Input

ENC
C+
Randomm Value

DEC

Alice KE9SLIWELBob SJAJIWE54S
Chris GKX3RPB9U
Eva QKS769WML
Dave GR83F80BUY

KE9SLIWELPQJ0D4NXH

message1

x mod p

OutputSJAJIWE54S
QKS769WML
GR83F80BUY
GKX3RPB9U
KE9SLIWEL

IW0JDLS76
RDQM4LXF8ZPF1EG䞉䞉䞉
EncryptionrE(m) = (gr mod p, mʀy mod p)
for random r

JV7D34S
PQ䡆䠌D4NXH

message1

IE8XJFN39
P849XKJSN

p:prime g : generator
Public key y = g
secret key x

PermuteCiphertext Space

KE9SLIWELPQJ0D4NXH

Change thelook of encryption

Decryption

IE8XJFN39 ….
P849XKJSN

E(m)=(c1,c2) m=c2/(c1)x mod p:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

25

Re-encryption

26

Mixnet based voting protocol

Input

SIG

ENC

Encrypt Ballot
And Sign

JV7D34S
ENC Data

ENC Data

DEC

DEC

DEC

BallotBallotopen

Ballotopen

Supervisor
ChecksThe process

IE8XJFN39 ….
P849XKJSN:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

ENC Data

open

VOTE

Shuffle inner
Envelopes
And open

PQ䡆䠌D4NXH

KE9SLIWELPQJ0D4NXH( c1ʀ gs mod p, c2 ʀ ys mod p )for random s

NAME

Ballot

VOTE

Change thelook of encryption

Ballot in
doubleenvelopes

ENC Data

IW0JDLS76
RDQM4LXF8ZPF1EG

Ballot data

SJAJIWE54S
QKS769WML
GR83F80BUY
GKX3RPB9U
KE9SLIWEL

VOTE

Re encryption E*(c1, c2) =Output
VOTE

Alice KE9SLIWELBob SJAJIWE54S
Chris GKX3RPB9U
Eva QKS769WML
Dave GR83F80BUY

Permute27

Shuffle Enc-Data
And then decrypt
Prove inZero-Knowledge
That process
is correct:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

28:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

30

Zero-knowledge Proof
One-way function: x Ѝ y= x2 mod n (n: public constant)[ LV P\
SDVVZRUG

Zero-knowledge Proofs

If you know sqrt of ymod n, then you
must be Alice!

A powerful tool to ensure correctness without
revealing the witness

x

18

ᵟᶊᶇᶁᶃ

ᵠᶍᶀ

y,n

Zero-knowledge Proofs

Zero-knowledge Proofs

Alice proves she knows x s.t. y=x2 mod nr

u= r2 mod n

x

ᵟᶊᶇᶁᶃ

ᵠᶍᶀ

y,n

Alice proves she knows x s.t. y=x2 mod n

Alice: pick a random number r
and give u= r2 mod n to Bob.
Bob picks a random bit b andgive it to Alice.
Alice computes v= rʀ͉b modn
Bob verifies ifuʀyb = v2 mod n holds:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\r

b= 0 or 1

x

ᵠᶍᶀ

y,n

Alice proves she knows x s.t. y=x2 mod n

Alice: pick a random number r
and give u= r2 mod n to Bob.
Bob picks a random bit b andgive it to Alice.
Alice computes v= rʀ͉b modn
Bob verifies ifuʀyb = v2 mod n holds:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

u= v2 mod n (b=0)
u ʀy = v2 mod n (b=1)

x

r

x

ᵟᶊᶇᶁᶃ

ᵠᶍᶀ

y,n

ᵟᶊᶇᶁᶃ

ᵠᶍᶀ

y,n

Alice: pick a random number r
and give u= r2 mod n to Bob.
Bob picks a random bit b andgive it to Alice.
Alice computes v= rʀ͉b modn
Bob verifies ifuʀyb = v2 mod n holds:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

33

34

Summary

Alice proves she knows x s.t. y=x2 mod n

Zero-knowledge
Can simulate theprotocol without x.

?

r

Why is this a Zero-knowledge Proof?

Completeness
If Alice knows x,
Bob accepts
Soundness
If Alice doesn’t know x, Bob
rejects with prob. ½

32

Zero-knowledge Proofsr(b=0)
rʀx (b=1)

ᵟᶊᶇᶁᶃ

y,n

:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

Alice proves she knows x s.t. y=x2 mod n

x

ᵠᶍᶀ

31

Zero-knowledge Proofs

r

ᵟᶊᶇᶁᶃ

Alice: pick a random number r
and give u= r2 mod n to Bob.
Bob picks a random bit b andgive it to Alice.
Alice computes v= rʀ͉b modn
Bob verifies ifuʀyb = v2 mod n holds

Cryptography and Transparency

Cryptography/ Digital signatures clarifies who said what.
Cryptography ensures that one can not cheat in the system.
Cryptography ensures that if anyone cheated, it can bedetected.

Alice: pick a random number r
and give u= r2 mod n to Bob.
Bob picks a random bit b andgive it to Alice.
Alice computes v= rʀ͉b modn
Bob verifies ifuʀyb = v2 mod n holds:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

Cryptography provides transparency to operations in IT
systems, even the software running on the other computeris not visible.

35

19:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

36

ICIAM: International Congress on Industrial and Applied Mathematics:DVHGD 8QLYHUVLW\ 6DNR /DERUDWRU\

20

37

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Data Mining for Labor Market Intelligence
Ee-Peng Lim

School of Computing and Information Systems, Singapore Management University,
Singapore

Global economy and technology disruptions have created major impacts to labormarkets in recent years. To fully understand these impacts to companies and rankand-ﬁle people, we need to introduce new labor market intelligence capabilities using
data mining. In this talk, we will review labor market intelligence and the underlying
data mining problems. We will also illustrate how data mining can be used to analysetrends in job supply as well as patterns in job seeking behavior from big data. Finally,
we will cover the challenges in labor market intelligence research and how one may
overcome these challenges.

21Labor Market

FMfI2021, Hanoi, December 2021

Data Mining forLabor Market Intelligence
Jobs(Demand)

LIM Ee Peng

Job
Applicants(Supply)

Lee Kong Chian Professor of Computer Science
Director, Living Analytics Research Centre
Singapore Management University

Universities

Companies
1

2

Labor Market Intelligence:
Using market data for intelligent applications

Why is Labor Market important?

• Economics of a country

GDP
Market-level

Company/Individual-level• Vibrancy of businesses

Market Trend
Analysis(supply/demand)• Wealth of citizens

Salary/Skills
Analysis(supply/demand)

Performance
Analysis(supply/demand)• Social impact of universities

3

4

Traditional Approach to Labor Market
Intelligence

Limitations of Existing Approaches• Data mining researchers are NOT THE FIRST to study labormarket intelligence.
• Labor economists are way ahead!

• Simplified model assumptions:E.g., every worker/firm behaves the same way.
E.g., worker decides jobs based on salary only.

macroeconomics
labour force participation
unemployment
matching theory
market friction

microeconomics
search theory• Focus on modeling variables derived from aggregated data or
survey data, not actual micro-data.
E.g., number of workers, average wage, etc.

labor market
equilibrium• Models for explanation but not models for applications
E.g., Economic models are good for explaining the trends andoutcomes, but insufficient for developing IT solutions.

wage (minimum wage)
monopsonistic market
5

6

22

Advantages of Data Mining Approach:Accuracy, Timeliness and Usability

Warning! But Opportunity -• Accuracy
Economic models are good for descriptive analytics but notpredictive analytics.

• Data mining research for labor market intelligence is still at very
nascent stage.
• Labor market related datasets for data mining are notstandardized nor publicly available.

• TimelinessAs new market data are collected, data mining models can be
updated immediately, especially when there are major
disruptions• Many new data mining techniques have not been applied yet.

• Usability:
Personalized solutions to job applicants and companies

7

8Market Trend Analysis
Market Trend
Analysis(supply/demand)

Salary and Skills Analysis

Past Research• Labor Force Survey• Employer Skills Survey• Census of Population• Time series modelling (e.g, autoregressive model)

Salary/Skills
Analysis(supply/demand)
Past Research• Labor Force Survey• Employer Skills Survey• Census of Population• Knowledge Bases (e.g., O*Net)

Modeling & Forecasting Research• Job datasets and websites• Univariate time series to multivariate/multimodal time series• Time series model using deep learning

Prediction & NLP Research••••Lim, Bryan, and Stefan Zohren. "Time-series forecasting with deep learning: a survey." Philosophical Transactions of
the Royal Society A 379.2194 (2021): 20200209.

Job website data
Salary Prediction for Jobs
Skill Modeling from Job Postings
Skill Extraction from Job Descriptions

9

10

Skill Entity Recognition (SER): A Deep
Learning Approach

Measuring the Popularity of Job Skills• To detect skill mentions in job
description.
• Related Work: dictionary-based

 limited dictionary size and heterogeneity• NLP-Based Approach Annotate skill data for 1800 job posts.
 Augment FLAIR, contextual string
embedding, with dimension removal andsectional labels
Stack a CRF-based SER on FLAIR.
 Our SER approach outperforms state-ofthe-art models such as BERT.
 Further link skill mentions to known skill
entities in knowledge graph (Wikipedia).

Xu, T., Zhu, H., Zhu, C., Li, P., & Xiong, H. (2018, April). Measuring the popularity ofjob skills in recruitment market: A multi-criteria approach. In Proceedings of the
AAAI Conference on Artificial Intelligence (Vol. 32, No. 1).
11

12

23

Skill Wikifier

Performance Analysis

https://research.larc.smu.edu.sg/skill-wikifier/demo

Performance
Analysis(supply/demand)

Past Research• Employee Engagement Survey• Productivity Survey• Employee Turnover Analysis• Wage and Employee Performance

Company/User Profiling and Recommendation Research••••

Datasets: Recruitment site data
Employee Turnover Prediction
Applicant User Profiling
Jobs/Skills Recommendation

14

13

Contagious Effect for Employee TurnoverPrediction

What other research gaps?

Teng, M., Zhu, H., Liu, C., Zhu, C., & Xiong, H. (2019). Exploiting the Contagious
Effect for Employee Turnover Prediction. Proceedings of the AAAI Conference on
Artificial Intelligence, 33(01), 1166-1173.

Market Trend
Analysis(supply/demand)

Salary/Skills
Analysis(supply/demand)

Performance
Analysis(supply/demand)

Modeling &Forecasting
Research

Prediction &NLP
Research

Company/User
Profiling &Recommendation
Research

Data Mining Models for Multiple Types of Analysis
15

16

On Learning User/Job Latent Attributes
from Job Application Data

Probabilistic Labor Model• Our Proposed Approach: Probabilistic Labor Market Modelthat captures behavior of people and jobs in the market bylearning their attributes, and how they may interact.

• Probability of user ‫ݑ‬௜ applying job ‫݌‬௝ShouldI apply?

• Users ܷ ൌ ሼ‫ݑ‬ଵ ǡ ‫ݑ‬ଶ ǡ ǥ ‫ݑ‬ெ ሽwith attributes gender, education level, age

Jobሾ ୫୧୬ ǡ  ୫ୟ୶ ሿ• Jobs ܲ ൌ ሼ‫݌‬ଵ ǡ ‫݌‬ଶ ǡ ǥ ‫݌‬ே ሽeach job ‫݌‬௝ ‫ ܲ א‬has an offer salary interval ሾ‫ݓ‬௝௠௜௡ ǡ ‫ݓ‬௝௠௔௫ ሿ• Applications ‫ – ܣ‬an ‫ ܯ‬ൈ ܰ matrixͳ   ‫ݑ‬௜      ‫݌‬௝‫ܣ‬௜௝ ൌ ቊͲ 

Description௦௔௟௔௥௬

ܽො௜௝ ൌ ܽ௜௝

Salary-based
probability

௧௢௣௜௖

ȉ ܽ௜௝

௔௖௖௘௦௦
ȉ ܽ௜௝

Topic-based Accessibilityprobability based probability

Hendrik Santoso Sugiarto and Ee Peng Lim. On Modeling Labor Markets for
Fine-grained Insights. Best Paper. The 5th Workshop on Data Science forSocial Good, 2020.
17

18

24

Salary Criteria

Salary Criteria• User ‫ݑ‬௜ is interested in job ‫݌‬௝ if
job’s offers salary ൒ her reserved salary ࢜࢏• User ‫ݑ‬௜ is interested in job ‫݌‬௝ if
job’s offers salary ൒ her reserved salary ࢜࢏(this is a range!)(cannot be observed)• But each user has different optimisms ࢓࢏ ‫ א‬ሾͲǡͳሿ.
If ‫ݑ‬௜ is extreme-optimistic (݉௜ ൌ ͳ) and ‫ݑ‬௜ will focus on ‫ݓ‬௝௠௔௫ െ ‫ݒ‬௜If ‫ݑ‬௜ is extreme-optimistic ሺ݉௜ ൌ Ͳ) and ‫ݑ‬௜ will focus on ‫ݓ‬௝௠௜௡ െ ‫ݒ‬௜
• Probability of ‫ݑ‬௜ interested in ‫݌‬௝ based on salary:௦௔௟௔௥௬ܽ௜௝ൌ ߪ ݉௜ ‫ݓ‬௝௠௔௫ െ ‫ݒ‬௜ ൅ ሺͳ െ ݉௜ ሻ ‫ݓ‬௝௠௜௡ െ ‫ݒ‬௜ߪ ‫ ݔ‬ൌ ͳȀሺͳ െ ݁ ି௫ ሻ

19

Topic and Accessibility Criteria

Learning the PLM Model• Probability of ‫ݑ‬௜ interested in ‫݌‬௝ based on topic:• Likelihood of application

௧௢௣௜௖

ܽ௜௝

ൌ ܿ‫݁݊݅ݏ݋‬ሺ࢟௜ ǡ ࢠ௝ ሻ௦௔௟௔௥௬

ܽො௜௝ ൌ ܽ௜௝

User topics Job topics

࢟௜ and ࢠ௝ are ‫ܭ‬-dimensional vectors.

௧௢௣௜௖

ȉ ܽ௜௝

• Objective Function:• Probability of ‫ݑ‬௜ noticing ‫݌‬௝ :20௔௖௖௘௦௦ȉ ܽ௜௝௔௖௖௘௦௦ܽ௜௝ൌ ‫ݍ‬௜ ȉ ‫ݎ‬௝User effort

Job visibility

21

22

Experiments

Probabilistic Labor Market (PLM) Model(Jobs)
Job topics• Evaluate PLM model against other models.
• Examine the user variables learned from data.

Offer salary Reserved salary Optimism
 Effort

Optimism• Singapore Jobs Dataset (SJD):

Topic
Criteria

Visibility

 # user: 33,866
 # jobs: 68,091
 # applications: 827,380

SalaryCriteria

Accessibility Criteria
User
topicsReservedsalaryEffort(Users)

23

24

25

Application Prediction Result

Experiments(Area under precision-recall curve metric - AUCPRC)• Model with topics better than model without topics• Models without topics:ሺሻ ൐  ൐ ሺሻ ൐  ൐ ሺሻ
• Models with topics (K=25):
 ൐ ሺሻ ൐ ሺሻ ൐  ൐ • Model evaluation using application prediction task Given a user and a job, predict if the user applies the job.
• Baseline models not using topics
 Optimism based model: ¾ uses optimism derived from the user’s previous applications

 Salary based models:   ǡ  ¾ assume that the job’s offer salary is average of ‫ݓ‬௝௠௜௡ and ‫ݓ‬௝௠௔௭ and the user’s
reserved salary is average or minimum of salaries of all jobs applied

Singapore Jobs Dataset

 Popularity based model: 

Wuzzuf Jobs Dataset¾assume active users apply popular jobs first PLM without topics but with Salary + Accessibility:   Non-negative Matrix Factorization:  Latent Dirichlet Allocation:  PLM variants:
Salary + Topics:  
Topics + Accessibility: ሺሻ

Number of topics• Baseline models using topics

25

26

Labour Segments

Labour Segments (Singapore Job Dataset)
Each major topic may correspond to a labor segment with
Job Seekers: ܷ௟ ൌ ሼ‫ݑ‬௜ ‫ܷ א‬ȁܿ‫࢟ ݁݊݅ݏ݋‬௜ ǡ ࢚௟ ൐ ͲǤͷሽ
Jobs:
ܲ௟ ൌ ሼ‫݌‬௝ ‫ܲ א‬ȁܿ‫ࢠ ݁݊݅ݏ݋‬௝ ǡ ࢚௟ ൐ ͲǤͷሽ

27

28

Reserved Salary Distribution

User effort and User Optimism

medianaverageFinancial• optimistic• more effort

29

Clerical• less optimistic• more effort

30

26

Gender Preference

Age Preference

Reserved salary of
male applicants
generally higher
than female
applicantsReserved salaryof younger
applicants
generally lower
than older
applicantsTrading,
research,marketingsegments are
dominated byyounger
applicants

HR, Accounting,
Secretarial,
Clerical segments
are dominated byfemale applicants
31

32

Potential Applications

Conclusion• For job seekers:• Labor market intelligence (LMI) is an important research areathat is still in nascent stage.

 Understand market situation
 Set reasonable salary expectation
 Determine the right amount of job seeking efforts• Future directions:• For employers:






 Understand market situation
 Set appropriate salaries to attract talent Increase job visibility• For policy makers:

Multimodal data
Cross-platform data
Knowledge graphs
Convergence of data mining techniques• New LMI solutions and findings based on data mining
techniques will contribute to improving social and economic
wellbeing.

 Understand market situation
 Control labor supply Attract job investments

33

34

Thank yoularc.smu.edu.sg

27

Forum “Math-for-Industry" 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Finite sample inference for generic autoregressivemodelsHien Duy NGUYEN
School of Mathematics and Physics, University of Queensland, St. Lucia, Australia
Department of Mathematical and Physical Sciences, La Trobe University, Bundoora,
Australia
Autoregressive models are a class of time series models that are important in both
applied and theoretical statistics. Typically, inferential devices such as confidence sets
and hypothesis tests for time series models require nuanced asymptotic arguments
and constructions. We present a simple alternative to such arguments that allow
for the construction of finite sample valid inferential devices, using a data splitting
approach. We prove the validity of our constructions, as well as the validity of relatedsequential inference tools. A set of simulation studies are presented to demonstrate theapplicability of our methodology.
1. Introduction
Let (Ω, F, Pr) be a probability space, and define a sequence of random variables(Xt (ω))t∈[T ] to be a time series, indexed by t ∈ [T ] = {1, . . . , T }, where Xt =
Xt (ω) ∈ X for some space X. We suppose that the time series (Xt )t∈[T ] is orderp ∈ N autoregressive and parametric, in the sense that for every A ⊆ Xp , Pr ω : (Xt (ω))t∈[p] ∈ A =f (x1 , . . . , xp ; θ0 ) dx1...p ,A

and for each B ⊆ X and t > p,

Pr (ω : Xt (ω) ∈ B|Ft−1 ) =Bf (xt |xt−p...t−1 ; θ0 ) dxt .

Here, θ0 ∈ T is a parameter that characterizes the marginal and conditional probability
density functions (PDFs)
f (x1 , . . . , xp ; θ0 ) and f (xt |xt−p...t−1 ; θ0 ) , for each t > p,

where xa...b = (xa , xa+1 , . . . , xb−1 , xb ), for a, b ∈ N such that a < b. The symbol
Ft = σ (X1 , . . . , Xt ) indicates the sigma algebra generated by the random variables(Xi (ω))i∈[t] . The characterization thus allows us to write the PDF of the time series
XT = (Xt )t∈[T ] asf (xT ; θ0 ) = f (x1 , . . . , xp ; θ0 )

T


t=p+1

28

f (xt |xt−p...t−1 ; θ0 ) .

In this work, we concern ourselves with the problem of drawing inference about
θ0 , given that we do not know its value. Specifically, we are concerned with theconstruction of 100 (1 − α) % confidence sets of the form C α (XT ) ⊆ T, wherePrθ0 (θ0 ∈ C α (XT )) ≥ 1 − α,for any α ∈ (0, 1). Here, Prθ indicates the probability measure under the assumptionthat the PDF of XT has form f (xT ; θ). We shall also denote the associated expectation
operator by Eθ .
Furthermore, we are interested in testing hypotheses of the form
(1)

H0 : θ0 ∈ T0 versus H1 : θ0 ∈ T1 ,where T0 , T1 ⊆ T and T0 ∩ T1 = ∅. Here, we wish to construct valid P -values PT ,where
sup Prθ (PT ≤ α) ≤ α.

θ∈T0In order to construct our inference devices, we follow the work of [15], who considered
the construction of finite sample valid confidence sets and hypotheses for independentand identically distributed data (IID), using a data splitting construction with genericestimators. Due to the lack of reliance on any estimator specific properties, the authorsof [15] refer to their inference procedures as universal inference (UI).
The UI construction consists of demonstrating that a split data likelihood ratio
construction is an E-value, in the sense of [14], and [8]; i.e., a positive random variablewith expectation less than or equal to 1. The UI construction is extremely flexible
and has been adapted for construction of inferential devices using composite likelihood
ratios [11] and empirical Bayesian likelihoods [10]. We note that in the simple case ofconfidence sets for linear first order autoregressive models, our constructions can becompared to the finite sample results of [13] and [3, Sec. 4.1].
Besides our constructions of conventional confidence sets and P -values, using thesame construction as that of [15], we also provide anytime valid confidence set andP -value sequences for sequential estimation from online data, in the spirit of [7]. We
demonstrate the applicability of some of our constructions via numerical examples.
The paper proceeds as follows. In Section 2, we present our finite sample confidence
set and P -value constructions, as well as their anytime valid counterparts. In Section
3, applications of some of our constructions are provided via numerical examples. Finalremarks are then provided in Section 4.
2. Finite sample inference devices
Let us split XT into two contiguous subsequences XT1 = (X1 , . . . , XT1 ) and XT2 =(XT1 +1 , . . . , XT ), where T1 ≥ p. We shall also write T2 = T − T1 . Further, let Θ̂T be a
generic random estimator, such thatΘ̂T = θ̂ XT1 ,29

for some function θ̂ : XT1 → T, and define the likelihood ratio statisticL Θ̂T ; XT,
RT (θ) =L (θ; XT )where
L (θ; XT ) =T

t=T1 +1

f (Xt |Xt−p...t−1 ; θ)is the conditional likelihood of [XT2 |XT1 ].

Lemma 1. For any θ ∈ T, Eθ [RT (θ)] ≤ 1.

Proof. Write X̃t−p...t−1 = X̃t−p , . . . , X̃t−1 , where X̃t = Xt , if t ≤ T1 , and X̃t = xt ,otherwise. ThenEθ [RT (θ)]
=Eθ Eθ RT (θ) |XT1


T

T


f
x
|
X̃
;
Θ̂

t
t−p...t−1
T
t=T1 +1


f
x
|
X̃
;
θ
dx2T
=Eθ
t
t−p...t−1
T
XT2
t=T1 +1
t=T1 +1 f xt |X̃t−p...t−1 ; θ

T



=Eθ
f xt |X̃t−p...t−1 ; Θ̂T dx2T
=Eθ
(i)



XT2 t=T +1

X···= Eθ 1 = 1,

1

X





f xT |X̃T −p...T −1 ; Θ̂T dxT · · · f xT1 +1 |X̃T1 −p+1...T1 ; Θ̂T dxT1 +1(ii)

where (i) is due to Tonelli’s Theorem and (ii) is by definition of conditional PDFs. □With Lemma 1 in hand, we can now construct 100 (1 − α) % confidence sets of the
form
(2)

C α (XT ) = {θ : Rn (θ) ≤ 1/α} .

Proposition 1. For any α ∈ (0, 1) and θ0 ∈ T,

Prθ0 (θ0 ∈ C α (XT )) ≥ 1 − α.

Proof. By Markov’s inequalityPrθ0 (Rn (θ0 ) ≥ 1/α) ≤ αEθ0 [Rn (θ0 )] = α,(i)where (i) is by Lemma 1. Then, we complete the proof by noting thatPrθ0 (θ0 ∈ C α (XT )) = 1 − Prθ0 (Rn (θ0 ) ≥ 1/α)
≥ 1 − α.

30□

To test hypotheses of form (1), we require an additional estimator




(3)
Θ̃T ∈ θ̃ ∈ T : L θ̃; XT ≥ L (θ; XT ) , for all θ ∈ T .

Then, we may construct the test statistic 
ST = RT Θ̃T

and its P -value PT = 1/ST .

Proposition 2. For any α ∈ (0, 1) and T0 ⊂ T,
sup Prθ (PT ≤ α) ≤ α.

θ∈T0Proof. For each θ ∈ T0 , we have L Θ̂T ; XTEθ [ST ] = Eθ  
L Θ̃T ; XT L Θ̂T ; XT≤ Eθ 
L (θ; XT )(i)
= Eθ [RT (θ)] = 1,(ii)

where (i) is by definition (3) and (ii) is due to Lemma 1. Finally, by Markov’s inequality,
we havePrθ (ST ≥ 1/α) ≤ α =⇒ Prθ (PT ≤ α) ≤ α,□

as required.
2.1. Anytime valid inference. Let
MT (θ) =T

t=p+1 fT

Xt |Xt−p...t−1 ; Θ̂t−1

t=p+1 f (Xt |Xt−p...t−1 ; θ),for each T ≥ p+1, and MT (θ) = 1, for each T ≤ p. We firstly show that (MT (θ))T ∈N∪{0} 
is a martingale adapted to the natural filtration FT = σ (X1 , . . . , XT ). Here, Θ̂TT ≥p+1is a non-anticipatory sequence of estimators of θ0 , such that Θ̂T is dependent only on
XT .
Lemma 2. For each T ∈ N and θ ∈ T, Eθ [MT (θ) |FT −1 ] = MT −1 (θ).

31

Proof. For T > p + 1,
Eθ [MT (θ) |FT −1 ] T
t=p+1 f X̃t |Xt−p...t−1 ; Θ̂t−1
 f (xT |XT −p...T −1 ) dxT

=
T
X
|X
;
θ
f
X̃
t
t−p...t−1
t=p+1


T −1
 

f
X
|X
;
Θ̂
t
t−p...t−1
t−1
t=p+1= T −1
f xT |XT −p...T −1 ; Θ̂T −1 dxTX
t=p+1 f (Xt |Xt−p...t−1 ; θ)=MT −1 (θ) .
(i)where X̃T = xT and X̃t = Xt , for t < T . Here, (i) is due to the properties of conditional
density functions. For T ≤ p + 1, the result holds by definition.
□
We now wish to test the hypotheses (1) in a sequential
manner. To do so, we first 
require an additional sequence of parameter estimates Θ̃T, where

(4)

Θ̃T ∈



T ≥p+1θ̃ ∈ T :Define

T


t=p+1


T



f Xt |Xt−p...t−1 ; θ̃ ≥f (Xt |Xt−p...t−1 ; θ) , for all θ ∈ T .
t=p+1

 
NT = MT Θ̃T

for T ≥ p + 1 and NT = 1 for T ≤ p.

Proposition 3. For each α ∈ (0, 1) and T0 ⊂ T,
sup Prθ sup NT ≥ 1/α ≤ α.
θ∈T0

T ≥0

Proof. By Lemma 2, (MT (θ))T ∈N is a Martingale, and hence by Lemma 3, we havePrθ sup MT (θ) ≥ 1/α ≤ αM0 (θ) ≤ α.
T ≥0Note that for each T and θ ∈ T0 ,
T



f
X
|X
;
Θ̂
t
t−p...t−1
t−1
t=p+1


NT = 
T
f
X
|X
;
Θ̃
t
t−p...t−1
T
t=p+1


T
f
X
|X
;
Θ̂
t
t−p...t−1
t−1
t=p+1
≤ T
(i)
t=p+1 f (Xt |Xt−p...t−1 ; θ)
= MT (θ) ,32

where (i) is due to definition (4). Thus, for each θ ∈ T0 , we havePrθ sup NT ≥ 1/α ≤ Prθ sup MT (θ) ≥ 1/α ≤ α.
T ≥0

T ≥0





□

We observe that if we define P̄T = 1/NT , then the sequence P̄T T ∈N is also valid,in the sense that
sup Prθθ∈T0

inf P̄T ≤ α

T ≥0

≤ α.

Now, we shall construct sequential confidence sets of the forms
DTα = {θ ∈ T : MT (θ) ≤ 1/α} .

Proposition 4. For any α ∈ (0, 1) and θ0 ∈ T,

Prθ0 (θ0 ∈ DTα , for all T ∈ N) ≥ 1 − α.

Proof. Note that {θ0 ∈ DTα } = {MT (θ0 ) ≤ 1/α} and soαPrθ0 (θ0 ∈ DT , for all T ∈ N) = Prθ0 sup MT (θ0 ) ≤ 1/α ≥ 1 − α,T ≥0where (i) is due to Lemmas 2 and 3.

(i)□Observe that by definition we also havePrθ0 θ0 ∈ D̄Tα ≥ 1 − α,where D̄Tα = Tt=1 DTα , for each α ∈ (0, 1) and T ∈ N.

3. Numerical examples

3.1. Normal autoregressive model. Let (Xt )t∈Z be a random sequence defined as
Xt = θ0 Xt−1 + Et ,(5)

where (Et )t∈Z is an IID sequence, with Et ∼ N (0, 1), for each t ∈ Z. We shall constructa confidence interval for θ0 using the finite sample (FS) procedure.
We take as data XT , and split the data into two halves XT1 = (X1 , . . . , XT1 ) and2
XT = (XT1 +1 , . . . , XT ), where T1 = T /2 (assuming that T is even, for convenience).
Let Θ̂T be an estimator of θ0 depending only on XT1 . We use Θ̂T to construct the ratio


T
f
X
|X
;
Θ̂
t
t−p...t−1
T
t=T1 +1
RT (θ) = T
t=T1 +1 f (Xt |Xt−p...t−1 ; θ)


T
ϕ
X
;
Θ̂
X
,
1
t
T t−1
t=T1 +1= T
t=T +1 ϕ (Xt ; θXt−1 , 1)
 1 T 

2
1  
2
Xt − Θ̂T Xt−1 − (Xt − θXt−1 ),= exp
2 t=T +1133

where

ϕ y; µ, σ

2





= 2πσ


2 −1/2

1 (y − µ)2
exp −
2 σ2



,

is the normal density function with mean µ ∈ R and variance σ 2 > 0.
Thus, by Proposition 1, we obtain 100 (1 − α) % confidence intervals (CIs) of form
(2):
(6)




T
2
1  
2
α
Xt − Θ̂T Xt−1 − (Xt − θXt−1 ) ≤ log (1/α) .
C (XT ) = θ ∈ R :2 t=T +1
1Here, we can use the typical least squares (LS) estimator(7)Θ̂T = arg min
θ∈R

T1

t=2

 T1

2

t=2
(Xt − θXt−1 ) = T1

Xt−1 Xt2
t=2 Xt−1.

We can compare the performance of CIs of form (6) to the typical asymptotic normal
CIs (cf. [2, Sec. 5.2]) for the LS estimator
ΘLS
T =

(8)
using the distributional limit
(9)

T

t=2

T

Xt−1 Xt2
t=2 Xt−1

,


 d


2
T 1/2 ΘLST − θ0 −→ N 0, 1 − θ0 .

To assess the relative performance of the FS and LS CIs, we perform a small
simulation study. We simulate r = 1000 samples of size T = 100 from model (5)with θ0 = 0.5 and construct 90% CIs. To compare the performances of the CIs, we
compute coverage proportion (CP) (proportion of the r CIs of each type that containθ0 ) and the average length (AL) of the CIs.
We obtain the results CPFS = 0.998 and CPLS = 0.895, and ALFS = 0.643 and
ALLS = 0.286. We thus observe that both the LS and FS CIs obtain the correct
nominal level of confidence, although the FS CIs are conservative with respect tocoverage. This conservativeness is also reflected in the lengths of the intervals, where
the FS CIs over twice as long as the LS CIs. However, this is expected given that the
FS CIs are constructed only by Markov’s inequality application, whereas the LS CIs
makes use of the information geometry of the normal distribution. Figure 1 provides avisualization of 20 pairs of FS and LS CIs from the simulation study. We observe thatin many cases, the FS CIs provide useful inference regarding the presence of non-zeroautocorrelation θ0 , even if the intervals can be larger than necessary.

34

1.0
0.5
−0.5

0.0

theta_0

5

10

15

20

Simulation

Figure 1. A visualization of 20 pairs of 90% CIs for θ0 = 0.5 in thenormal autoregressive model. The FS CIs are colored black and LS CIs
are colored red.
3.2. Cauchy autoregressive model. We now consider model (5) with Et ∼ Cauchy (0, 1),
which implies that the ratio statistic has form


T
f
X
|X
;
Θ̂
t
t−p...t−1
T
t=T1 +1
RT (θ) = T
t=T1 +1 f (Xt |Xt−p...t−1 ; θ)


T
κ
X
−
Θ̂
X
t
T
t−1
t=T1 +1= T
t=T1 +1 κ (Xt − θXt−1 )=T

1 + (Xt − θXt−1 )22 ,T =T1 +1 1 + Xt − Θ̂T Xt−1

where κ (y) = π −1 {1/ (1 + y 2 )} is the PDF of a the law Cauchy (0, 1). This implies a
100 (1 − α) % FS CI for θ0 of the form




T
2
 
1
1 + (Xt − θXt−1 )α.
C (XT ) =

2 ≤

α

T =T1 +1 1 + Xt − Θ̂T Xt−1

We again use the LS estimator Θ̂T to construct the FS CI and compare ourconstruction to the LS CI using the distributional limit (9) as an approximation, since

35

1.0
0.5
−0.5

0.0

theta_0

5

10

15

20

Simulation

Figure 2. A visualization of 20 pairs of 90% CIs for θ0 = 0.5 in the
Cauchy autoregressive model. The FS CIs are colored black and LS CIs
are colored red.
the Cauchy model does not satisfy the required regularity conditions of [2, Sec. 5.2].
The comparison is made via the same simulation study as described in Section 3.1.
We obtain the results CPFS = 0.995 and CPLS = 0.944, and ALFS = 0.236 and
ALLS = 0.285. We notice now that the LS CIs no longer achieve the nominal 90%
confidence level, and are now also conservative, although not as conservative as the FS
CIs. Interestingly, even though the FS CIs are more conservative, they are on average
shorter than the LS CIs. We observe this via Figure 2, which visualizes 20 pairs of FS
and LS CIs from the simulation study.
4. Unit root test
We assume again Model (5), with Et ∼ N (0, 1). However, we now wish to test the
hypotheses(10)

H0 : θ0 = 1 versus H1 : θ0 ∈ (−1, 1) .

This is the classical normal unit root test setting of [4], which is usually tested using
the LS estimator (8) as the test statistic.
Under the null hypothesis, it is known that the LS estimator has a non-normal
asymptotic distribution that is highly irregular and requires numerical integration or
simulation in order to approximate its quantiles and density (see, e.g., [1, 5, 12]).

36

Table 1. Unit root test results at the α = 0.1 level of significance.
FS Asymptotic
1.000
1.000
1.000
1.000
0.990
1.000
0.904
1.000
0.005
0.106

θ0
0.00
0.50
0.90
0.95
1.00

However, to perform our FS test, we can simply construct the test statistic



T
2
1  
2
Xt − Θ̂T Xt−1 − (Xt − Xt−1 ),(11)ST = RT (1) = exp
2 t=T +1
1

where we use (7) for Θ̂T . By Proposition 2, PT = 1/ST is a P -value, satisfyingPrθ0 =1 (PT ≤ α) ≤ α.
We can assess the performance of the FS test based on statistic (11) versus the usual
test, based on (8), using the quantiles provided in [1, Tab. 1]. We simulate r = 1000
samples of size T = 1000 and test (10) with θ0 ∈ {0, 0.5, 0.9, 0.09, 1}. We then compare
the asymptotic test to the FS test on the basis of proportion of rejection (PR) out of
the r samples at the α = 0.1 level of significance. Our results are presented in Table 1.
From Table 1, we observe that the FS test is more conservative than the asymptotic
test, as to be expected from the previous results, along with the Markov’s inequality
construction. However, the test does not require knowledge of any special distribution,and can more easily implemented, as a tradeoff.
5. Final remarks
Remark 1. The anytime valid inference results of Propositions 3 and 4 can be statedin terms of stopping times of the test and confidence event sequences. This can beachieved via [6, Lem. 3].
Remark 2. It is noteworthy that the process of splitting the data may be somewhat
arbitrary. However, one alleviate the need of making a choice by averaging over the
results of choices of splits. That is, let (T1,i )i∈[n] be a sequence of n values T1,i ∈{p + 1, . . . , T − 1}, for each i ∈ [n], and let Θ̂T,ibe a sequence of estimators,i∈[n]where Θ̂T,i depends only on the data (Xt )t∈[T1,i ] . Then, the averaged ratio statistic


T
n
f
X
|X
;
Θ̂
t
t−p...t−1
T,i1  t=T1,i
R̄T (θ) =T
n i=1
t=T1,i f (Xt |Xt−p...t−1 ; θ)is an E-value, in the sense that Eθ R̄T (θ) ≤ 1. Corresponding versions of Propositions
1 and 2 then follow.
Here, a choice must still be made regarding the n valued sequence (T1,i )i∈[n] . However,
one can make all possible choices, in the sense of taking (T1,i )i∈[n] = (p + 1, . . . , T − 1).

37

Then, we would have a ratio statistic in the form


T
T −1
f
X
|X
;
Θ̂

t
t−p...t−1
T,i
t=i
1
,
R̄T (θ) =TT − p − 1 i=p+1
t=i f (Xt |Xt−p...t−1 ; θ)
where Θ̂T,i
is a sequence of estimators with Θ̂i depending only on (Xt )t∈[i] ,i∈{p+1,...,T −1}

for each i ∈ {p + 1, . . . , T − 1}. This statistic is also an E-value and requires no user
input regarding the choice of split. However, it is a much more expensive statistic than
RT (θ), since it requires T − p − 1 estimators to be computed, whereas RT (θ) requires
only one. The user must thus make a tradeoff between computation and user input.
Since the average of E-values is an E-value, the same discussion can be maderegarding the choice of estimator Θ̂T . One can choose different estimators Θ̂T andaverage over the RT (θ) statistics corresponding to each estimator in order to produce
a new statistic that remains an E-value.
Remark 3. Our text focuses on ratio statistics RT (θ) that are constructed using
conditional likelihood objects L (θ; XT ). However, we may replace the conditional
likelihoods with conditional composite likelihoods or conditional integrated likelihoods,
in the manner of [11] and [10], respectively. This can be useful in situations where thelikelihoods L (θ; XT ) are intractable or difficult to compute.
Appendix
The following result is often called Ville’s Lemma and a proof can be found in [9,Thm. 3.9].
Lemma 3. Let (YT )T ∈N∪{0} be a non-negative supermartingale, then, for each α > 0,
Pr sup YT ≥ 1/α ≤ αE [Y0 ] .
T ≥0References[1] K M Abadir. The limiting distribution of the autocorrelation coefficient under a unit root. Annalsof Statistics, 21:1058–1070, 1993.
[2] T Amemiya. Advanced Econometrics. Harvard University Press, Cambridge, 1985.
[3] B Bercu, B Delyon, and E Rio. Concentration Inequalities for Sums and Martingales. Springer,
Cham, 2015.
[4] D A Dickey and W A Fuller. Distribution of the estimator for autoregressive time series with aunit root. Journal of the American Statistical Association, 74:427–431, 1979.
[5] G B A Evans and N E Savin. Testing for unit roots: 1. Econometrica, 49:753–779, 1981.
[6] S R Howard, A Ramdas, J McAuliffe, and J Sekhon. Time-uniform, nonparametric,
nonasymptotic confidence sequences. Annals of Statistics, 49:1055–1080, 2021.
[7] R Johari, P Koomen, L Pekelis, and D Walsh. Peeking at A/B tests. In Proceedings of KDD,
pages 1517–1525, 2017.
[8] W M Koolen and P Grunwald. Log-optimal anytime-valid E-values. International Journal of
Approximate Reasoning, to appear, 2021.
[9] T Lattimore and C Szepesvari. Bandit Algorithms. Cambridge University Press, Cambridge, 2020.
[10] H Nguyen and M Gupta. Finite sample inference for empirical Bayesian methods. Technical
Report hal-03363121, HAL, 2021.

38

[11] H D Nguyen, J Bagnall-Guerreiro, and A T Jones. Universal inference with composite likelihoods.
In Proceedings of the 63rd ISI World Statistics Congress. International Statistical Institute, 2021.
[12] M M Rao. Asymptotic distribution of an estimator of the boundary parameter of an unstableprocess. Annals of Statistics, 6:185–190, 1978.
[13] V Vovk. Strong confidence intervals for autoregression. ArXiv, (arXiv:0707.0660v1), 2007.
[14] V Vovk and R Wang. E-values: calibration, combination, and application. Annals of Statistics,
49:1736–1754, 2021.
[15] L Wasserman, A Ramdas, and S Balakrishnan. Universal inference. Proceedings of the National
Academy of Sciences, 117:16880–16890, 2020.

39

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Inversion Analysis for Medical Imaging
Yu Jiang

School of Mathematics, Shanghai University of Finance and Economics, China

Inversion analysis for medical imaging is one of the important ﬁelds in the ﬁeld of
inverse problem research. The main purpose of solving this kind of inverse problem isto reconstruct the information that can be used for disease diagnosis from the information obtained from medical images. This talk will mainly cover the latest progress ofsome medical imaging technologies, such as magnetic resonance elastography, optical
tomography related inversion analysis technologies.

40

Magnetic Resonance Elastography, MRE

Inversion Analysis for
Medical Imagingzz

MRI + elastographymeasure the viscoelasticity of human tissue(Muthupillai et al., Science, 269, 1854-1857, 1995.䠅

(֜ enable us to virtually realize a doctor’s palpation

Diagnosis:
 the stage of liver fibrosis early stage cancer: breast cancer, pancreatic cancer,
prostate cancer, etc.
 neurological diseases: Alzheimer’s disease,hydrocephalus, multiple sclerosis, etc.
 Aid of surgery, postoperative observation, evaluation oftreatment
 Nondestructive testing: biological material, polymer material

Yu Jiang

Shanghai University of Finance and Economics
FMfI 2021
Joint work with Gen Nakamura (Hokkaido University)& Kenji Shirota (Aichi Prefectural University)

2

Two types of data analysis (inversion)

MRE measurements: phase image
MRI signal` Model independent data analysis

i.e. Don’t need to model tissues, but just assume that thewaves are superposition of sinusoidal waves with attenuation

䠎D FFT
real part: R

imaginary part: I`

magnitude image`

phase image

Local frequency estimate (LFE, Mayo Clinic)
LWV/LAV method (Nakamura-Yoshikawa)`Computational and Mathematical Methods in Medicine, 2013, 912920` Model dependent data analysis

i.e. Need to model tissues (Jiang – Nakamura, SIAM J. APPL.
MATH. 71(6), 2011, 1965–1989)`

Modified Integral Method (Jiang - Nakamura)`

Journal of Physics: Conference Series 290, 2011, 012006

Least square method (regularization scheme)` Newton type regularization scheme (Jiang – NakamuraShirota)``

Scalar model`

J. Inverse Ill-Posed Probl. 2019; Inverse Problems 2021

Modified Stokes model` Isotropic+ nearly incompressible

Isotropic incompressible stationary scalar model:` Modified Stokes model:` ߤ: storage modulus (elastic); ߟ: loss modulus (viscosity)

ɏ: density, ݂: frequency of external vibration;` Here, we assumed:

`
`
`

41

For Ɋǡ ߟ ‫ܮ א‬ஶ Î ‫ܝ‬ǡ ‫ܪ א ݌‬ଵ ൈ ‫ܮ‬ଶ
Jiang, et. Al., SIAM J. Appl. Math. 71, pp. 1965-1989
H. Ammari, Quar. Appl. Math., 2008: isotopic constant elasticity

Inverse Problem of MRE

Modified Stokes model` 2D numerical simulation (Freefem++)`Plane strain assumption` By knowing ‫ߗ א ܝ‬Îߤ: storage modulus (elastic); ߟ: loss modulus (viscosity)?

mm

Ux_realCurl operator

Numerical differentiation` Modified Stokes model (soft tissues: nearly`

incompressible, isotropic media):`(point wisely)``` L
Locallyll hhomogeneous (constants):
(
t t )

`

Numerical differentiation is ill-posed or unstable!!!
MRE data: high noise level, ~10%

Curl operator: filter of the pressure term (longitudinal wave)

Levenberg-Marquardt method

Lest square problem` Iterative scheme:` Lest square problem:

For Ɋǡ ߟ ‫ܮ א‬ஶ Î ‫ܝ‬ǡ ‫ܪ א ݌‬ଵ ൈ ‫ܮ‬ଶ``

` Iterative method:`` Landweber iteration scheme (Ammari et al.)` Newton type regularization scheme: Levenberg–: Fréchet derivative` Satisfies the tangential cone condition

Marquardt method (Jiang-Nakamura-Shirota, Inverse
Problems (2021))` Converge to a solution if have a “good” initial guess close

to the true value

42

Implement LM-method numerically

Numerical test (smooth function)` FreeFEM++ (version 4.9)

Need to assume Ɋǡ ߟ ‫ܪ א‬ଶ`: Hood-Taylor element P2-P1/P3-P2: Hsieh-Clough-Tocher (HCT) element, C-class` in FEM space``` linear system

Numerical test (smooth function, ͸ʹ.ͷHz )` 6% relative noise (H1 sense)

Numerical test (smooth function, ͸ʹ.ͷHz )Numerical test (smooth function, 125 Hz )` 6% relative noise (H1 sense)

Numerical test (smooth function, 125Hz )43

Conclusion and future works`     Ɋǡ ߟ ‫ܪ א‬ଶ Ǣ`    Ɋǡ ߟ ‫ܮ א‬ஶ Ǣ

Thank you for your attention.

    ߟǢ` Ǥ`44

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

A simple mathematical model on spread of
Covid-19 with the eﬀect of vaccination and its
application to Japan
Takashi Tsuchiya

National Graduate Institute for Policy Studies, Tokyo, JapanThe spread of Covid-19 causes serious damages to Japanese society since 2020. However, the number of new cases is decreasing drastically with the progress of vaccination
reaching 70% to 80% in ratio as of November 2021. In this talk, a simple mathematical model is presented to describe the spread of Covid-19 used to predict the numberof day-by-day new cases in Tokyo taking the eﬀect of vaccination into account. The
dynamics of infection are described with a simpliﬁed version of SIR model, where theperiod of infection of a patient is assumed to be a constant instead of obeying to an
exponential distribution in SIR model. Another feature is it takes account of potential
spreaders without symptom. The model works fairly well in spite of its simplicity.
In Japan, the timing of the next (sixth) wave of Covid-19 is of great public interest.
We discuss the possibility of herd immunity relying on vaccination, and predict the
future based on the model and data.

45

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

A mathematical model for COVID-19 transmission
dynamics with a case study of Myanmar
Aung Zaw Myint

Department of Mathematics, University of Mandalay, Myanmar

We propose a compartmental mathematical model to predict and control the transmission dynamic of COVID-19 disease in Myanmar. We compute the basic reproduction number threshold. We perform local and global stability analysis for infection
equilibrium in terms of basic reproduction number, and we conduct a sensitivity analysis in our corona-virus model to determine the relative importance of model parameters
to epidemic transmission. Moreover, numerical simulation demonstrates that the disease transmission rate more than eﬀective to mitigate the basic reproduction number.

46

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Some Applications of Mathematics in Medical
Works
Jessada TANTHANUCH

School of Mathematics, Institute of Science, Suranaree University of Technology,
Thailand

Mathematics, often called the “Queen of the Sciences”, is one of the basic sciences.
However, this major role of the basic science is able to apply to many medical works.
This presentation shows the applications of mathematics research to biomedical engineering applications. The overall concepts of some research by School of Mathematics
and School of Biomedical Innovation Engineering, Suranaree University of Technology,
Nakhon Ratchasima, Thailand, are given. The research works are the modelling of
knee shape, the modelling of the blood ﬂow to heart, the modelling to predict the patients postoperative WOMAC score after total knee replacement, the applications ofsupport vector machine, twin parametric support vector machine for the medical image classiﬁcation problems, the development of image processing technique to enhancequality of ultrasound and x-ray images and teeth classiﬁcation, and the case study ofusing 3D printing in the preoperative planning for the surgery.
References[1] Thongking, W., Mitsomwang, P., Sindhupakorn, B., Tanthanuch, J. (2022) Analysis and Classiﬁcation of Abnormal Vertebral Column by Convolutional Neural Network Algorithm., Suranaree
Journal of Social Science., Vol 16., No. 1., (January-June 2022)., in press.
[2] Nerysungnoen, B., Chatwongwan, W., Nuengkota, P., Puntanakun, S. and Tanthanuch, J. (2020),
The Application of Using 3D Printing for Operation Planning in Maharat Nakhon Ratchasima
Hospital: Case report of Changing Mandibular Bone, Journal of Medicine and Health Science, Vol 27 No. 3: December 2020., pp 166-178. (in Thai)
https://he01.tci-thaijo.org/index.php/jmhs/article/download/246846/167503/[3] Tanthanuch, J., Kaptsov, E.I., Meleshko, S.V. (2021) Equation of Rayleigh Noise Reduction Modelfor Medical Ultrasound Imaging: Symmetry Classiﬁcation, Conservation Laws and Invariant Solutions., Journal of Vibration Testing and System Dynamics, Vol 5 Issue 3, September 2021,
pp. 237-247.
[4] Tanthanuch, J., Kaptsov, E.I. and Meleshko, S.V. (2019) Equation of Rayleigh Noise Reduction
Model for Medical Ultrasound Imaging: Symmetry Classiﬁcation and Conservation Laws in Cylindrical Coordinates, AIP Conference Proceedings. Vol. 2153, Issue 1, 020022 (2019):
https://doi.org/10.1063/1.5125087

47

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Engineered algorithms for large-scale single-cell
RNA sequencing and multimodal data analysis
Stefan Canzar

Gene Center, Ludwig Maximilian University of Munich, Germany

Experimental methods for sequencing DNA or RNA of single cells have transformed
biological and medical research. The throughput of this technology has dramaticallyincreased over the last few years, such that today the expression of genes in millions
of cells can be measured in a single experiment. The computational interpretation of
the produced data, however, often exceeds the capacity of existing algorithms. We
have therefore developed method Sphetcher, an eﬃcient algorithm that computes amuch smaller set of cells that represent the transcriptional space of the original dataas accurately as possible. Sphetcher can compute such a so-called sketch of millions ofcells in minutes and facilitates the identiﬁcation of rare cell types.
Single-cell sequencing in addition allows to reconstruct trajectories that describe
dynamic changes in gene expression that occur during the diﬀerentiation of cells. We
have developed method Trajan that allows to compare such trajectories of, e.g., diﬀerentiating immune cells that are involved in the response to an infection.
In algorithm Specter we combine measurements of multiple types of molecules toreﬁne cell types. We showed that Specter is able to resolve subtle transcriptomic diﬀerences between subpopulations of memory T cells based on their combined expressionof mRNAs and surface proteins. For the joint visualization of such modalities, we extended t-SNE and UMAP, the most popular methods for the visualization of biomedical
data.

48

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Mathematical modelling for COVID-19 in the
Victorian Public Service
Michael Lydeamore

Department of Econometrics and Business Statistics, Monash University, Australia(joint work with COVID-19 Modelling and Analytics team, Government of Victoria)

The COVID-19 pandemic has put infectious diseases modelling in the spotlight.
Many institutes and governments globally have very suddenly had a desire and needfor accurate modelling and analytics on a rapidly evolving situation. During 2020, I was
seconded to the Victorian Department of Health, and formed a modelling and analytics
team that regularly provided situational reporting, analysis and policy relevant advice
to high level decision makers and ministers.
I will discuss three pieces of work that were inﬂuential at very diﬀerent stages of
the Victorian COVID-19 experience. The ﬁrst was a model that was created in a timeof little knowledge, but was inclusive of many operational details that normally would
not be considered in model construction. This model was used in numerous decisions,
including Victoria’s PPE planning and hospital equipment procurement.
The second model contains much more detail, including age and complex contact
patterns, and was used to inform the gradual easing of restrictions before Victoria’s
second wave including school and workplace re-opening.
The ﬁnal piece of work is a data visualisation dashboard known as the “Mystery
Case Tracker”. This tool brings together infection timelines, contact patterns and geographical coding into a dashboard utilised by the outbreak team to rapidly understandand classify places of risk.
As well as the pieces of work, I’ll discuss how analytics and modelling are broadly
thought of in these settings, and how approaching issues with an analytics mindset canbe helpful in solving problems rapidly.

49Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Modelling COVID-19 on a bipartite contact
network of 5 million individuals for the Elimination
Strategy in Aotearoa New Zealand
Emily Harvey

M.E. Research & Te Pūnaha Matatini, New Zealand(joint work with James Gilmour - Department of Physics, University of Auckland,
Oliver MacLaren - Department of Engineering Science, University of Auckland, DionO Neale - Department of Physics, University of Auckland & Te Pūnaha Matatini,Frankie Patten-Elliott - Department of Physics, University of Auckland & Te Pūnaha
Matatini, Steven Turnbull - Department of Physics, University of Auckland & Te
Pūnaha Matatini, David Wu - Department of Engineering Science, University of
Auckland)Many of the models used for rapid policy advice during the COVID-19 pandemic
rely on simplifying assumptions about the homogeneity of populations and the impact
of non-pharmaceutical interventions on transmission. In the context of an elimination
strategy, with small case numbers, such approximations become increasingly poor representations of reality. We have built a stochastic model of infection dynamics that runs
on an empirically-derived, bipartite contact network that explicitly represents each of
the 5 million people in Aotearoa NZ. This model includes mechanistic representationof testing, contact tracing, and isolation processes, as well as targeted ‘Alert Level’changes. The model has been used to inform government responses to SARS-CoV-2
outbreaks in Aotearoa NZ during 2020 and 2021. We ﬁnd that the heterogeneity andnetwork structure in our model leads to qualitatively diﬀerent behaviour, comparedwith a “well-mixed” model, in a number of scenarios. We highlight some key diﬀerences between this model and such well-mixed ODE and branching process models.

50

Questions

Modelling COVID-19 on a bipartite contact
network of 5 million individuals• How many people will get infected, and when?
• Who will get infected?

for the Elimination Strategy in Aotearoa NZ• What will happen to the people who get infected?
• Where/how will people get infected? (What factors drive infection risk?)

Presenter: Emily Harvey emily@me.co.nz• What capacity levels will be needed in test-trace-isolate and in the health

Team: Dion O’Neale, Oliver Maclaren, James Gilmour, Joshua Looker,
Frankie Patten-Elliott, Joel Trent, Steven Turnbull, David Wu & others

system to meet assumed levels of performance?

Traditional Epidemic Models
Susceptible

ExposedInfectious

Building an Interaction Network
Recovered• Explicitly represent each individual along with individual level attributes• Assumes a well-mixed, homogeneous, population• Heterogeneous interaction structure → contagion spreads on an explicit
contact network for the whole of Aotearoa NZ

Susceptible

A

?

≡

E
Infectious

BCD

Bipartite Network Model

Why bipartite networks ?

A





Explicitly represents:- individuals

g1

E

B

g2
D













- groups (interactions)





















through groups or contexts - theplaces interaction occurs

• We build the network in layers
for each type of ‘group’ fromempirical data sources

51





To represent a school with 1000
students we need:• 1000 + 1 nodes & 1000 links for• Network interactions are

C

Recovereda bipartite representationor• 1000 nodes & 1000 x 999 links
for a one-mode representation

Empirical Multilayer Network

Current Empirical Multilayer Network

Individuals:~5 million (2018 census) individuals who have age, ethnicity,
sex, and location (usual residence SA2) from Census 2018

GroupsDwelling
Nodes

Individual
Nodes‣ Workplaces (StatisticsNZ IDI - tax & Census)

Instead we extract distributions, counts, and look to
identify correlations between factors to probabilisticallyreconstruct a (set of) interaction network(s).

SchoolNodes‣ Schools (Ministry of Education & StatisticsNZ IDI)
‣ Dwellings (StatisticsNZ IDI - Census)

Individuals

We typically can not use linked microdata directly forbuilding layers of the network (or for linking them).

Workplace
Nodes

Layers:Community
Nodes‣ Community (electronic transactions, contact surveys,
telco & movement data)

IDI: Statistics NZ Integrated Data Infrastructure (linked microdata)

Adding a contagion process

Contagion model states• We use a modied Gillespie algorithm (hence Markovian) with extensions toallow for non-Markovian dynamics including scheduled (delayed) processes,
and algorithmic speed ups.

Susceptible

ExposedPre-symptomatic

Symptomatic

Recovered ordeceased

EU

PU

IU

RU

Infection
Unknown(undetected)

SU• Contagion spread is stochastic with explicit representations of contact tracing,DX

testing and quarantine/isolation processes (which is where we use the delayed
processes).

• For more info contact Oliver Maclaren oliver.maclaren@auckland.ac.nz
Infected states

Note: not shown here are the states representing asymptomatic cases,
hospitalisation, and critical care.

Contagion model states
ExposedSusceptible

Pre-symptomatic

Symptomatic

Infection
Unknown(undetected)

SUEU

Conrmed(testing)

PU

PC

IU

IC

Contagion model states
Recovered ordeceased

Susceptible

ExposedPre-symptomatic

Symptomatic

Recovered ordeceased

EU

PU

IU

RU

EQ

PQ

IQ

DX

PC

IC

RT

Infection

RU

Unknown(undetected)

SU

DX

Quarantined(contact tracingor symptoms)

SQ

RTConrmed(testing)

Infection

Infected states

Infected states

Note: not shown here are the states representing asymptomatic cases,
hospitalisation, and critical care.

Note: not shown here are the states representing asymptomatic cases,
hospitalisation, and critical care.

52

Aotearoa NZ context 2020

Examples: same disease, different network
Proportion of Pop

Recovered

Infected

Deadtime (days)

Jeﬀeries, et al. (2020) COVID-19 in New Zealand and the impact of the national response:a descriptive epidemiological study, Lancet Public Health

Elimination Strategy

Aotearoa NZ context 2020

Auckland August
Outbreak 2020
Auckland shifted to Alert Level 3,
rest of Aotearoa NZ to Alert Level 2

Michael Baker, Amanda Kvalsvig, Nick Wilson
https://theconversation.com/6-months-after-new-zealands-rst-covid-19-case-its-time-for-a-more-strategic-approach-144936

Elimination Strategy context

Interaction of non-pharmaceutical interventions

Old variant

New variant

Alert Level 2.5

Alert Level 3

A key focus of modelling wason early detection ofcommunity outbreaks, andcontrol (elimination) of themonce detected.

Alert Level 4

Outbreak Size at Detection

300“Would Alert Level 2.5 have been suﬃcient to give elimination of the August 2020 outbreak?”

200

100

0

Small outbreak size at detection→ good odds of elimination.

53

0

20

40

60

0

20

Time to Detection

40

6

9

7

10
10

8

60

Interaction of increased transmissibility variants:
Alpha

Impact of delays (applying AL3)
Probability of Elimination
Based on Cases at Detection and Delay in Intervention
1.00

Alert Level 3
baseline transmissibility

Probability of Elimination

0.75

Alert Level 3
50% increased transmission

Alert Level 4
50% increased transmission

Delay (Days)
1
2

0.50

3
5
7

0.25

0.00
0

25

50

Cases at Detection

75

100

Interaction of increased transmissibility variants:
Alpha

August 2021 (Delta outbreak)• The observed growth rate (Reﬀ) at diﬀerent Alert Levels does not scale linearly withtransmissibility of diﬀerent variants
Alert Level 3
baseline transmissibility

Alert Level 3
50% increased transmission

Alert Level 4
50% increased transmission

Farah Hancock, Data journalist, RadioNZ

August 2021 (Delta outbreak)August 2021 (Delta outbreak)Elimination strategy

Farah Hancock, Data journalist, RadioNZ

54

Network model
Daily conrmed cases

Daily conrmed cases

Branching process modelAugust 2021 (Delta outbreak)August 2021 (Delta outbreak)

Infection Context

Dwellings
Casual community
Close community
Schools
Workplaces

Time after detection

Proportion of infections

Average new infections daily

Adjusting ‘control’ level C(t) in Branching process model

Trent, J., (2021)

Time after detection

Effect of relaxing Alert Levels: Delta, September 2021

Effect of relaxing Alert Levels: Delta, September 2021
The right result, but not entirely for the right reason

Dwellings
Casual community
Close community
Schools
Workplaces

Effect of opening schools: Delta, November 2021

Workplace heterogeneity (Alert Level intervention)

Interactions matter most when they make the contact network more connected

Workplaces

Individuals

Dwellings

55

Workplace heterogeneity (Alert Level intervention)

Workplace heterogeneity (Alert Level intervention)

Workplaces

Workplaces

50% reduction in transmission
within workplaces

50% of workplaces are closed(not working on site)

Individuals

Individuals

Dwellings

Dwellings

Workplace heterogeneity (Alert Level intervention)

Workplace heterogeneity (Alert Level intervention)
76% of workplaces are closed (Alert Level 4), remaining
ones have transmission reduction measures that lead toa 37.5% reduction in transmission

All workplaces open, but have transmission reduction
measures that lead to an 85% reduction intransmission (equivalent in the average sense)

Workplaces

50% of workplaces are closed(not working on site)

Individuals

Elimination

Dwellings

No Elimination

Where else might this have an effectWorkplace heterogeneity (Alert Level intervention)• Vaccination clustering Age (eligibility) and/or Attitudes

Heterogeneous workplace closure conceptual diagram• Ability to stay home from work if symptomatic
Schools

Workplaces

Individuals
Original contact network

50% edge removal
Dwellings

56

Effect of interventions and heterogeneity

Thank you for listening

The eﬀect (and eﬀectiveness) of interventions depends on what other interventions
are applied and on the specics of the infection tree:• The eﬀect of contact tracing (e.g. on reduction in Reﬀ) depends on the Alert Level
status, the level of testing, the time to detection and other features of theinfection tree

• The eﬀect of an increase in transmissibility of the virus on Reﬀ depends on the
network structure and number and ‘riskiness’ of contacts

And thanks to TPM, MBIE, DPMC,& HRC for funding.

• Eﬀectiveness of Alert Level changes and other interventions depend on who is
infected in the initial outbreak

Heterogeneity plays a large role.

57Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

SEIR network models for Coronavirus disease(COVID-19) in Vietnam
Doanh Nguyen-Ngoc
UMMISCO & ACROSS, IRD/France and Thuyloi University, Vietnam(joint work with Alexis Drogoul, UMMISCO, IRD/France and in collaboration withother colleagues)
In this talk, we will introduce some SEIR network models incorporating diﬀerent
spatial scales from province, district and ward levels to cell level to explore the spreadof diﬀerent waves of Coronavirus disease (COVID-19) in Vietnam and to support the
Rapid Response Team, Vietnam National Committee Against Covid-19.

58

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Mathematical model based prediction andapplication to COVID-19
Shingo Iwami

Division of Biological Science, Graduate School of Science, Nagoya University, Japan

If it becomes possible to capture the nonlinear dynamics behind phenomena witha mathematical model and its numerical analysis, it will be possible to predict the
future which might be limited. For example, when applied to medical data, it can be
expected to evaluate and predict treatment eﬀects and prognosis with high accuracy.
In this talk, I will present an example of how the development of a mathematical modelthat explains clinical data of COVID-19 patients has essentially made it possible topropose treatments and design clinical trials based on predictions.

59Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Securing Vaccine Delivery Against Physical Threats
Mai Anh Tien

School of Computing and Information Systems, Singapore Management University,
Singapore(joint work with Arunesh Sinha, School of Computing and Information Systems,
Singapore Management University)
Vaccine delivery in under-resourced locations with security risks is not just logistically challenging but also life threatening. The current COVID pandemic spread andthe need to vaccinate has added even more urgency to this issue. In this paper, we propose a framework to plan vaccination drives that balance physical security and desired
vaccination coverage with limited resources. We set up the problem as a Stackelberg
game between a defender and adversary, where the set of vaccine centers is not ﬁxed
a priori. This results in a mixed combinatorial and continuous optimization problem.
As part of solving this problem, we provide a novel contribution by identifying generalduality conditions of switching max and min when discrete variables are involved. We
perform experiments to show eﬀects of various parameters on the problem and showthat the solution proposed is scalable in practice.

60

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Towards Minimax Optimal Best Arm Identiﬁcation
In Linear Bandits
Vincent Y. F. Tan

National University of Singapore, Singapore(joint work with Junwen Yang, National University of Singapore)

We study the problem of best arm identiﬁcation in linear bandits in the ﬁxedbudget setting. By leveraging properties of the G-optimal design and incorporating it
into the arm allocation rule, we design a parameter-free algorithm, Optimal Designbased Linear Best Arm Identiﬁcation (OD-LinBAI). We provide a theoretical analysisof the failure probability of OD-LinBAI. While the performances of existing methods(e.g., BayesGap) depend on all the optimality gaps, OD-LinBAI depends on the gapsof the top d arms, where d is the eﬀective dimension of the linear bandit instance.
Furthermore, we present a minimax lower bound for this problem. The upper andlower bounds show that OD-LinBAI is minimax optimal up to multiplicative factors
in the exponent. Finally, numerical experiments corroborate our theoretical ﬁndings.

61

Outline
Towards Minimax Optimal Best Arm Identification in
Linear Bandits
1

Problem setup and preliminaries

2

Algorithm

3

Main results

4

Numerical experiments

Junwen Yang, Vincent Tan
National University of Singapore

December 10, 2021

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

1 / 36

Junwen Yang, Vincent Tan (NUS)

OutlineOD-LinBAI

December 10, 2021

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Problem setup
Linear bandits

1

Problem setup and preliminaries

2

Algorithm

3

Main results

4

Numerical experiments

An arm set A = [K ], which corresponds to arm vectors{a(1), a(2), . . . , a(K )} ⇢ Rd .

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

3 / 36

Junwen Yang, Vincent Tan (NUS)

Problem setup

Problem setup

Linear bandits

Linear bandits

An arm set A = [K ], which corresponds to arm vectors

OD-LinBAI

December 10, 2021

4 / 36

An arm set A = [K ], which corresponds to arm vectorsd{a(1), a(2), . . . , a(K )} ⇢ R .

{a(1), a(2), . . . , a(K )} ⇢ Rd .

At each time t, the agent chooses an arm At from the arm set A andthen observes a noisy reward

At each time t, the agent chooses an arm At from the arm set A andthen observes a noisy reward

Xt = h✓⇤ , a(At )i + ⌘t ,Xt = h✓⇤ , a(At )i + ⌘t ,where ✓⇤ 2 Rd is the unknown parameter vector and ⌘where ✓⇤ 2 Rd is the unknown parameter vector and ⌘t is

independent zero-mean 1-subgaussian random noise.

t isindependent zero-mean 1-subgaussian random noise.

Let E denote the set of all the linear bandit instances defined above.
Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

4 / 36

Junwen Yang, Vincent Tan (NUS)62

OD-LinBAI

December 10, 2021

4 / 36

Problem setup

Problem setup

Best arm identification in the fixed-budget setting

Best arm identification in the fixed-budget setting

To maximize the probability of identifying the best arm with no morethan T arm pulls.

To maximize the probability of identifying the best arm with no morethan T arm pulls.
The agent uses an online algorithm ⇡ to decide the arm At to pull ateach time step t, and the arm iout 2 A to output as the identified
best arm by time T .

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

5 / 36

Junwen Yang, Vincent Tan (NUS)

Problem setup

OD-LinBAI

December 10, 2021

5 / 36

Dimensionality-reduced arm vectors
Dimensionality-reduced arm vectors

Best arm identification in the fixed-budget setting

What if the corresponding arm vectors a(1), a(2), . . . , a(K ) do notspan Rd ?

To maximize the probability of identifying the best arm with no morethan T arm pulls.
The agent uses an online algorithm ⇡ to decide the arm At to pull ateach time step t, and the arm iout 2 A to output as the identified
best arm by time T .
We seek to minimize"

Pr iout 6= arg max h✓⇤ , a(j)iJunwen Yang, Vincent Tan (NUS)

j2A

OD-LinBAI#

December 10, 2021

5 / 36

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

Dimensionality-reduced arm vectors

Dimensionality-reduced arm vectors

Dimensionality-reduced arm vectors

Dimensionality-reduced arm vectors

December 10, 2021

6 / 36

What if the corresponding arm vectors a(1), a(2), . . . , a(K ) do notspan Rd ?

What if the corresponding arm vectors a(1), a(2), . . . , a(K ) do notspan Rd ?

If the arm vectors do not span Rd , the agent can work with a set of0
reduced-dimensionality vectors {a0 (1), a0 (2), . . . , a0 (K )} ⇢ Rd0
(d 0 < d) that spans Rd .

If the arm vectors do not span Rd , the agent can work with a set of0
reduced-dimensionality vectors {a0 (1), a0 (2), . . . , a0 (K )} ⇢ Rd0
(d 0 < d) that spans Rd .
0

Let B 2 Rd⇥d be a matrix whose columns form an orthonormal basisof the subspace spanned by a(1), a(2), . . . , a(K ).

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

6 / 36

Junwen Yang, Vincent Tan (NUS)63

OD-LinBAI

December 10, 2021

6 / 36

Dimensionality-reduced arm vectors

Dimensionality-reduced arm vectors

Dimensionality-reduced arm vectors

Dimensionality-reduced arm vectors

What if the corresponding arm vectors a(1), a(2), . . . , a(K ) do notspan Rd ?

What if the corresponding arm vectors a(1), a(2), . . . , a(K ) do notspan Rd ?

If the arm vectors do not span Rd , the agent can work with a set of0
reduced-dimensionality vectors {a0 (1), a0 (2), . . . , a0 (K )} ⇢ Rd0
(d 0 < d) that spans Rd .

If the arm vectors do not span Rd , the agent can work with a set of0
reduced-dimensionality vectors {a0 (1), a0 (2), . . . , a0 (K )} ⇢ Rd0
(d 0 < d) that spans Rd .

00

Let B 2 Rd⇥d be a matrix whose columns form an orthonormal basisof the subspace spanned by a(1), a(2), . . . , a(K ).

Let B 2 Rd⇥d be a matrix whose columns form an orthonormal basisof the subspace spanned by a(1), a(2), . . . , a(K ).

Such an orthonormal basis can be calculated efficiently with the
reduced singular value decomposition, Gram–Schmidt process, etc.

Such an orthonormal basis can be calculated efficiently with the
reduced singular value decomposition, Gram–Schmidt process, etc.
Set a0 (i) = B > a(i) for each arm i 2 A.

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

6 / 36

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

Least Squares Estimator

Least Squares Estimator

Ordinary Least Squares (OLS) Estimator

Ordinary Least Squares (OLS) Estimator

Let A1 , A2 , . . . , An 2 A = [K ] be the arms pulled by the agent and
X1 , X2 , . . . , Xn 2 R be the corresponding noisy rewards.

December 10, 2021

6 / 36

Let A1 , A2 , . . . , An 2 A = [K ] be the arms pulled by the agent and
X1 , X2 , . . . , Xn 2 R be the corresponding noisy rewards.
Suppose that the corresponding arm vectors{a(A1 ), a(A2 ), . . . , a(An )} span Rd .
(This is not always true in linear bandits.)

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

7 / 36

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

Least Squares Estimator

Least squares estimators

Ordinary Least Squares (OLS) Estimator

Statistical Property of the Least Squares Estimator

December 10, 2021

7 / 36

By applying properties of subgaussian random variables, the confidencebounds for the ordinary least squares estimator can be derived as follows.

Let A1 , A2 , . . . , An 2 A = [K ] be the arms pulled by the agent and
X1 , X2 , . . . , Xn 2 R be the corresponding noisy rewards.
Suppose that the corresponding arm vectors{a(A1 ), a(A2 ), . . . , a(An )} span Rd .
(This is not always true in linear bandits.)
The ordinary least squares (OLS) estimator of ✓⇤ is given by✓ˆ = V −1nXa(At )Xt

t=1

where V =Pn

t=1 a(At )a(At )Junwen Yang, Vincent Tan (NUS)> 2 Rd⇥d is invertible.

OD-LinBAI

December 10, 2021

7 / 36

Junwen Yang, Vincent Tan (NUS)64

OD-LinBAI

December 10, 2021

8 / 36

Least squares estimators

G-optimal design

Statistical Property of the Least Squares Estimator
By applying properties of subgaussian random variables, the confidencebounds for the ordinary least squares estimator can be derived as follows.

G-optimal design
The G-optimal design problem aims at finding a probability
distribution ⇡ : {a(i) : i 2 A} ! [0, 1] that minimises

Proposition 1 (Lattimore and Szepesvári (2020, Chapter 20))g (⇡) = max ka(i)k2V (⇡)−1
i2A

If A1 , A2 , . . . , An are deterministically chosen without knowing therealizations of X1 , X2 , . . . , Xn , then for any a 2 Rd and δ > 0,
s
"
✓ ◆#
1
Pr h✓ˆ − ✓ ⇤ , ai ≥ 2kak2V −1 log δ,δwhere V (⇡) =P>
i2A ⇡(a(i))a(i)a(i) .

where kak2V −1 = a> V −1 a.
Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

8 / 36

Junwen Yang, Vincent Tan (NUS)

G-optimal design

OD-LinBAI

December 10, 2021

9 / 36

G-optimal design
The following theorem states the existence of a small-support G-optimal
design ⇡ and the minimum value of g .

G-optimal design
The G-optimal design problem aims at finding a probability
distribution ⇡ : {a(i) : i 2 A} ! [0, 1] that minimisesg (⇡) = max ka(i)k2V (⇡)−1i2A

where V (⇡) =P

i2A ⇡(a(i))a(i)a(i)>.

g (⇡) is closely related to the confidence bounds in the best arm
identification problem.

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

9 / 36

Junwen Yang, Vincent Tan (NUS)

G-optimal design

OD-LinBAI

December 10, 2021

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December 10, 2021

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Outline

The following theorem states the existence of a small-support G-optimal
design ⇡ and the minimum value of g .
1

Problem setup and preliminaries

2

Algorithm

3

Main results

4

Numerical experiments

Theorem 1 (Kiefer and Wolfowitz (1960))
If the arm vectors {a(i) : i 2 A} span Rd , the following statements areequivalent:(1)⇡ ⇤ is a minimiser of g .

(2)⇡ ⇤ is a maximiser of f (⇡) = log det V (⇡).

(3)

g (⇡ ⇤ ) = d.

Furthermore, there exists a minimiser ⇡ ⇤ of g such thatd(d + 1).
| Supp (⇡) | 
2
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OD-LinBAI

Algorithm: informal

Informal Algorithm: First Step
Begin with K arms

Algorithm 1 Optimal Design-based Linear Best Arm Identification (OD-LinBAI)
Input: time budget and the arm set.
1: Initialization.
2: for each phase do
3:
Find dimensionality-reduced arm vectors for the active arms.
4:
Find a G-optimal design for the active arms and pull arms according
to it.
5:
Estimate the expected rewards of the active arms by OLS.
6:
Elimination.
. To maintain a set of active arms
7: end for
Output: the best arm.

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Informal Algorithm: First StepOD-LinBAI

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December 10, 2021

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Informal Algorithm: First Step
Begin with K arms

Begin with K arms

Dimensionality Reduction

Dimensionality Reduction

Find G-opt Design
Pull arms according toJunwen Yang, Vincent Tan (NUS)

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December 10, 2021

13 / 36Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

Informal Algorithm: r th Step where 1 < r  blog2 dc

Informal Algorithm: First Step
Begin with K arms

Dimensionality Reduction

Find G-opt Design
Pull arms according toRetain best

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

arms

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Informal Algorithm: r th Step where 1 < r  blog2 dc

Informal Algorithm: r th Step where 1 < r  blog2 dc

Dimensionality Reduction

Dimensionality Reduction

Find G-opt Design
Pull arms according toJunwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

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Informal Algorithm: r th Step where 1 < r  blog2 dc

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Algorithm: initialization (Part 1)
Algorithm 2 Optimal Design-based Linear Best Arm Identification (OD-LinBAI)
Input: time budget T 2 N, arm set A = [K ], and K arm vectors{a(1), a(2), . . . , a(K )} ⇢ Rd .
1: Initialize t0 = 1, A0
A and d0 = d.
2: For each arm i 2 A0 , set a0 (i) = a(i).
3: Calculate. To ensure the total time budget consumed  T

Dimensionality Reduction

Find G-opt Design
Pull arms according toRetain best

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

arms

m=

December 10, 2021

14 / 36Junwen Yang, Vincent Tan (NUS)

Algorithm: dimensionality-reduced arm vectors (Part 2)

OD-LinBAI✓

T
log2 d◆December 10, 2021.

15 / 36

if r = 1 thenFind a G-optimal design ⇡r : {ar (i) : i 2 Ar −1 } ! [0, 1] with| Supp (⇡r ) |  d(d + 1)/2.
14:
else
15:
Find a G-optimal design ⇡r : {ar (i) : i 2 Ar −1 } ! [0, 1].
16:
end if
17:
Set
X
Tr (i).
Tr (i) = d⇡r (ar (i)) · me and Tr =13:

6:i2Ar −1ar (i) = Br> ar −1 (i).

18:

end ifOD-LinBAI=⇥

12:

Set dr = dim (span ({ar −1 (i) : i 2 Ar −1 })).
if dr = dr −1 then7:
For each arm i 2 Ar −1 , set ar (i) = ar −1 (i).
8:
else
9:
Find matrix Br 2 Rdr −1 ⇥dr whose columns form a orthonormal
basis of the subspace spanned by {ar −1 (i) : i 2 Ar −1 }.
10:For each arm i 2 Ar −1 , project arm vectors onto a lowerdimensional subspace, i.e.,
5:Junwen Yang, Vincent Tan (NUS)r =1

dlog2 deAlgorithm: G-optimal design (Part 3)

4: for r = 1 to dlog2 de do

11:no dlogP
2 de−1 ⌃⌥d
T − min K , d(d+1)−
22r

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Choose each arm i 2 Ar −1 exactly Tr (i) times.

Junwen Yang, Vincent Tan (NUS)67

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Algorithm: OLS (Part 4)

19:

Algorithm: elimination (Part 5)

Calculate the OLS estimator of this phase:✓ˆr = Vr−1

tr +T
r −1
X

ar (At )Xt

Let Ar be the set of dd/2r e arms in Ar −1 with the largest estimatesof the expected rewards.
22:
Set tr +1 = tr + Tr .
23: end for
Output: the only arm iout in Adlog2 de .
21:

t=tr

with
Vr =

X

Tr (i)ar (i)ar (i)> .

i2Ar −1

20:

For each arm i 2 Ar −1 , estimate the expected reward:p̂r (i) = h✓ˆr , ar (i)i .

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OutlineOD-LinBAI

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Notations
Some Other Notation1

Problem setup and preliminaries

2

Algorithm

3

Main results

4

Numerical experiments

For any arm i 2 A, letp(i) = h✓⇤ , a(i)idenote the expected reward.
For convenience, we assume that p(1) > p(2) ≥ · · · ≥ p(K ).
For any suboptimal arm i, we denote∆i = p(1) − p(i)as the optimality gap. For ease of notation, we also set ∆1 = ∆2 .

Junwen Yang, Vincent Tan (NUS)

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20 / 36Junwen Yang, Vincent Tan (NUS)

Hardness Quantities

Hardness Quantities

Recall suboptimality gap

Recall suboptimality gap∆i = p(1) − p(i) = h✓⇤ , a(1)i − h✓⇤ , a(i)i .

OD-LinBAI

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21 / 36∆i = p(1) − p(i) = h✓⇤ , a(1)i − h✓⇤ , a(i)i .

Hardness Quantities
First hardness quantity for linear banditsX
H1,lin =∆−2i
1i dJunwen Yang, Vincent Tan (NUS)

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Hardness Quantities

Hardness quantities

Recall suboptimality gap
Table: Comparisons of di↵erent hardness quantities: H1 , H2 , H1,lin and H2,lin∆i = p(1) − p(i) = h✓⇤ , a(1)i − h✓⇤ , a(i)i .

H1 =Hardness Quantities

P1i KFirst hardness quantity for linear banditsX
H1,lin =∆−2i

H1,lin =∆−2iP

1i d1i d

∆−2
i

1
1 H
H2  log(2K )H2 = max ∆i 2
2i Ki

H2,lin = max ∆i 2
2i diH

1  H1,lin
 log(2d)2,linSecond hardness quantity for linear bandits
H2,lin = max

1
1  HH1,lin
 Kd

i

2
1  HH2,lin Kd

2id ∆2i

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December 10, 2021

22 / 36Junwen Yang, Vincent Tan (NUS)

Upper bound

OD-LinBAI

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Existing Works
Comparison to Existing Art
BayesGap (Ho↵man et al., 2014):

Theorem 2 (Error Probability of OD-LinBAI)I

For any linear bandit instance ⌫ 2 E, OD-LinBAI outputs an arm iout
satisfying
◆
✓
◆
✓
4K
m
Pr [iout 6= 1] 
+ 3 log2 d exp −d
32H2,linI

Not parameter-free (require
knowledge of the problem instance)⇣⇣the⌘⌘
Error probability: exp −⌦ HT1

where m was defined in the algorithm and scales as
✓
◆
T
i
m=⇥
and
H2,lin = max 2 .
2id ∆ilog2 dJunwen Yang, Vincent Tan (NUS)

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24 / 36Junwen Yang, Vincent Tan (NUS)

Existing Works

Existing Works

Comparison to Existing Art

Comparison to Existing Art

BayesGap (Ho↵man et al., 2014):

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BayesGap (Ho↵man et al., 2014):

I

I

Not parameter-free (require
knowledge of the problem instance)⇣⇣the⌘⌘T
I Error probability: exp −⌦H1

IPeace (Katz-Samuels et al., 2020):

Not parameter-free (require
knowledge of the problem instance)⇣⇣the⌘⌘
Error probability: exp −⌦ HT1

Peace (Katz-Samuels et al., 2020):

I

I

Not parameter-free (require the knowledge of the problem instance)
I Not minimax optimalI

Not parameter-free (require the knowledge of the problem instance)
Not minimax optimal

LinearExploration (Alieva et al., 2021):
I

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Error probability: exp − ⌦( H̃ log)
K

Junwen Yang, Vincent Tan (NUS)

69

2

2

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Existing Works

Lower Bound

Comparison to Existing Art
BayesGap (Ho↵man et al., 2014):
I
I

Not parameter-free (require
knowledge of the problem instance)⇣⇣the⌘⌘
Error probability: exp −⌦Setup for the Lower Bound Results

T
H1

For any linear bandit instance ⌫ 2 E, we denote the hardness quantity
H1,lin of ⌫ as H1,lin (⌫).

Peace (Katz-Samuels et al., 2020):
I
I

Not parameter-free (require the knowledge of the problem instance)
Not minimax optimal

LinearExploration (Alieva et al., 2021):
I��
T
Error probability: exp − ⌦( H̃ log
)
K
2

2

GSE (Azizi et al., 2021):
I

�
�
T ∆2
Error probability: exp − ⌦( d log 1K )2

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Lower BoundOD-LinBAI

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Lower boundTheorem 3 (Minimax Lower Bound)

Setup for the Lower Bound Results

If T ≥ a2 log(6Td)/900, thenFor any linear bandit instance ⌫ 2 E, we denote the hardness quantity
H1,lin of ⌫ as H1,lin (⌫).

✓
◆
1
240T
exp −
.
6
a

⇡
6= 1] ≥min max Pr [iout⇡Let E(a) denote the set of linear bandit instances in E whose H1,lin isbounded by a (a > 0), i.e.,⌫2E(a)

E(a) = {⌫ 2 E : H1,lin (⌫)  a}.

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

26 / 36Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

Lower boundComparison of Upper and Lower Bounds

Theorem 3 (Minimax Lower Bound)

Tightness of Upper and Lower Bounds
I

✓

◆

2700T
H1,lin (⌫) log2 d

◆◆

1
240T
⇡
6= 1] ≥ exp −min max Pr [iout⇡ ⌫2E(a)6aFurther if a ≥ 15d 2 , then✓✓⇡
6= 1] · exp
min max Pr [iout⌫2E(a)

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As the time budget T tends to infinity,

If T ≥ a2 log(6Td)/900, then⇡December 10, 2021

OD-LinBAI

Upper bound:✓✓exp −⌦

T
H2,lin log2 d◆◆Lower bound:✓✓exp −OT
H1,lin log2 d◆◆Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

.
I

1
≥ .
6

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Comparison of Upper and Lower Bounds

Comparison of Upper and Lower Bounds

Tightness of Upper and Lower Bounds

Tightness of Upper and Lower Bounds

As the time budget T tends to infinity,
I

Upper bound:✓T
H2,lin log2 d◆◆✓✓
exp −OT
H1,lin log2 d◆◆

exp −⌦
I

Lower bound:

As the time budget T tends to infinity,

✓

I

I

H2,lin  H1,lin

Upper bound:

Lower bound:✓✓exp −⌦

T
H2,lin log2 d◆◆✓✓
exp −OT
H1,lin log2 d◆◆H2,lin  H1,lin
Hence, OD-LinBAI is minimax optimal.

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

28 / 36Junwen Yang, Vincent Tan (NUS)

OutlineOD-LinBAI

December 10, 2021

28 / 36

Synthetic dataset 1: a hard case[cos( ⇡4 ), sin( ⇡4 )]>a(K ) = [cos( 3⇡), sin( 3⇡
)]>
4
4

1

Problem setup and preliminaries

2

Algorithm

3

Main results

4

Numerical experiments✓ ⇤ = a(1) = [1, 0]>

One best arm, one worst arm and K − 2 almost second best arms.
a(i) = [cos(⇡/4 + φi ), sin(⇡/4 + φi )]> with φi ⇠ N (0, 0.092 ) for
i = 2, 3, . . . , K − 1
H1 ⇡ H2 ⇡Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

29 / 36Junwen Yang, Vincent Tan (NUS)

Synthetic dataset 1: a hard caseKK
H1,lin ⇡ H2,linddOD-LinBAI

December 10, 2021

Synthetic dataset 1: a hard case

1

1

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0

0
0

5

10

15

20

25

0

10

20

30

40

0
25

50

OD-LinBAI

December 10, 2021

0.2

10

2

5 10

2

10

3

2 10

3

0
50

102

5 102 103

4 103

Figure: Error probabilities for di↵erent time budgets T with K = 25, 50.

Figure: Error probabilities for di↵erent numbers of arms K with T = 25, 50.

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Real-world dataset: Abalone Dataset (Dua and Gra↵,2017)

Real-world dataset: Abalone Dataset
1
0.9
0.8

Description the Abalone Dataset

0.7
0.6

The age of each abalone is usually hard to determine from physicalmeasurements.

0.5
0.4

K = 400 : 400 groups of 8 attributes (such as sex, length, diameter,etc.) of the abalone

0.3

d = 9: ✓⇤ 2 R9 comes from the linear regression

0.1

0.2

To identify the abalone with the largest age with using the 8
attributes from physical measurements

0
2
5 10

10

3

2 10

3

5 10

3

10

4

Figure: Error probabilities for di↵erent budgets T .

Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

33 / 36Junwen Yang, Vincent Tan (NUS)

OD-LinBAI

December 10, 2021

34 / 36

References
Tor Lattimore and Csaba Szepesvári. Bandit algorithms. Cambridge University Press, 2020.
Jack Kiefer and Jacob Wolfowitz. The equivalence of two extremum problems. Canadian
Journal of Mathematics, 12:363–366, 1960.
Matthew Ho↵man, Bobak Shahriari, and Nando Freitas. On correlation and budget constraints
in model-based bandit optimization with application to automatic machine learning. In
Artificial Intelligence and Statistics, pages 365–374. PMLR, 2014.

Thanks for listening!

Julian Katz-Samuels, Lalit Jain, Kevin G Jamieson, et al. An empirical process approach to theunion bound: Practical algorithms for combinatorial and linear bandits. Advances in Neural
Information Processing Systems, 33, 2020.
Ayya Alieva, Ashok Cutkosky, and Abhimanyu Das. Robust pure exploration in linear bandits
with limited budget. In International Conference on Machine Learning, pages 187–195.
PMLR, 2021.
MohammadJavad Azizi, Branislav Kveton, and Mohammad Ghavamzadeh. Fixed-budget
best-arm identification in contextual bandits: A static-adaptive algorithm. arXiv preprint
arXiv:2106.04763, 2021.
Dheeru Dua and Casey Gra↵. UCI machine learning repository, 2017. URL
http://archive.ics.uci.edu/ml.

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December 10, 2021

36 / 36Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Global and Local Prediction Methods of COVID-19
Time Series with Machine Learning
Amir Mosavi

Obuda University, Hungary(joint work with Sina Ardebili, Annamaria R. Varkonyi-Koczy)This presentation is devoted to the advancement of the machine learning-based
methods for accurate prediction of the Covid-19 outbreak prediction. Advancementof the novel models for time-series prediction of COVID-19 is of utmost importance.
Machine learning (ML) methods have recently shown promising results. The present
study aims to engage an artiﬁcial neural network-integrated by grey wolf optimizer for
COVID-19 outbreak predictions by employing the global and local dataset. For thecase study, the training and testing processes have been performed by time-series data
related to January 22 to September 15, 2020 and validation has been performed by timeseries data related to September 16 to October 15, 2020. Results have been evaluatedby employing mean absolute percentage error (MAPE) and correlation coeﬃcient (r)values. ANN-GWO provided a MAPE of 6.23, 13.15 and 11.4% for training, testingand validating phases, respectively. According to the results, the developed model
could successfully cope with the prediction task.

73

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Deep learning in diagnostic applications: the good,the bad, and the ugly.
Yaniv Gal

MoleMap Ltd, New ZealandArtiﬁcial Intelligence (AI) in general, and deep learning (DL) in particular, haverecently gained popularity in both academic and commercial applications due to its
ability to automatically identify meaningful features in the data and calculate a complex
decision boundary between in the constructed feature space.
The medical device industry has recognised the potential of deep learning to support
clinicians’ diagnosis and gradually integrate deep learning into the clinical workﬂow to
enforce diagnostic decisions and signiﬁcantly reduce the likelihood of human error. Skincancer diagnosis is an example for such application, where dermatologist rely mainly (orsolely) on visual inspection in order to diagnose suspicious skin lesions. Deep learning,which can extract subtle features from dermoscopic images provides accurate diagnosis
to support the clinicians in their decision and reduce error rates.
While the question of whether AI will ever replace the clinician’s decision making
is still in debate, it is undeniable that the diagnostic performance of these algorithms
is continuously improving and in some case is comparable or even surpasses humanspecialists. Moreover, advances in deep-learning training techniques and improved
network architectures now allow training these models with less data and yet, lower
the risk of overﬁtting, which makes these algorithms even more accessible and increases
their attractiveness. However, once a model is trained on labelled data, it is impossible
to explain its decisions on new data, in a way that will be meaningful for the user (i.e.
clinician). This lack of “explainability” in deep learning creates a landscape whereclinical decisions that are supported by AI require the clinician to either blindly trust
the trained model or monitor the automated decision to a level that diminishes theutility that it brings, until enough trust is gained.
Furthermore, when a trained AI model is given a sample that it was not trainedon (i.e. unknown class), it is impossible to predict the output of the system and suchcases often lead to a wrong diagnostic result that is presented by the AI model withhigh score, implicitly suggesting that the result should be trusted.
This talk reviews some of the beneﬁts, pitfalls, and challenges of using modern
AI models in real-world diagnostic applications. We use the AI technology that iscurrently developed by Kāhu, New Zealand, for skin cancer detection as a case study,
and describe what is a desirable solution to the above challenges may be.

74Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Language models in industry and around the world
Caleb Moses

Dragonﬂy Data Science, Wellington, New ZealandLanguage Models have been a strong focus for research in the AI industry following the publication of the Bidirectional Encoder Representations from Transformers(BERT) neural network architecture published by Google in 2018. More recently, major tech companies have been engaged in an arms race to build ever more complex
language models trained on increasingly massive text datasets, also aimed at as many
languages as possible.
I will discuss the latest language model trends and their implications for the digital economy, as well as their ethical implications. In addition, I will discuss nonEnglish language models and their diﬀerences as well as the current situation for underresourced minority languages around the world.

75

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Option pricing with transaction costs –mathematical modelling in new digital economy
Xiaoping Lu

School of Mathematics and Applied Statistics, University of Wollongong, Australia

Mathematics plays an important role in modern ﬁnance, particularly in pricing
options, which are ﬁnancial derivatives with complicated structures. When transaction
costs are considered, there is no longer a unique fair price between the buyer andthe writer of an option, as both parties wish to recover the costs incurred in trading
the underlying stocks from the prices that they are willing to pay or receive for the
option. Mathematically, transaction costs make option pricing problems much more
complicated, especially for American options and options under stochastic volatility.
In this talk, we shall discuss the valuation problems for options with transaction costs,
and examine how transaction costs aﬀect option prices and the optimal exercise policy
for American options.

76

Acknowledgement
Option pricing with transaction costs–mathematical modeling in new digital economy
Most of the work reported here are carried out with Dr D. Yan andProf. S.-P. Zhu

Xiaoping Lu
School of Mathematics and Applied Statistics
University of Wollongong

2 / 36

What is an option?

European or American?
There are two main styles of options:An option is a contract between two parties, a buyer (holder) anda seller (writer), that gives the buyer right but not obligation withthe following conditions:

European Option:
The holder can only exercise their right at the expiration date.
This type of options are characterized by their time to expiry
and exercise price.

at a prescribed time in the future, the expiry date, the holderof the option may
buy (sell) a prescribed asset, known as the underlying, for

American Option:a prescribed price, known as the exercise price or strike price.

It allows the holder to exercise before the expiry when theunderlying asset price is deemed proﬁtable (optimal).

An option giving the right to buy is called a call option, and theright to sell is a put option.

Because of this ﬂexibility, the price of an American option is
dictated by the concept of optimal exercise price.

3 / 36

Value of an option

4 / 36The Black–Scholes model

In an option contract, the holder has right, but not obligation
to buy or sell. Therefore, the option has value for the holder,which must be paid for.

Under the Black–Scholes (B–S) framework, the value of an option,
denoted by V (S, t), is governed by the following partial diﬀerential
equation (PDE)

The writer does have a potential obligation, as he/she has tosell or buy if the holer decides to buy or sell the asset. The
writer needs to be compensated for this obligation.

1
∂V
∂2V
∂V
+ σ 2 S 2 2 + rS
− rV = 0,
∂t
2
∂S
∂S
where S is the current price of the underlying stock;t, the current time;
σ, the volatility of the underlying;r, the risk-free interest rate.

However, how much would one pay for this right? that is,what is the value of an option?
Accurate evaluation of an option would give answer to theabove question, and more.

This equation is referred to as the B–S PDE. The solution of the
PDE subject to appropriate conditions gives rise to the fair priceof the corresponding option.

The Black–Scholes model is the foundation of modern
ﬁnancial theory.

5 / 3677

6 / 36

Boundary and ﬁnal Conditions

Optimal boundary for American option
For an American option, there exists an optimal exercise price,
Sf (t), which divides the domain of S into a continuation/holding
region and an exercise region. For an American put

The B–S PDE can be solved analytically for the value of European
options, subject to conditions on S and t. For a European put:at the time of expiry, i.e. at t = T , the payoﬀ of the option isthe value ofV (S, T ) = max(K − S, 0).

V (Sf (t), t) = K − Sf (t), S = Sf (t),which is derived from the option value being simply the
intrinsic value when the optimal exercise price is reached.
While in the exercise region

where K is the pre-determined exercise price or the strike.
If S = 0, at any time the option will take the present value ofK received at expiry T ,V (S, t) = K − S, 0 ≤ S < Sf (t).

V (0, t) = Ke−r(T −t) .

∂V(Sf (t), t) = −1, is needed to ensure
Another condition,∂Ssmoothness between the option value and the payoﬀ function
at S = Sf (t).

As S → ∞ the option is unlikely to be exercised, so
V (S, t) → 0, S → ∞.
7 / 368 / 36

A complete B–S PDE system for American put
⎧
1
∂V
∂2V
∂V
⎪
⎪
+ σ 2 S 2 2 + rS
− rV = 0,
⎪
⎪
∂t
2
∂S
∂S
⎪
⎪
⎪
⎪
V
(S,
T
)
=
max(K
−
S,
0),
⎨
V (Sf (t), t) = K − Sf (t),
⎪
⎪
⎪
∂V
⎪
⎪
⎪ ∂S (Sf (t), t) = −1,⎪⎪⎩
V (S, t) = 0, S → ∞.

B–S model assumptions
The asset price S follows the lognormal walkdS = μSdt + σSdWt ,where μ is the drift rate, dWt is√normally distributed with amean 0 and standard deviation dt.

(1)

The risk-free interest rate r and the asset volatility σ are
known.
There are no transaction costs associated with hedging aportfolio.

In theory, the solution of PDE system (1) gives the value of theAmerican put V and the optimal exercise price Sf at any time t(0 < t < T ) and at stock price S ≥ Sf .

The underlying pays no dividends (can be dropped).
There is no arbitrage possibilities.

However, the problem is highly nonlinear due to the fact that Sf is
unknown, constituting a moving boundary problem, for which ananalytic solution is not attainable except in special cases.

Trading of the underlying can be done continuously.
Short selling is permitted.
9 / 36

10 / 36

Limitations of B–S model

Option pricing with transaction costs

Under the assumptions, the B–S PDE or a variant incorporating
dividends or time-dependent r or σ is satisﬁed by any derivative
security whose price depends only on current S and t.

In the presence of transaction costs, both the holder/buyer
and the writer/seller of an option would want to recover theircosts from the price they are willing to pay or receive.

The B–S model set up the foundation for modern derivative
pricing. However, there are some limitations:

The market is incomplete due to the market friction — there
is no longer a unique price for the buyer and the writer of an
option.

The value of an option is independent of μ.
=⇒ Two investors with diﬀerent views on μ agree on thevalue of the option.

Instead, the holder price ph and the writer price pw form a
bid-ask spread,ph < pBS < pw ,Transaction costs are not taken into consideration. In
practice, trading of underlying assets attracts fees.

where pBS is the B–S price.
Continuous hedging is impossible, as it could lead to unreal
high writer price and possible negative buyer price.

The model is based on a continuous hedging strategy.
=⇒ The strategy could lead to an unrealistic high option
selling price, when transaction costs are considered.

The standard B–S model is no longer suitable.

11 / 36

12 / 36

78

The Leland Model

The discrete hedging strategy

The assumptions are those for the B–S model, except

Consider a ﬁnancial market, where investors have access to bothstocks and options.

it is a discrete hedging strategy.

A hedged portfolio of a writer of an option, who is shorting one
unit of option and long holding Δ unit of the stock, can be
expressed asΠ = −V + ΔS

the portfolio is re-balanced every δt, where δt is a
non-inﬁnitesimal constant interval.
The costs associated with the trading of the underlying areproportional to the value of the transaction.

The change in the value of the hedged portfolio in a time-step δt is

the costs for a transaction of ν shares valued at S are κ|ν|S,
where κ is a constant rate and ν is the number of shares
traded: bought ν > 0 and sold ν < 0.

δΠ = −δV + ΔδS − κS|ν|.

Leland, Option pricing and replication with transaction costs, J.
Finance, 40(5) (1985).

(2)

On the other hand, the portfolio of a holder is Π = V − ΔS, andthe change in the value of the portfolio in a time-step is

Wilmott et al., Hedging option portfolios in the presence oftransaction costs, Adv. Futures Opt Res., 7 (1994).

δΠ = δV − ΔδS − κS|ν|.

(3)13 / 36

The discrete hedging strategy

The discrete hedging strategy
The number of stocks held at time t + δt due to hedging is
Δt+δt = ∂V
∂S (S + δS, t + δt).

The random walk in discrete time is given by√δS = μSδt + σSφ δt,

So the number of traded stocks iswhere φ is drawn from a standard normal distribution.

ν=Applying Itô’s lemma to V , we obtain for the writer


√
∂V
φ δt
δΠ = σS Δ −
∂S


∂V
∂V
1 2 2 ∂2V 2
σ S+− μΔS δt − κS|ν|φ + μS
−
2
∂S 2
∂S
∂t

ν ≈ σS

15 / 36

The discrete hedging strategy

∂2V √
φ δt.
∂S 2

Thus, the expected transaction costs during one time step δt is 2 ∂ V  √
2κσS 2  2  δtE [κS|ν|] =π∂S

16 / 36

The modiﬁed B–S PDE

Under the hedging strategy, the writer of the option is expected tomake as much from his/her portfolio as if he/she puts the moneyin the bank, thenE [δΠ] = rΠδt.

Thus, the following PDE (Wilmott et al. 1994) is derived 2 ∂ V 
∂V
∂2V
2
∂V
1
+ σ2S 2 2 ±κσS 2  2  + rS
− rV = 0, (4)
∂t
2
∂S
πδt
∂S
∂S

This leads to∂V∂V(S + δS, t + δt) −(S, t)∂S∂S

Applying Taylor’s expansion of Δt+δt , we obtain

Following the hedging strategy, we choose Δ = ∂V
∂S , which is thenumber of the stock held at time t.

∂V
∂2V
1
+ σ2S 2 2 +∂t2∂S

14 / 36

where the ‘+’ sign is for the writer of the option, and the ‘−’ forthe buyer.

 2 ∂ V 2∂VκσS 2  2  + rS
− rV = 0πδt∂S∂SThis equation is a modiﬁed B–S PDE with an extra term due to thetransaction costs, which can be used to price a portfolio of options.

Following similar argument, the governing equation for the holderof the option can be derived as 2 ∂ V 
∂V
∂2V
2
∂V
1
+ σ2S 2 2 −κσS 2  2  + rS
− rV = 0
∂t
2
∂S
πδt
∂S
∂S

The pricing equation is nonlinear because of the transaction costs
term so analytical solution is not available.
However, it is not diﬃcult to solve (4) with simple ﬁnite diﬀerence
methods.

17 / 3679

18 / 36

The Laland Equation

Complications of American options
The Leland model can be used to price American options withappropriate boundary conditions.

For a single option, (4) reduces to the Leland equation
∂2V
∂V
1
∂V
+ σ̃ 2 S 2 2 + rS
− rV = 0,
∂t
2
∂S
∂S



8 κ
where σ̃ 2 = σ 2 1 ± πδt
σ .

However, the optimal exercise boundary associated with an
American option again makes the problem more complicated.

(5)

Unlike in the case of European options, the price for the writer ofan American option cannot be determined independently.
The holder of an American option has the right to decide if
and when to exercise their right.
— Mathematically, a holder’s problem is a moving boundary
problem.

The Leland equation (5) only diﬀers from the standard B–S PDEin the adjusted volatility term, σ̃, which is referred to as theLeland number.

Once the holder makes the decision, the writer has obligation
to fulﬁll the holder’s right.
—The writer’s problem is one with known time-dependent
boundary (the holder’s optimal exercise price).

As a result, for a European option, the B–S formula could be used
to determine the option prices for the writer and the buyer,
respectively.

19 / 36

20 / 36

Pros and cons of discrete hedging strategy

Utility indiﬀerence pricing
Utility indiﬀerence price — the price at which the investor isindiﬀerent, in the sense that his or her expected utility under
optimal trading maintains the same, whether he or she trades
stocks in the market with or without an option in the portfolio.

This strategy is basically a modiﬁed B–S model, which is easy to
implement, especially for a single option case.
However, there are some inherent problems:Perfect hedging is not possible, which leads unavoidable
hedging errors.

Utility indiﬀerence pricing is pioneered by Davis et al. for thepricing of European options with proportional transaction costs.

—More frequent re-balancing would reduce the hedging error,
but increase transaction costs.

Davis et al., European option pricing with transaction costs, SIAM
J. Control and Optim., 31(2) (1993).

Investors’ risk preferences are not incorporated in the model,
which could lead to the mispricing of an option.

The method by Davis et al. was later extended for pricing American
options by others, however, only Zakamouline provided analysis on theearly exercise boundary as well as prices for the holder and writer.

—The only choice an investor could make is the re-balancing
interval.

Zakamouline, American option pricing and exercising withtransaction costs, J. Comput. Finance, 8(3) (2005).

21 / 36

22 / 36

Pros and cons of utility indiﬀerence method

Our new utility indiﬀerence method

Utility indiﬀerence formulation takes into consideration of theinvestors’ risk preferences in terms of their utility functions.

For a pricing method to work well in practice, it should not be toocomplicated.

The core idea of utility indiﬀerence method is utility maximization.

To achieve a balance, we propose a new method based on utility
maximization, following the discrete hedging idea to re-balance
one’s portfolio at regular interval only.

Mathematically, one needs to solve three-dimensional
Hamilton-Jacobi-Bellman (HJB) equations corresponding to theinvestor’s portfolios with or without an option.

Yan and Lu (2021), Utility-indiﬀerence pricing of European options
with proportional transaction costs, Journal of Computational andApplied Mathematics.

Since an analytical solution is not attainable, the HJB equations
have to be solved numerically.

The key point in our approach is that we deduce the expected
number of stocks traded in one time-step, instead of treating it asa process. This reduces the HJB equation to two dimensional forthe portfolio without an option, thus, provides considerable
time-savings.

The analysis in Zakamouline (2005) is insightful, but it is also very
sophisticated and computationally expensive. As a result, a key
aspect of the utility indiﬀerence pricing is to ﬁnd an eﬃcient
numerical technique or approximate approach.

23 / 36

24 / 3680

Our utility indiﬀerence formulation

Our utility indiﬀerence formulation

Consider a market with two assets, a risk-free bond, which earns aconstant interest rate r, and a risky stock which follows log-normal
dynamics(6)
δS = μSδt + σSδBt ,√Applying Itô’s lemma to ν, and keeping the terms of O( δt), we
obtain(ω − 1) σωW
S δBt
ν=
.
1 + sign(ν)κω

where Bt is a one-dimensional Brownian motion.

Therefore, the expected transaction costs in a time-step is
�
�
√
√
�
�
2
2
κ δt
E κS|ν| =σω(1−ω)W ≈κσω(1−ω)W δt.
π 1 + sign(ν)κωπ

An investor, whose total current wealth is denoted as W , starts
with a known initial endowment W0 . The change of the investor’s
total wealth in one time step δt can be expressed as
δW = [ω(μ − r) + r]W δt + ω(t)σW δBt − κS|ν|.

Then

(7)

δW = [ω(μ − r) + r]W δt + ω(t)σW δBt −where�ω ∈ [0,� 1] is the fraction of the wealth in the risky stock,, the number of stocks traded at t + δt.
ν = δ ωWS�√2κσω(1 − ω)W δt.
π(8)25 / 36

Our utility indiﬀerence formulation

26 / 36

Our utility indiﬀerence formulation
The value function Q is subject to the following conditions⎧⎪⎪⎨Q(W, T ) = U (W (T )),Q(0, t) = U (0),⎪⎪⎩ lim Q(W, t) = U (W ).

The investor without option maximizes their expected utility ofterminal wealth W by choosing an optimal trading strategy withthe value function
� �
��
�
�
Q(W, t) = max Et U W (T ) ��W (t) = W ,ω∈[0,1]

(10)

W →∞

Mathematically the utility maximization problem is to ﬁnd ω ∗which satisﬁes (9).

where U (·) is the investor’s utility function.

The function Q should satisfy the following HJB equation:
��
�
�
∂Q
∂Q
2
(μ − r)ω − πδtκσω(1 − ω) W ∂W
(9)∂t + maxω∈[0,1]
�
2
∂Q
∂ Q
+ 12 σ 2 ω 2 W 2 ∂W
+ rW ∂W= 0.
2

The utility function is concave, which leads to�∂Q2κσ) ∂W(μ − r − πδtω∗ = − �
∂Q
∂2Q
2
2 πδt
κσ ∂W
+ σ 2 W ∂W
2
27 / 36

Our utility indiﬀerence formulation

28 / 36

Utility indiﬀerence priceAn investor who buys or sells an option, in addition to investing in
risk-free bond and the risky stock, also aims to maximize theirterminal wealth subject to certain conditions.

Let S0 be the initial stock price at time t = 0. By deﬁnition, theutility indiﬀerence price, the price for the holder of the option withinitial wealth W0 , is the value ph satisfying the following equation

The HJB equation for a portfolio with a European option is
�
�
��
∂QE
∂QE
2
+ max
κσω(1 − ω) W
(μ − r)ω −
∂t
πδt
∂W
ω∈[0,1]�
1 2 2 2 ∂ 2 QE
∂ 2 QE2+ σ ω W+ ωσ SW2∂W 2∂S∂W∂ 2 QE
∂QE
1
∂QE
+ μS
+ σ2S 2
= 0.
(11)
+ rW
∂W
∂S
2
∂S 2

QE−h (W0 − ph , S0 , 0) = Q(W0 , 0).

(12)

Similarly, the writer price of the option, pw , is given by
QE−w (W0 + pw , S0 , 0) = Q(W0 , 0).

(13)

It is clear that the utility indiﬀerence price depends on the utility
function, therefore, it will reﬂect investors’ risk preferences. Thiscannot be achieved under the Black-Scholes framework.

The value function QE should satisfy the ﬁnal condition for theholder or writer, respectively.
29 / 3681

30 / 36

Utility indiﬀerence price for American options

Numerical examples

The same principle applies for American options. However, due tothe embedded optimality problem, not only we have to consider
the portfolio for the holder and that for the writer separately, butalso take into consideration if the holder would excise their right.

For our utility indiﬀerence approach, the following exponential
utility is applied
U (y) = 1 − e−λy ,where λ is a constant that represents the degree of risk preference,with λ > 0 being risk averse, λ = 0 risk-neutral, and λ < 0 risk
seeking.

In utility indiﬀerence pricing, the holder takes the pay-oﬀ andcontinues to optimize the portfolio, whereas the writer pays the
pay-oﬀ, also continues to optimize the portfolio until expiry.

Unless otherwise mentioned, all of the calculations are carried out
for the following parameters: the risk-less interest rate r = 0.06,
volatility σ = 0.45, strike price K = 10, expiry T = 0.5 (years),
1(year) with various values of otherhedging frequency δt = 10
parameters and initial values.

As a result, the HJB system is more complicated, and much morecalculations need to be done. Here we omit the lengthy equations,only present the results. More details can be found in
Lu, Yan, and Zhu (2022), Optimal exercise of American puts withtransaction costs under utility maximization, Applied Mathematics andComputation.

The HJB equations are solved by a policy iteration scheme.

31 / 36

32 / 36Comparison of European put prices

Comparison of American put prices
Table 2: American put price for λ = 0.05, μ = 0.1 and κ = 0.08%.

Table 1: European put price for λ = 0.05, μ = 0.1 and κ = 0.08%.
S0

B-S price

8
9
10
11
12

2.16882
1.567141.103120.760020.51476

Holder priceHedging Indiﬀerence
2.16469
2.16864
1.562021.56685
1.09760
1.102780.754640.75969
0.509900.51447

Writer priceHedging Indiﬀerence
2.172942.181121.57224
1.57697
1.10863
1.110310.765390.765080.51961
0.51819

S0

B-S

8
9
10
11
12

2.24869
1.614211.130880.776430.52451

Holder priceHedging Indiﬀerence
2.24495
2.24835
1.609211.61380
1.12535
1.130430.77100
0.77599
0.519590.52411

Writer priceHedging Indiﬀerence
2.252342.25576
1.619131.62054
1.13634
1.13568
0.781810.779850.52939
0.52682

Similar trend for the writer and holder prices is observed forAmerican options. This is because when the investor’s risk
preference is considered (utility indiﬀerence approach) the
asymmetry between the writer and holder of an option contract is
reﬂected in their prices.

It can be observed that the writer price and holder price deviate
similar amount from the B–S price for hedging strategy, but thewriter price in utility indiﬀerence pricing deviates more than theholder price from the B–S price.

33 / 36

34 / 36Comparison of optimal exercise prices

Conclusion
Pricing options with transaction costs (market incomplete) is
more diﬃcult than pricing without transaction costs (market
complete), which is especially true for American options dueto the holder’s early exercise right.
Utility indiﬀerence pricing is better in terms of optimization,
taking into consideration of investors’ risk preferences. But it
involves lengthy computation, suitable for simple vanilla
options whose prices depend only on stock price and time.
Hedging strategy is easy to implement, suitable for options
whose interest rate or stock volatility is also a randomvariable, for example, as in our paper
Lu, Zhu & Yan (2021). Nonlinear PDE model for European
options with transaction costs under Heston stochastic volatility.
Communications in Nonlinear Science and Numerical Simulation.

Figure 1: μ = 0.1, r = 0.02, σ = 0.2, λ = 0.1, T = 1 and κ = 1%.

The holder would exercise earlier with transaction costs.
35 / 36

36 / 3682

Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Blackwell game and its applications in online
prediction tasks
Kohei Hatano

Kyushu University / RIKEN AIP

We review the Blackwell game, which is a classical game and a multi-objective
extension of the Von Neumann’s min-max game, online convex optimization(OCO),
the standard framework of online prediction in the machine learning literature, anddiscuss their relationship. Then we will show some examples of online prediction tasks
such as online load balancing, which seemingly do not ﬁt to OCO, can be reduced toBlackwell games and resulting algorithms.

83Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

Mutuality between AI and Optimization
Nguyen Dinh Hoa

International Institute for Carbon-Neutral Energy Research & Institute ofMathematics for Industry Kyushu University, JapanAI and optimization are often considered from diﬀerent perspectives by diﬀerent
research communities. However, it is undeniable that they are closely related to eachother, where optimization is a core part of many machine learning algorithms, while
AI can be employed to support optimization schemes. This talk presents examples ofsuch mutuality with illustrations in energy systems. Moreover, directions for future
research are also introduced.

84

1

2

Outline Introduction

Mutuality between AI and Optimization

 Optimization for AI
 AI for Optimization

Nguyen Dinh Hoa

 Conclusions

International Institute for Carbon-Neutral Energy ResearchInstitute of Mathematics for Industry
Kyushu University
Emails: hoa.nd@i2cner.kyushu-u.ac.jp, hoa@imi.kyushu-u.ac.jp
FMfI 2021
Hanoi, December 13–16, 2021

3

Self Introduction

4

My Research Education:
 2010–2013: PhD, The University of Tokyo, Tokyo, Japan.
 2007–2009: MEng, Chulalongkorn University, Bangkok, Thailand.
 2002–2007: BSc (Talented Program), Hanoi University of Technology, Hanoi, Vietnam.

 Career:

Optimization

 12/2021–now: Associate Professor, I2CNER and IMI, Kyushu University, Fukuoka, Japan.

Power &
EnergyScience 11/2017–3/2018: Visiting Scholar, University of Illinois at Urbana-Champaign, USA.
 12/2016–12/2021: Assistant Professor, I2CNER and IMI, Kyushu University, Fukuoka, Japan.

Clean,
resilient,
autonomous,
comfortablesystems

 1/2015–12/2016: Postdoctoral Researcher, Toyota Technological Institute, Nagoya, Japan.
 10/2013–1/2015, 10/2009–9/2010: Lecturer, Hanoi University of Technology, Hanoi, Vietnam.
Control
TheoryAI

ICT
Others(materials,

photonics,
etc.)

Outline5 Introduction

Optimization for AI
data Optimization for AI

model structure

 AI for Optimization

Optimization

model parameters Many AI algorithms aim to learn models or ‘relations’ (dynamics, behaviors,etc.), given a finite set of data ‫ݔ‬௜ , ‫ݕ‬௜ ௜ୀଵ,…,௡ .

 Conclusions

 Central to those algorithms is to find optimal model parameters via solving௡1min ෍ ݈(‫ݕ‬௜ , ݂ఏ ‫ݔ‬௜ )ఏ ݊௜ୀଵ݂ఏ ȉ : model or relation to be foundߠ: vector of parameters݈ ȉ : loss function

85

6

7

Optimization for AI‫ݔ‬௜ , ‫ݕ‬௜௜ୀଵ,…,௡݂ఏ ȉ௡1min ෍ ݈(‫ݕ‬௜ , ݂ఏ ‫ݔ‬௜ )ఏ ݊

8Example 1: Electric Demand Prediction௜ୀଵNode ݆

Data processing
Problem solving

Centralized AI

Node ݅

Node ݆

problem solving problem solving

Aggregation

 Different norms can be used
 Distinct solving methods
can be employed

Our proposed RBF

temperature

Node ݅

Optimization Problem

humidity

 How to solve this programming and problem nature lead to different types of AI
algorithms: centralized, decentralized, or fully decentralized (distributed).
Node ݅

Data Training

Historicaldataߠtime samples
fixedelectricity
consumption

Node ݆

to be found‫ ݕ‬min ‫݇ ݕ‬‫ݐ‬problem solving problem solving
decoupled
inputsNode ݇

Node ݊

Fully decentralized AI

ˠ Federated learning:
aggregation of model
parameters instead of data

ˠ Multi-agent based‫ ݓ‬ଶ ,ܾ ଶ
ͤ RBF: radial basis function

Traditional RBF

problem solving problem solving

Decentralized AI

Parametersെ‫݇ ݐ‬

coupledinputsࡸ૛ – optimizationேmin ‫ܬ‬ଶ = ෍ ‫ ݇ ݕ‬െ ‫݇ ݐ‬௞ୀଵଶ

A. T. Nguyen, D. H. Nguyen, "A Machine Learning-based Approach for The Prediction of Electricity Consumption“,2019 Asian Control Conference (ASCC 2019), Fukuoka, Japan.

9

10

Example 1: Electric Demand Prediction

Example 1: Electric Demand Prediction
Model is trained using electric demand in Tokyo during 25/9 – 25/10/2018

Output Predictionmin ‫ ݇ ݕ‬െ ‫݇ ݐ‬

Forecast data
temperature

Predicted
electricity
consumption

humiditytime samples
fixed

found

ˠADMM: alternating direction method of multipliers

D. H. Nguyen, “L_1 Optimization for Sparse Structure Machine Learning Based Electricity Demand Prediction”,
in “Optimization in Large Scale Problems: Industry 4.0 and Society 5.0 Applications”, Springer Optimization and
Its Applications, Springer Nature Switzerland AG, 2019.

A. T. Nguyen, D. H. Nguyen, "A Machine Learning-based Approach for The Prediction of Electricity Consumption“,2019 Asian Control Conference (ASCC 2019), Fukuoka, Japan.

11

Example 2: Occupancy Analysis

Example 1: Electric Demand Prediction
Electricity demand prediction for Tokyo on October 26, 2018

A dataset of electricity
consumption andconsumer occupancy

Prediction at different times
and using different numbersof neurons

Featurere vvectors
construction
SVM model
odel training
Occupancy
cy predictionpPrediction interval using
standard deviation method

Useful for demand response and energy management strategies

D. H. Nguyen, “L_1 Optimization for Sparse Structure Machine Learning Based Electricity Demand Prediction”,
in “Optimization in Large Scale Problems: Industry 4.0 and Society 5.0 Applications”, Springer Optimization and
Its Applications, Springer Nature Switzerland AG, 2019.

D. H. Nguyen, "Residential Energy Consumer Occupancy Prediction based on Support Vector Machine",
Sustainability, vol. 13(15), 8321, 2021.

86

12

Example 2: Occupancy Analysis

13

14

Example 2: Occupancy Analysis

Essence of SVM: kernelfeature map

New data are labeled by

D. H. Nguyen, "Residential Energy Consumer Occupancy Prediction based on Support Vector Machine",
Sustainability, vol. 13(15), 8321, 2021.

D. H. Nguyen, "Residential Energy Consumer Occupancy Prediction based on Support Vector Machine",
Sustainability, vol. 13(15), 8321, 2021.

15

Outline16

AI for Optimization
 A lot of optimization problems whose optimal spaces cannot be foundanalytically or in polynomial time (NP-hard), e.g., non-convex or
combinatorial programming.

 Introduction
 Optimization for AI
 AI for Optimization

 AI can be a useful aid to solution finding: Conclusions

 Simulated annealing (SA) Gaussian processes (GP) Nature and bio-inspired algorithms (GA, PSO, ACO, etc.)
…

17

18

Answer 1: Heuristic Variation of Parameters
6

Energy trading between prosumers (producer + consumer)

6

Maximum
capacity for
WUDGLQJ§N:

Load demands [kW]5

Objective function for a trading period
Bilateral trading constraint
Trading capacity constraint

Power [W]200

Question: How to ensure successful trading of all prosumers withpreferred energy prices and amounts?

0
-200
-400
-600
-800

ˠ Answers are based on optimal solutions derived from KKT conditions

-1000

0

2

4

6

8

10

12

14

Iterations

D. H. Nguyen, "Optimal Solution Analysis and Decentralized Mechanisms for Peer-to-Peer Energy Markets",
IEEE Transactions on Power Systems, vol. 36(2), pp. 1470-1481, 2021.

16

18

20

22

24

26

After learning
Maximum traded
SRZHU§
maximum
capacity

3

2

2

1

1

5

10

15

1500
1000
500
0
-500
-1000
-1500
-2000
-2500

0

2

4

6

8

10

12

14

16

Iterations

D. H. Nguyen, "Optimal Solution Analysis and Decentralized Mechanisms for Peer-to-Peer Energy Markets",
IEEE Transactions on Power Systems, vol. 36(2), pp. 1470-1481, 2021.

87

0

20

Time [h]Power [W]

400

4

2000

Before learning
Maximum traded
power is << 2 kW
available

600

Time-varying, bipartite graph

5

3

0

Diagram of the IEEE European Low Voltage Test Feeder
800

4

18

20

22

24

26

Solar power [kW]Example 3: Objective Function Parameter Learning

19

OutlineAnswer 2: Analytical Cooperative Learning

20

 Introduction
 Optimization for AI
Consensus of preferred priceranges from prosumers

 AI for Optimization
 Conclusions

Preferred range of power trading by prosumer ݅

D. H. Nguyen, "A Cooperative Learning Approach for Decentralized Peer-to-Peer Energy Trading Markets And
Its Structural Robustness Against Cyberattacks", IEEE Access, vol. 9, pp. 148862-148872, 2021.

Conclusions

21

22

 Optimization is a core of many AI algorithms Æ finding better solving
schemes for optimization problems will advance AI algorithms

Thank you for listening!

 AI can be leveraged to help solve difficult optimization problems
 AI and optimization, together with data science, are essential parts in

Q&A

a lot of current and future systems and applications (energy, materials,
biology, medicine, etc.) Æ blend to make the best use of themContact: hoa.nd@i2cner.kyushu-u.ac.jp88Forum “Math-for-Industry” 2021
December 13-16, 2021, Vietnam Institute for Advanced Study in Mathematics

What can we ﬁnd from Big Data with random
Noise?
Jin Cheng

School of Mathematical Sciences, Fudan University & Shanghai Key Laboratory ofContemporary Applied Mathematics, China

The rapid development of science and technology has produced a large amount ofdata with random noise. How to extract useful information from big data eﬀectively is
one of the fundamental problems in artiﬁcial intelligence, machine learning and otherﬁelds. From the perspective view of mathematics, there are some essential diﬃculties
to be overcome. We consider the following two kinds of problems, Problem 1: How
to use a large number of data with “large” random errors to construct more accuracy
functions; Problems 2: How to obtain useful information in some areas that cannot be
observed or diﬃcult to observe data.
To solve these two diﬃcult problems, we obtain that: 1. Tikhonov regularization
based theory and algorithms for big data with random noise. The “more” data canbe used to reduce the noise level of the data; 2. Theory and algorithms of how to use
the physical mechanism “diﬀerential equation”; to reconstruct the unknown functionin the place where the data may not be observed or is diﬃcult to observe. We can also
construct the indicator functions which can be used to describe the accuracy between
the approximate solution and the true solution.

89

91

Optimal control problem in linear elasticity
NGUYEN Quang Huy, School of Applied Mathematics andInformatics, Hanoi University of Science and Technology
The impact of extreme weather events on calorie intake – income
relationship: semiparametric estimates for Vietnam
TRINH Huong Thi, Department of Mathematics and Statistics,
Thuongmai University, Hanoi
Differential Geometry Formulation of Hanging Membranes
Yoshiki JIKUMARU, Institute of Mathematics for Industry, Kyushu
University, Japan
Poster Session Committee Members
Shizuo Kaji and Nguyen Ha Nam

92

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96

On the non-connectivity of moduli spaces of line arrangementsBenoît Guerville-Ballé

IMI - Kyushu Univeristy, Japan
FMfI 2021, December 13-16, 2021

Moduli space of complex line arrangements
Definition

L4

A line combinatorics C = (L, P) is the data of an ordered finite set L and a subset P of the power set of L which verify:• for all P ∈ P, #P ≥ 2,

• for all L1, L2 ∈ L, it exists a unique P ∈ P such that L1 ∈ P and L2 ∈ P.

P4Line arrangements in CP2 are classically studied as simpler case of singular plane algebraic curves.
They are defined2
as a finite collection
 of distinct lines in CP . The incidence structure of a line arrangement A = {L1, . . . , Ln}, given by
P = {P ⊂ A |L∈P L = ∅, ∀P  Q ⊂ A,
L∈Q L = ∅}, forms naturally a line combinatorics. This line combinatorics, isnamed the combinatorics of A and it is denoted by C(A). It can be described as the incidence graph ΓA of the lines of A, with vertices
composed by the lines and singular points, joined by an edge if P ∈ L.

P3

P1

Definition

The realization space R(A) of a line arrangement A (or of its combinatorics C(A)) is the set of all arrangements whichhave isomorphic combinatorics, or equivalently whose incidence graphs are isomorphic to ΓA . The moduli space M(A)of A is the quotient of R(A) by the action of PGL3(C):M(A) = { B | C(B ) ∼ C(A) } / PGL3(C).

Notation: The connected component of M(A) which contains the arrangement A is denoted by M(A)A .

Main Question:

L3
L2

P2

P1

P2

P3

P4

L1

L2

L3

L4

A
L1

ΓA

How to construct arrangements or combinatorics which have a non-connected moduli space?

Small examples: Up to 7 lines, the moduli space of a line arrangement is path-connected. The first example of a line arrangement with a non-connected moduli space is the MacLane arrangement [1].
It is fomed by 8 lines each contains three triple points. For 9 line arrangements, see [2], there are three types of arrangements with a non-connected moduli space: those which contains a MacLane
arrangement, the Falk-Sturmfels arrangements and the Nazir-Yoshinaga arrangements.

The splitting-polygon structure

Applications

Let A = {L1, . . . , L5} be the arrangement of 5 lines defined by the equations:Definition• Let C = (L, P) be a line combinatorics and let 3 ≤ r ≤ #A. A plinth Ψ in C
is form by two tuples: the support S = (S1, . . . , Sr ) ⊂ L and the pivot-points/ Pi and Si+1 ∈/ Pi.
(P1, . . . , Pr ) ⊂ P such that, for each Pi, we have Si ∈A line arrangement A is said to have a plinth if its combinatorics does.

• Q1λ is a generic point of S1 which is determined by aparameter λ ∈ C.

• E1λ is the line which passes through Q1λ and P1.

• Q2λ is the intersection point of S2 and E1λ.

• Eiλ is the lines which passes through Qiλ and Pi.

E2λ

• P2

λ is the intersection points of E λ and S• Qi+1i+1.
i• R1λ is the intersection point of Erλ and S1

We denote by Aλ the arrangement A ∪ {E1λ, . . . , Erλ}.
Definition

E1λ

R1λ
◦

◦

Q3λ
◦

Q1λ

E1λ : λx + y − λz = 0,

S3

L4

Q3λ

E3λ

3. each line Eiλ contains #A + r − #Pi − 2 singular points in Aλ.

Notation: The combinatorics of the arrangement Aλ is denoted by C(A)Ψ when E λ form a splittingpolygons on Ψ.

L5
◦

•

P2,5

◦
Q2λ

L2

•
•

◦

E2λ

P3,5
P3,4

L3

Q1λ

◦

E1λ

In addition of the MacLane arrangements, the splitting-polygons structure allows to reconstruct
the Falk-Sturmfels arrangements and the Nazir-Yoshinaga arrangements.

, [3])Let (S1, . . . , Sr ) and (P1, . . . , Pr ) be a projectively rigid plinth Ψ of an arrangement Aof M(A)A . If E λ1 and E λ2 are two distinct splitting-polygons on Ψ, then M(C(A)Ψ)
splits over M(A)A .
More precisely, the line arrangements Aλ1 and Aλ2 are in different connectedcomponents of M(C(A)Ψ).

[1] S. MacLane. Some interpretations of abstract linear dependence in terms of projective
geometry. Am. J. Math., 58:236-240, 1936.

R1λ

L1

2. for all i, j ∈ {1, . . . , n}, we have Eiλ ∈/ A, and Eiλ = Ejλ,References√

S1

The tuple E λ = (E1λ, . . . , Erλ) forms a splitting-polygon on the plinth Ψ if:
1. Q1λ = R1λ,Theorem (E3λ : (λ − 1)x + λy − λz = 0.

Let λ1 = 1+i2 3 and λ2 = 1−i2 3 be the two roots of ∆Ψ. The arrangements Aλ1 and Aλ2 share thesame combinatorics C(A)Ψ which is the MacLane combinatorics. By the previous Theorem, theylie in different connected components of the moduli space M(C(AΨ)).

• P1

S2

E2λ : (1 − λ)x − y + λz = 0,

The points Q1λ and R1λ are equal if and only if the lines L1, E1λ and E3λ are collinear. Algebraicallyspeaking, we have a splitting-triangle if and only if the following determinant vanishes.
0 λ (λ − 1)
λ  = λ2 − λ + 1.
∆Ψ(λ) = 0 1
1 −λ −λ 
√

Q2λ ◦

L3 : x − z = 0

Let Q1λ = [1 : λ : 0] be a generic point of L1. Following the previous construction, we deduce thatthe equations of E1λ, E2λ and E3λ are:

E3λ

• P3

L2 : x = 0
L5 : y − z = 0

We consider on A the projectively rigid plinth Ψ defined by (L1, L2, L4) for the support and({L3, L4}, {L3, L5}, {L2, L5}) for the pivot-points.

• A plinth Ψ is said to be projectively rigid in M(A)A (resp. in M(A)), if for all
arrangement A in M(A)A (resp. in M(A)), it exists a projective transformationτ ∈ PGL3(C) such that τ(Si) = Si and τ(Pi) = Pi, for all i ∈ {1, . . . , r}.
Let A be a line arrangement such that, for a fixed integer
3 ≤ r ≤ #A, the lines (S1, . . . , Sr ) ∈ A and the singularpoints (P1, . . . , Pr ) ∈ Sing(A) form a plinth Ψ.

L1 : z = 0
L4 : y = 0

The line combinatorics of A is given by{L1, L2, L3}, {L1, L4, L5}, {L2, L4}, {L2, L5}, {L3, L4}, {L3, L5} .

Theorem (, [3])The tuples (L1, L2, L4) and {{L3, L4, L6}, {L1, L6, L8}, {L2, L5, L8}} form a projectively
rigid plinth Φ of the MacLane combinatorics.
The moduli space of the combinatorics C(A)Ψ,Φ is not path-connected and it admits
4 connected components.

[2] F. Ye. Classification of moduli spaces of arrangements of nine projective lines. Pac. J. Math.,
265(1):243-256, 2013.
[3] B. Guerville-Ballé On the non-connectivity of moduli spaces of arrangements: the splittingpolygons structure. arXiv:2111.00399.

97:
lle.com
Contact & Infot-guervilleba
 www.benoi@math.cnrs.fr
erville-balle benoit.gu

98

j=1

r



(dj + 1) − 2 (p − 1)/2,(1)

E2

Therefore, equal characteristic deformations between Artin-Schreier curves give full
information about the geometry of the moduli space AS g ! For example, suppose p = 5
and g = 14. One can explicitly construct some deformations between the curves indifferent stata of AS 14 (see [1, Theorem 3.7]) to obtain the following diagram.
There is an edge between two strata if one is contained in the closure of the other.
Hence, it follows from the diagram that the moduli space AS 14 is connected. Moreover, the irreducible components of AS 14 are the closures of the following strata:
Γ{5,4},Γ{3,3,3}, Γ{4,3,2}, Γ{5,2,2}, and Γ{3,2,2,2}. Furthermore, the intersection ofΓ{3,3,3} and Γ{3,2,2,2} is Γ{9}, the intersection of Γ{4,3,2} and Γ{5,2,2} is Γ{7,2}, . . .

E1

−
→
→
−
→ is contained in theGiven two partitions E 1 and E 2 of d + 2. The stratum Γ−E1→ if and only if there exists a deformation over k[[t]] from a point inclosure of Γ−E2→ to one in Γ−→ .
Γ−is the genus of Y . Equation (1)shows that all the Artin-Schreier k-curves with thesame genus gY have the same rj=1(dj + 1). We denote by AS g the moduli spaceof Artin-Schreier curves of genus g.
The moduli space AS g can be partitioned by locally closed strata corresponding to→−
the partitions of d + 2 [5]. In particular, the partition E = {e1, . . . , er } of d + 2→, which is the collection of all the curves in AS g
is associated with the stratum Γ−Ewhich are branched at s number of points {P1, P2, . . . , Ps}, with ramification jumpei − 1 at Pi. The following well-known fact will relate the geometry of AS g withequicharacteristic deformations of Artin-Schreier covers.
Equicharacteristic deformations and the geometry of AS g

gY =Moreover, f (x) is unique up to adding an element of the form bp − b, where b ∈ k(x).
Hence, we might assume that f (x) is “reduced”. Suppose {P1, . . . , Pr } is the set ofpoles of f (x) in P1k , and dj ̸≡ 0 (mod p) is the order of the pole of f (x) at Pj . Then
dj is also the ramification jump at Pj , andy p − y = f (x) ∈ k(x).

An Artin-Schreier (AS) curve is a Z/p-Galois cover of the projective line over a field
k in characteristic p. Any such cover, say ϕ : Y −→ P1k , is defined by the equation

Moduli space of Artin-Schreier curves

1dx
11, x7(x−1)
5




1
0

5[t5]


0

0

3[t10]

4[0]2. When p = 5, AS g is connected for any g ≥ 14 and g = 0, 2. It is disconnected otherwise.

1. When p = 2, 3, the moduli space AS g is connected for any g.

The moduli space AS g is connected when g is sufficiently large. In particular,

The above result implies that the equicharacteristic deformations problem can be translated tothe exact Hurwitz tree problem. By studying exact differential forms (in characteristic p > 0),
we obtain the following result (c.f. [1, Theorem 1.1] [2, Theorem 1.2]).
The connectedness of the moduli space

We say the above tree is of type {12} → {4, 3, 5}. At each vertex v of the tree, the firstcomponent of the dual is the depth Swan conductor at the boundary of the disc corresponds tov. If the depth is positive, the second component is the differential conductor at the boundary ofthe disc. When the depth is 0, the second component is a polynomial in k[x−1], which represents
the degeneration type of χ. The rational number below each edge e is the “thickness” of thecorresponding annulus. The integer at a leaf is the conductor (the ramification-jump-plus-one)of the corresponding branch point.
Exact Hurwitz tree and equicharacteristic deformations→−→−
Suppose E 1 = {e} ≺ E 2 = {e1, . . . , er }. Then there is a deformation from each curve of→ to a curve of Γ−→ if and only if there exists a differential Hurwitz tree of type {e} →Γ−E1E2{e1, . . . , er }.

0, t111dx
17, x4(x−1)
3Then the Hurwitz tree associated with χ has the following form.

=: f (x, t).
y5 − y = 5
x (x − t10)2(x − t5)5

x + 2t10

Thanks to a local-global principle, it suffices to study the good reduction of the formal disc overR := k[[t]]’s covers! Suppose we are given an order-p-cover of the disc. One can derive from that
action a minimal semi-stable model for which the specializations of the fixed points are distinct
and lie in the smooth locus of the special fiber. Moreover, one can enrich the dual graph of themodel with the depth Swan conductor and the differential conductor (see [4, Definition 3.2.10])of each component of the special fiber. The description leads to a combinatorial-differential object
called a differential Hurwitz tree. For instance, suppose χ is a Z/5-action that arises from theGalois group of the extension

The moduli space of AS 14

Differential Hurwitz Tree{5, 2, 2}(π1et(C))p ∼= (π1et(Cη ))p{3, 2, 2, 2}{4, 3, 2}{5, 4} {3, 3, 3} {7, 2}{9}[3, XIII, Corollaire 2.12], where (G)p denotes the “prime-to-p” part of a group G.
However, the p-part of π1et(A1k ) is no longer trivial as there always exists an étale Z/pcover defined by the equation y p −y = x. That cover, also known as an Artin-Schreier
cover, is the simplest example of a wildly ramified Galois cover, which distinguishes
characteristic p from characteristic zero.

We know from classical topology that the group π1(A1C) is trivial. In algebraic geometry, the étale fundamental group π1et is the direct analog of π1. For instance, if X is a
(X(C)). Regarding positive
connected scheme of finite type over C, then π1et(X) ∼= π1
characteristic, Grothendieck proved that every curve C over a field k of characteristicp > 0 lifts to a curve C over W (k), hence in characteristic zero. Moreover,Wildly ramified covers of curves

Vietnam Institute for Advanced Study in Mathematics

Dang Quoc Huy

Flat families of cyclic covers (of curves)

i=1

n


(ιj,i + 1)(pi − pi−1)

(2)

Ψ

X
Spec R

Y

Y

Λ

λ

Z

Z
.

[5] Rachel Pries and Hui June Zhu. “The p-rank stratification of Artin-Schreier curves”. In: Ann.
Inst. Fourier (Grenoble) 62.2 (2012), pp. 707–726. issn: 0373-0956. doi: 10.5802/aif.2692.
url: http://dx.doi.org/10.5802/aif.2692.

[4] Shigeki Matsuda. “On the Swan conductor in positive characteristic”. In: Amer. J. Math. 119.4(1997), pp. 705–739. issn: 0002-9327. url: http : / / muse . jhu . edu / journals / american _journal_of_mathematics/v119/119.4matsuda.pdf.

[3] Alexander Grothendieck. Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à 5.
Vol. 1960/61. Séminaire de Géométrie Algébrique. Institut des Hautes Études Scientifiques, Paris,
1963, iv+143 pp. (not consecutively paged) (loose errata).

[2] Huy Dang. “Hurwitz trees and deformations of Artin-Schreier covers”. In: arXiv e-prints,
arXiv:2002.03719 (Feb. 2020), arXiv:2002.03719. arXiv: 2002.03719 [math.AG].

[1] Huy Dang. “Connectedness of the moduli space of Artin-Schreier curves of fixed genus”. In:
Journal of Algebra 547 (2020), pp. 398–429. issn: 0021-8693. doi: https : / / doi . org / 10 .
1016 / j . jalgebra . 2019 . 11 . 034. url: http : / / www . sciencedirect . com / science /article/pii/S0021869319306659.

References

In particular, the canonical morphism ASW d1,...,dm,dm+1,...dn → ASW d1,...,dm
maps closures surjectively to closures.

ψXSpec kSuppose ϕ : Z −→ X is a cyclic G-Galois cover of curves over k, and ψ : Y −→X
is its H-Galois sub-cover (where H is a quotient of G). Suppose, moreover, that
→ XR is a deformation of ψ over a complete discrete valuation R of
Ψ : YR −→ XR of ϕ over R thatcharacteristic p. Then there exists a deformation Φ : ZR −contains Ψ as a sub-cover. That is, one can always fill in the following commutative
diagram of cyclic covers,

Denote by ASW d1,...,dn the moduli space of Z/pn-covers whose j-th sub-covers
jhave i=1 di(pi − pi−1) as the degree of the different. The moduli space can bepartitioned into strata which are parameterized by r × n matrices like the one above!
In addition, by generalizing the Hurwitz tree technique, we prove the following.
Cyclic covers deform in towers

matrix as below to record the ramification data of the cover.
ι1,1 + 1 ι1,2 + 1 . . . ι1,n + 1ι2,1 + 1 ι2,2 + 1 . . . ιn,n + 1. . . . . . . . . . . . . . . . . . . . . . . . . .ιr,1 + 1 ιr,2 + 1 . . . ιr,n + 1

nwhere ιj,i is the i-th upper jump at Pj . It follows
r from (2) that Z/p -covers of fixed(ιj,i + 1). We hence use an r × n
genus on each sub-cover have the same di := j=1

deg(DPj ) =where ℘(y) := F (x) − x is the Artin-Schreier-Witt isogeny. Suppose {P1, . . . , Pr }is the set of poles of the fi’s. Then the degree of the different at Pj is

Similarly, a Z/pn-cover ϕn : Yn −→ P1k can be represented by�℘(y1, . . . , yn) = f1(x), . . . , fn(x) ∈ Wn(k(x))

Toward the moduli space of cyclic covers

99

100

An algorithm for counting the number of solutions for brick Wang tiling
Yang Hang, Graduate School of Mathematics, Kyushu university, yang.hang.685@s.kyushu-u.ac.jp
Abstract
Wang tiling problem is an important problem in Graph Theory and Combinatorics. Brick Wang tiling [3] is a problem in the range of Wang tiling with permissive restrictions, andin many situations there exist multiple solutions. We develop an algorithm to get the specific number of valid solutions of brick Wang tiling. We discuss the validity conditions of
the method. We use Mathematica to implement our algorithm and check the data.
1 Wang Tile

5 Our results

A Wang tile is a square tile with each edge colored. An edge in a Wang tile is also
called a leg. Wang tiles are pasted side by side to form atile graph. In a tiling problem, adjacent tiles must havethe same color on the shared leg.
correct Wang tiling
The tiling problem was first introduced by Hao Wang in
1961 [5]. He discussed the question whether an infinite
plane can or cannot be covered by a given set of Wang
incorrect Wang tilingtiles as described above.
Lately, Wang tiling problem is studied widely and aperiodic Wang tile sets that tile
the plane were discovered [4,6]. Some special types of Wang tile, such as brick
Wang tile, were illustrated in [3].

Lemma 1 Assume that 1,2,3,4 are the four legs of one tile. We can get the vector
𝑁𝑁4 corresponding to the leg 4 by the formula 𝑁𝑁4 = (𝑁𝑁1 𝑁𝑁3 ) (¬𝑁𝑁2 ) + (¬𝑁𝑁1 𝑁𝑁3 ) 𝑁𝑁2 if
we know the vectors 𝑁𝑁1 , 𝑁𝑁2, 𝑁𝑁3 corresponding to the legs 1,2,3 respectively. (Weassume leg 2 is opposite to leg 4 . In formula 𝑁𝑁1 𝑁𝑁3 is the inner product.)Lemma 2 A circle is equivalent to a path described by matrix 𝑀𝑀𝑂𝑂𝑂𝑂𝑂𝑂
𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑟𝑟𝑟 𝑀𝑀2 𝐹𝐹𝑟𝑟𝑟 ) 𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑟𝑟𝑟 𝑀𝑀2 𝐹𝐹𝑏𝑏𝑏 ) 𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑟𝑟𝑟 𝑀𝑀2 𝐹𝐹𝑔𝑔𝑔 )= 𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑏𝑏𝑏 𝑀𝑀2 𝐹𝐹𝑟𝑟𝑟 ) 𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑏𝑏𝑏 𝑀𝑀2 𝐹𝐹𝑏𝑏𝑏 ) 𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑏𝑏𝑏 𝑀𝑀2 𝐹𝐹𝑔𝑔𝑔 )𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑔𝑔𝑔 𝑀𝑀2 𝐹𝐹𝑟𝑟𝑟 ) 𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑔𝑔𝑔 𝑀𝑀2 𝐹𝐹𝑏𝑏𝑏 ) 𝑡𝑡𝑟𝑟(𝑀𝑀1 𝐹𝐹𝑔𝑔𝑔 𝑀𝑀2 𝐹𝐹𝑔𝑔𝑔 )2 Application of Wang Tile
Wang tiling has applications in computer graphics [7] . Generating wall patterns is
one of its applications [2]. Brick Wang tiles are a special set of Wang tiles
introduced by Derouet-Jourdan in 2016 to model wall patterns. In their algorithm,
a set of proper fractions is regarded as color set and the border of bricks isdetermined by proper fractions [2]. In [3] the result is generalized by providing alinear algorithm to decide and solve the tiling problem for arbitrary planar regionswith holes.

computer graphics

wall patterns generated by Wang tilewhere 𝑡𝑡𝑡𝑡 is the trace of matrix.

reflection

3 Brick Wang tiling Problem

A Brick Wang tile is a Wang tile with one pair of opposite legs of the same color
and the other pair of different colors.
A Brick Wang tiling Problem consists of:
1 a tile graph 𝑇𝑇𝐺𝐺
non-brick Wang tilebrick Wang tile
2 a color set 𝐶𝐶
3 valid prototile set 𝑊𝑊 = {𝑤𝑤, 𝑒𝑒, 𝑠𝑠, 𝑛𝑛|𝑤𝑤, 𝑒𝑒, 𝑠𝑠, 𝑛𝑛 ∈ 𝐶𝐶, (𝑤𝑤 = 𝑒𝑒⋀𝑠𝑠 ≠ 𝑛𝑛)⋁(𝑤𝑤 ≠ 𝑒𝑒⋀𝑠𝑠 = 𝑛𝑛)}
In tiling problem we assign colors to legs to ensure that prototiles formed by 4 legsof each tile are in the prototile set, that also means to ensure each tile is a brick
Wang tile.
Our target is to enumerate the specific number of valid solutions in a certain Brick
Wang Tiling problem.
We also have the following problem setting:
1. Color set={ red , blue , green }2. Boundary legs [1] are colored in advance.
3. Tile graph is connected.

Proposition 3 The number of valid solutions is 𝑁𝑁𝑢𝑢 𝑁𝑁𝑣𝑣 when 𝑁𝑁𝑢𝑢 and 𝑁𝑁𝑣𝑣 are different
vectors of one certain (shared) leg got from different sides.
Proposition 4 The number of valid solutions is t𝑟𝑟(𝑀𝑀) when there is only one circle
described by matrix 𝑀𝑀.
Vectors are got linearly

Algorithm 5a When a tile
Graph is a tree, we choose a
leg 𝑢𝑢 𝑣𝑣 (subjectively) and let
this leg divide the tree into
two smaller trees. We gettheir vectors 𝑁𝑁𝑢𝑢 𝑁𝑁𝑣𝑣 by
Applying Lemma 1 repeatedly. Then we
utilize Proposition 3 to get the
number of valid solutions.
𝑁𝑁𝑢𝑢 and 𝑁𝑁𝑣𝑣 are get by this progress linearly
Algorithm 5b If there exist circles, then we apply Lemma 2 repeatedly and reduce
the number of circles until there is only one circle. Then by Proposition 4 we get the
number of valid solutions.

This tile graph has 6 valid tilings

4 Model and tools
The enumeration vector ( 𝑁𝑁𝑙𝑙 = (𝑟𝑟𝑙𝑙 , 𝑏𝑏𝑙𝑙 , 𝑔𝑔𝑙𝑙 ) )is introduced to describe the possibilities
that one leg can be tiled by red, blue
or green respectively [1].
We define operations on vectors. A
special operation is ¬ (¬𝑁𝑁𝑙𝑙 = (𝑏𝑏𝑙𝑙 + 𝑔𝑔𝑙𝑙 ,𝑟𝑟𝑙𝑙 + 𝑔𝑔𝑙𝑙 , 𝑟𝑟𝑙𝑙 + 𝑏𝑏𝑙𝑙 ) when 𝑁𝑁𝑙𝑙 = (𝑟𝑟𝑙𝑙 , 𝑏𝑏𝑙𝑙 , 𝑔𝑔𝑙𝑙 ) ) isused to describe the situation where one leg must have a different color from its
opposite leg.
Matrices composed by enumeration vectors are used to denote the paths
composed of a series of uncolored legs. Two objects (paths, circles, etc.) areequivalent as long as they have the same matrix. Hence we reduce the number ofcircles in a tile graph by replacing one circle with its equivalent path.

6 Conclusion
1. We formalized a brick Wang tiling problem and introduced an algorithm forcounting the number of solutions.
2. We introduced an enumeration vector and its operations for counting solutions
formally. We showed the correctness of our algorithms using graph theory.
3. We implemented our algorithm using Mathematica.

References[1] H. Yang, Brick Wang tiling problem and its enumeration (unpublished)[2] A. Derouet-Jourdan, M. Salvati, T. Jonchier, Generating Stochastic Wall Patterns On-the-fly with Wang Tiles,
EUROGRAPHICS 2019 Volume 38(2) 255-265 (2019), DOI: 10.1111/cgf.13635[3] A. Derouet-Jourdan, S. Kaji, Y. Mizoguchi, A linear algorithm for Brick Wang tiling, Japan Journal of Industrial and Applied
Mathematics (2019)36:749C761[4] E. Jeandel, M. Rao, An aperiodic set of 11 Wang tiles (2015), arXiv:1506.06492[5] H. Wang, Proving theorems by pattern recognition-II. Bell Syst Tech J 40(1), 1-41 (1961)[6] K. Culik II, An aperiodic set of 13 Wang tiles. Discrete Math, 160(1C3), 245C251 (1996)[7] M.F Cohen., J. Shade., S. Hiller, O. Deussen., Wang tiles for image and texture generation. ACM Trans. Graph.22(3),287C294 (2003)[8] T. Matsushima, Y. Mizoguchi, A. Derouet-Jourdan, Verification of a brick Wang tiling algorithm, EPiC Series in Computing
Volume 39, 107–116 (2016).

7 Future works
1. The correctness of our algorithm is showed only for a limited class of graphs. We
need to extend the target class of graphs and prove its correctness.
2. We will prepare a formal proof of our algorithm using a formal theorem proversystem such as Coq [8] .
3. We are considering an application area for a formal verified class of Wang tile
patterns using a brick Wang tiling.

101The ground state of the semi-relativistic Pauli-Fierz
Hamiltonian
Takeru Hidakaȋ   ǡ Ȍǣ Ǧ̷ǤǦǤ Ǥ

Abstract

The existence of the ground state for the massless semi-relativistic Pauli-Fierz model in quantum electrodynamics is considered.
(Joint work with F. Hiroshima and I. Sasaki)

This work was supported by, JSPS KAKENHI Grant Number JP16H03942, JP20H01808, JP16K17612 and JSPS KAKENHI Grant Number JP20K03628.

Background

Nonrelativistic Pauli-Fierz model

We are interested in the spectrum of Hamiltonians of quantum field theory. The
existence of the ground state implies the stability of quantum systems.

The Pauli-Fierz Hamiltonian describes low energy electrons minimallycoupled to a quantized radiation field . The Hamiltonian is given by

Definition of ground states
A ground state Φ is an eigenvector of the Hamiltonian associated with the minimum of
the spectrum E. E is called the ground state energy. By definition,(ψ,Hψ)≧E(ψ, ψ) (∀ψ), HΦ=EΦ, Φ≠ͲǤHere ሺψ,Φሻstands for the inner product ψ and Φ.

Particle-field interaction models

and the existence of the ground state of H for all m≧0 is shown in [GLL 01].

Nelson model
The Nelson model describes a system of quantum mechanical particles linearly
coupled to a scalar Bose field. The Hamiltonian of the Nelson model is of the form

No ground states exist if an infrared regular condition is failed and m=0 [Hirok 06].
electronphoton(boson)Semi-relativistic Pauli-Fierz model

nucleusLet and ǡ be a particle Hamiltonian and a free field Hamiltonian respectively,
where  is an artificial mass of boson. The decoupled Hamiltonian, Ͳǡǡis given by
The spectrum ofͲǡ is well known.

Spectrum ofMassive (m>0) caseground state

Massless (m=0) caseͲǡembedded eigenvalues

Spectrum of an Hamiltonian for a particle-field interaction modelm>0 case

m=0 case

Strategy2.
3.

References[GLL 01] M. Griesemer, E. H. Lieb and M. Loss, Ground states in non-relativistic
quantum electrodynamics, Invent. Math. 145 (2001) 557--595.
[HH 15] T. Hidaka and F. Hiroshima, Self-adjointness of the semi-relativistic PauliFierz Hamiltonian, Rev. Math. Phys. 27 (2015) 1550015 18pp.
[HH 16] T. Hidaka and F. Hiroshima, Spectrum of the semi-relativistic Pauli-Fierz
model I, J. Math. Anal. Appl. 437 (2016) 330--349.
[HHS 21] T. Hidaka, F. Hiroshima and I. Sasaki, Spectrum of the semi-relativistic
Pauli-Fierz model II, to appear in J. Spectr. Theory.
[Hirok 06] M. Hirokawa, Infrared catastrophe for Nelson's model—non-existenceof ground state and soft-boson divergence, Publ. Res. Inst. Math. Sci. 42, (2006)
897–922.
[Hir 14] F. Hiroshima, Functional integral approach to semi-relativistic Pauli-Fierz
models, Adv. Math. 259 (2014) 784--840.
[KMS 11] M. Könenberg, O. Matte and E. Stockmeyer, Existence of ground statesof hydrogen-like atoms in relativistic QED I: The semi-relativistic Pauli-Fierz
operator, Rev. Math. Phys. 23 (2011) 375--407.

Perturbation for embedded eigenvalues is not trivial.

1.

The existence of the ground state of the SRPF Hamiltonian is initially proven by
Könenberg, Matte and Stockmeyer for M>0 and m≧0 [KMS 11]. We show that H isthe self-adjoint operator for all M≧0 and m≧0 in [HH 15]. When m>0, theexistence of the ground state Φ for some confining potentials is proven in [HH16],
and it is also shown that Φ  ሾ  ͳͶሿǤThe SRPF
Hamiltonian has two singularities: ൌͲand ൌͲǤ We consider the massless SRPF
Hamiltonian:The existence of the ground state for the massless SRPF Hamiltonian is proven in[HH21]. The uniqueness of the ground state is shown by [Hir 14].

Add interaction

ground state?

The semi-relativistic Pauli-Fierz (SRPF) Hamiltonian is defined by

Define a Hamiltonian as a self-adjoint operator acting in a given Hilbert
space.
Prove the existence of ground states when m>0.
m→0+Prove the existence of ground states when m=0.

102

FEM Study on the Elastic
Deformation Process of Materials in
Industry
Phuong Cuc Hoang1, Thi Thanh Mai Ta2HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
School of Applied Mathematics and Informatics
cuc.hp185332@sis.hust.edu.vn, mai.tathithanh@hust.edu.vn

1. Introduction
Elastic deformation is a temporary deformation when amaterial is subjected to force within its elastic limits. Thismeans that the shape of the material reverses itself after
the removal of force or load. The study of elastic deformation is a matter of concern in the industry to make objective
assessments of the structure of the material.
Weak Formulation in Hilbert Space: Let V be a Hilbert
space, α be a continuous bilinear form on V × V ,i.e., α ∈L(V × V ; R) and L be a continuous linear form on V ,,i.e.
L ∈ L(V ; R). Find u ∈ V such that:α(u, v) = L(v)∀v ∈ V.

Finite Element Method: is a powerful numerical methodthat uses computational power to calculate approximate solutions of structural mechanics problems. It is widely usedin all majors engineering industries.
The FEM approaches this problem by splitting the body into
a number of small elements that are connected together at
nodes. This process is called discretization and the collection of nodes and elements is called the mesh. Discretization is useful because the equilibrium requirement now only
needs to be satisfied over a finite number of discrete elements, instead of continuously over the entire body.
2. Setting of the problem
We have the domain: Ω ⊂ R2 and f : Ω → R2 .
Let A(u) : Ω → R2,2 be the stress tensor.
Let ε(u) : Ω → R2,2 be the strain rate tensor defined as:1ε(u) = (∇u + ∇uT ).
2

A(u) = λtr(ε(u))I + 2µε(u),where λ and µ are the so-called Lamé coefficients, and I is
the identity matrix.
We have:2• The coefficient (λ + µ) describes the compressibility of3the medium; very large values correspond to almost incompressible materials.
• Young modulus E and Poisson coefficient ν:
1 λ
3λ + 2µand ν =.
E=µλ+µ2λ + µOr:Eν
Eand λ =.
µ=2(1 + ν)(1 + ν)(1 − 2ν)1• The Poisson coefficient is such that −1 ≤ ν < , and2
owing to the assumption λ ≥ 0, we infer ν ≥ 0. An almost incompressible material corresponds to a Poisson1
coefficient very close to .
2
The equilibrium conditions under the external load f can be
expressed as:in

3. Numerical method
Variational formulation
Take the scalar product of the equilibrium equation with atest function v : Ω → R2:
− div(A(u)).vdx = f.vdx.
Ω

Ω

∂Ω

Ω

Ω

Deformation of the swing arm machine under the force applied to the rim of the part
Steel Material

ΓN

And:
A(u) = λtr(ε(u))I + 2µε(u) = λdiv(u)I + 2µε(u).
Therefore:
A(u).∇vdx = 2µε(u) : ε(v)dx + λdivudivvdx.
Ω

Aluminum MaterialΩThe weak formulation of elastic equation:
Seek u ∈ (HΓ1D (Ω))d such that:α(u, v) = L(v)Deformation of the material composite bar under the influence of gravity∀v ∈ (HΓ1D (Ω))d,with the continuous bilinear form:α(u, v) = 2µε(u) : ε(v)dx + λdivu.divvdx,( a ) Original( b ) Aluminum( c ) Steel

(a)

(b)

(c)

Ω

and the continuous linear form:
L(v) = f.vdx + gN .vds.
Ω

ΓN

A triangulation is generated on Ω using buildmesh function. This computational domain is regular and its elements have no inner common vertices. The geometric angles of thetriangle > 0, and as the edge of thetriangle move towards 0, the area oftriangles also gradually moves toward 0.
4. Assembling the stiffness matrices(a) Original.
(b) Circle is steel, outside is aluminum.
(c) Circle is aluminum, outside is steel.
6. Conclusion and perspective• When the object is acted on by the same force of equalmagnitude, it will cause other deformations depending on
the Young modulus E of the object.
• Objects with high E have less deformation than objects
with low E.

The basic formulation is:{F } = [K].{u},where:• {F } is the force vector that also includes moments.

• [K] is the stiffness matrix of the entire structure - global
stiffness matrix.
• {u} is the vector of displacements.

The global stiffness matrix is constructed by assembling individual element stiffness matrices.
According to Maxwell’s Reciprocity Theorem, the stiffness
matrix is symmetric.

Ω.

Adding boundary conditions, we get a linear elastic equation system:
−div(A(u)) = f in Ω,
u = 0 on ΓD ,

A(u)n = gN on ΓN .

ΩΩWe have u = 0 on ΓD :
A(u).∇vdx = f.vdx + gN .vds.

The meshAccording to Hooke’s law, the stress tensor is related to thestrain rate tensor by the relation:div(A(u)) + f = 0

Apply Green’s formula:− div(A(u)).vdx = A(u).∇vdx − v.A(u)nds.

5. Numerical experiments
The parameters
No. Materials E (MPa)ν
1 Aluminum 0.69 E+5 0.33
2
Steel
0.20 E+6 0.291
Deformation of the bridge under the influence of gravity
Steel Material• We can combine multiple materials in one object to improve the performance of the product.

7. Forthcoming Research
We will develop the study of the elastic deformation of objects made from a variety of materials with complex shapes
in 2D, 3D and applications in industry.
8. Acknowledgements
The authors wish to express our gratitude to the Vietnam
Institute for Advanced Study in Mathematics (VIASM) andUniversity Kyushu, Japan for giving us the precious opportunity to present our research.
References[1] Alexandre Ern and Jean-Luc Guermond, Theory andPractice of Finite Elements, volume 159 of Applied
Mathematical Sciences. Springer, New York, 2004.
[2] Roberto Font and Francisco Periago, The Finite Element Method with FreeFEM++ for beginners, 2013.
[3] FreeFEM Documentation, A system of the elasticity.

Aluminum Material

103

The complexity of the parity argument with potential
Takashi Ishizuka
Graduate School of Mathematics, Kyushu University, Japanishizuka.takashi.664@s.kyushu-u.ac.jp
Introduction
Computational Complexity for Search Problems :We are interested in the complexity class TFNP (Total Functions in NP).

Overview of Complexity Class TFNP
TFNP

Every search problem in TFNP satisfies the following
TOTALITY: For every input, a solution always exists.
CHECKABLITY: Checking the correctness of each solution is easy.

PPA

P OTENTIAL O DD

The class PPAD is a subclass of TFNP and closely related to Game
Theory [Das09, Pap94, Yan09]. E NDOF L INE is a PPAD-complete
problem.

E NDOF L INE
E NDOF P OTENTIAL L INE
PPAD
EOPL

PLS

Related Works:
Goldberg and Hollender [GH18] showed that the following variants ofE NDOF L INE have the same complexity:
 given k sources and l = k sinks, find another sink or source;
 given k sources, k sinks or k other souces.

E NDOF L INE [Pap94]
Input:
 An implicit digraph with potential G(1, m, S, P, V) = (Σn, E)
Task: Find a string x ∈ Σn satisfying one of the following: P((S(x)) = x;
 S(P(x)) = x = π.

Our Contribution:
 We show that the following variants of E NDOF P OTENTIAL L INE,
defined below, have the same complexity:

 given k sources, find another degree-1 vertex or a non-increasing arc;
 given k sources, find k distinct vertices that are at least one of a sink, othersource, and a non-increasing arc.

π

 We consider the complexity of weighted variants of O DD, called
P OTENTIAL O DD.

The blue vertex is a known source. The red vertices are solutions.
BasicsComplexity Class TFNP:
We consider a polynomial-time computable and polynomial-balanced
relation R ⊆ {0, 1}∗ × {0, 1}∗:
 We can decide whether (x, y) ∈ R in P for all (x, y); and For each (x, y) ∈ R, |y| ≤ poly(|x|).
R has totality if for each x ∈ {0, 1}∗, there is a y ∈ {0, 1}∗ s.t. (x, y) ∈ R.
Total Search problem R
Input: a string x ∈ {0, 1}∗
Task: Find a string y ∈ {0, 1}∗ such that (x, y) ∈ R.

Polynomial-time reduction:
Let R, Q be search problems.
R is reducible to Q, denoted by R ≤P Q, iff there are two polynomial-time
computable functions f and g such that for every input x of R, f (x) is an input of Q;
 for each solution y to Q w.r.t. f (x), g(x, y) is a solution to R w.r.t. x.
Complete Problem:
Let C be a complexity class.
A search problem R is C-complete iff R ∈ C and Q ≤P R ∀Q ∈ C.

Complexity Class EOPL:
The class of all problems that are reducible to E NDOF P OTENTIAL L INE.
E NDOF P OTENTIAL L INE [FGMS20]
Input:
 An implicit digraph with potential G(1, m, S, P, V) = (Σn, E)
 A known source π ∈ Σn s.t. S(π) = π = P(π) and V(π) = 1
Task: Find a string x ∈ Σn satisfying one of the following: P((S(x)) = x;
 S(P(x)) = x = π; S(x) = x, P((S(x))) = x, and V(S(x)) − V(x) ≤ 0.
1
π

4
2
4

5

5

7

6
3
6

8

The blue vertex is a known source. The red vertices are solutions.

Main Results [Ish21]
M ULTIPLE -S OURCE E NDOF L INE
Input:
 An implicit digraph with potential G(1, m, S, P, V) = (Σn, E)
 A set Π ⊆ Σn s.t. ∀π ∈ Π, S(π) = π = P(π) and V(π) = 1
Task: Find a string x ∈ Σn satisfying one of the following: P((S(x)) = x;
 S(P(x)) = x = π; S(x) = x, P((S(x))) = x, and V(S(x)) − V(x) ≤ 0.

P OTENTIAL O DD
Input:
 An implicit graph G(d, m, N, V) = (Σn, E) an odd-degree vertex π ∈ Σn
Task: Find a vertex x ∈ Σn satisfying one of the following:
 x = π and x has odd-degree; V(x) ≥ V(y) for every y ∈ N(y) with {x, y} ∈ E;
 V(x) ≤ V(y) for every y ∈ N(y) with {x, y} ∈ E.

Theorem M ULTIPLE -S OURCE E NDOF L INE is also EOPL-complete,i.e., this problem has the same complexity as E NDOF P OTENTIAL L INE.

Theorem If d ≤ 3, then P OTENTIAL O DD is EOPL-complete.
 If d ≥ 4, then P OTENTIAL O DD is PPA ∩ PLS-complete.

References[Das09] C. Daskalakis. “Nash equilibria: Complexity, symmetries, and approximation,” Comput. Sci. Rev.,
Vol. 3(2), pp.87-100, 2009.
[FGMS20] J. Fearnley, S. Gordon, R. Mehta, and R. Savani. “Unique end of potential line,” J. Comput.
Syst. Sci., Vol. 114, pp.1-35, 2020.
[GH18] P. W. Goldberg and A. Hollender. “The Hairy ball problem is PPAD-complete,” J. Comput. Syst.
Sci., Vol. 122, pp.33-62, 2021.

[Ish21] T. Ishizuka. “The complexity of the parity argument with potential,” J. Comput. Syst. Sci., Vol.120,
pp.12-41, 2021.
[Pap94] C. Papadimitriou. “On the Complexity of the Parity Argument and Other Inefficient proofs ofExistence,” J. Comput. Syst. Sci., Vol. 48(3), pp.498-532, 1994.
[Yan09] M. Yannakakis. “Equilibrium, fixed points, and complexity classes,” Comput. Sci. Rev., Vol. 3(2),pp.71-85, 2009.

The work was partly supported by JSPS KAKENHI Grant Number 21J10845, Japan.

104Differential Geometry Formulation of Hanging Membranes

Yoshiki Jikumaru (Institute of Mathematics for Industry, Kyushu University, y-jikumaru@imi.Kyushu-u.ac.jp)
Joint work with Yohei Yokosuka (Kagoshima University)

1. Introduction$QWRQL*DXGLIDPRXVIRUKLVGHVLJQRIWKH6DJUDGD)DPLOLDSURSRVHG
a mechanically efficient structure obtained by “a reversed hanging chain”.

3UHYLRXVUHVHDUFKRIKDQJLQJPHPEUDQHV VXUIDFHV 
1. Structures: A. Gaudi (Sagrada Familia, 1880’s [6]), H. Isler (roof design, 1960’s [4])$ZHOONQRZQPHWKRG7KUXVW1HWZRUN$QDO\VLV>@,QRXUSRVWHUZHFRQVLGHUKDQJLQJPHPEUDQHV
with “good” mechanical properties, and related numerical analysis

2. Hanging chains (catenary, classical)

3. Hanging membranes [1]

𝑋𝑋: [𝛼𝛼, 𝛽𝛽] → 𝑅𝑅 , 𝑋𝑋 𝑠𝑠 = (𝑥𝑥 𝑠𝑠 , 𝑧𝑧(𝑠𝑠))𝑠𝑠 DUFOHQJWK𝒆𝒆 = 𝑋𝑋𝑋XQLWWDQJHQW𝜈𝜈XQLWQRUPDO
𝑻𝑻 = 𝑇𝑇𝒆𝒆LQWHUQDOIRUFHDORQJWDQJHQWGLUHFWLRQ
𝒒𝒒ORDGLQJSHUXQLWOHQJWK FRQVWDQWYHFWRU
𝑑𝑑𝑻𝑻(TXLOLEULXPHTXDWLRQ−𝑻𝑻 + 𝑻𝑻 + 𝑑𝑑𝑻𝑻 + 𝒒𝒒𝑑𝑑𝑑𝑑 = 𝟎𝟎 ⇔ + 𝒒𝒒 = 𝟎𝟎
2

𝑑𝑑𝑑𝑑

𝑻𝑻 + 𝑑𝑑𝑻𝑻

𝒒𝒒 𝑑𝑑𝑑𝑑

𝑋𝑋: 𝑀𝑀 → 𝑅𝑅 3 , 𝑋𝑋 = 𝑋𝑋 𝑥𝑥, 𝑦𝑦 LPPHUVLRQ VXUIDFHLQℝ3
𝐼𝐼 = 𝐴𝐴12 𝑑𝑑𝑥𝑥 2 + 𝐴𝐴22 𝑑𝑑𝑦𝑦 2 VW IXQGDPHQWDOIRUP
𝑒𝑒1 = 𝑋𝑋𝑥𝑥 /𝐴𝐴1 𝑒𝑒2 = 𝑋𝑋𝑦𝑦 /𝐴𝐴2 XQLWWDQJHQW𝜈𝜈 = 𝑒𝑒1 × 𝑒𝑒2 XQLWQRUPDO
𝑻𝑻(𝟏𝟏) = 𝑇𝑇𝐴𝐴2 𝒆𝒆𝟏𝟏 𝑑𝑑𝑑𝑑𝑻𝑻(𝟐𝟐) = 𝑇𝑇𝐴𝐴1 𝒆𝒆𝟐𝟐 𝑑𝑑𝑑𝑑LQSODQHVWUHVVUHVXOWDQWV
𝒒𝒒ORDGLQJSHUXQLWDUHD FRQVWDQWYHFWRU
(𝟏𝟏)
(𝟐𝟐)
(TXLOLEULXPHTXDWLRQ𝑻𝑻𝑥𝑥 𝑑𝑑𝑑𝑑 + 𝑻𝑻𝑦𝑦 𝑑𝑑𝑑𝑑 + 𝒒𝒒𝑑𝑑𝑑𝑑 = 0

𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑−𝑻𝑻(TXLOLEULXPRIIRUFH FKDLQ*DWHZD\DUFK6W/RXLV0LVVRXULWKH8QLWHG6WDWHV>@(TXLOLEULXPHTXDWLRQIRUWKHKDQJLQJFKDLQ,QWHUQDOVWUHVVUHVXOWDQW 𝑇𝑇 + 𝒒𝒒, 𝑋𝑋 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.
(TXLOLEULXPHTXDWLRQ𝜅𝜅 𝒒𝒒, 𝑋𝑋 + 𝑏𝑏 − 𝒒𝒒, 𝜈𝜈 = 0𝜅𝜅FXUYDWXUHRI𝑋𝑋

𝛽𝛽𝛼𝛼([DFWVROXWLRQ𝑞𝑞 = 0, 𝛾𝛾 ⟹ 𝑧𝑧 = 0 cosh
𝑇𝑇
𝛾𝛾

𝒅𝒅𝒅𝒅

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(TXLOLEULXPHTXDWLRQ2𝐻𝐻 𝒒𝒒, 𝑋𝑋 + 𝑏𝑏 − 𝒒𝒒, 𝜈𝜈 = 0
𝐻𝐻PHDQFXUYDWXUHRI𝑋𝑋

9DULDWLRQDOSULQFLSOHIRUWKHKDQJLQJFKDLQ)XQFWLRQDO𝐸𝐸 𝑋𝑋 = ‫𝒒𝒒 𝛼𝛼׬‬, 𝑋𝑋 + 𝑏𝑏 𝑑𝑑𝑑𝑑𝑋𝑋𝜀𝜀 = 𝑋𝑋 + 𝜀𝜀 ⋅ 𝛿𝛿𝛿𝛿 + 𝑂𝑂 𝜀𝜀 2 YDULDWLRQ RI𝑋𝑋

6DJUDGD)DPLOLD Façana GHODPassió >@)XQFWLRQDO𝐸𝐸 𝑋𝑋 = ‫𝒒𝒒 𝑀𝑀׬‬, 𝑋𝑋 + 𝑏𝑏 𝑑𝑑𝑑𝑑𝑋𝑋𝜀𝜀 = 𝑋𝑋 + 𝜀𝜀 ⋅ 𝛿𝛿𝛿𝛿 + 𝑂𝑂 𝜀𝜀 2 YDULDWLRQ RI𝑋𝑋𝜈𝜈, 𝛿𝛿𝛿𝛿 𝑑𝑑𝑑𝑑𝛿𝛿𝛿𝛿 = − න

𝛾𝛾
𝑥𝑥
𝑇𝑇0

𝑀𝑀

2𝐻𝐻 𝒒𝒒, 𝑋𝑋 + 𝑏𝑏 − 𝒒𝒒, 𝜈𝜈

4. Numerical analysis for hanging membranes(TXLOLEULXPHTXDWLRQLQFRQIRUPDOFRRUGLQDWHV − 𝒒𝒒, 𝑋𝑋 + 𝑏𝑏 𝑋𝑋𝑥𝑥 𝑥𝑥 + − 𝒒𝒒, 𝑋𝑋 + 𝑏𝑏 𝑋𝑋𝑦𝑦
“discretization”𝑦𝑦𝜈𝜈, 𝛿𝛿𝛿𝛿 𝑑𝑑A+ 𝒒𝒒𝐴𝐴1 𝐴𝐴2 = 0

2XUGLVFUHWL]HGHTXDWLRQ>@෍

𝑗𝑗∶ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑖𝑖

𝑤𝑤𝑖𝑖𝑖𝑖 = −𝑤𝑤𝑖𝑖𝑖𝑖 𝑋𝑋𝑗𝑗 − 𝑋𝑋𝑖𝑖 + 𝒒𝒒 = 𝟎𝟎

1
𝐴𝐴𝑖𝑖

𝒒𝒒,

𝑋𝑋𝑖𝑖 + 𝑋𝑋𝑗𝑗
+ 𝑏𝑏
2

”Force density” [5]3ODQHYLHZ

3HUVSHFWLYHYLHZ(OHYDWLRQYLHZ

References:[1] Y. Jikumaru and Y. Yokosuka, in preparation.
[2] V. V. Novozhilov, Thin shell theory, 2nd edn., Groningen, Noordhoff, 1964.
[3] P. Block and J. Oschendorf, Thrust Network Analysis: A New Methodology for Three-dimensional Equilibrium, J. Int. Assoc. Shell Spat. Struct. 48(3) (2007) 167-173.
[4] J. Chilton, The Engineer’s Contribution to Contemporary Architecture: Heinz Isler, Thomas Telford: London (2000) 5–159.
[5] H.-J. Schek, Force density method for form finding and computation of general networks, Comput. Methods Appl. Mech. Engrg., 3 (1974) 115–134.
[6] A. Gaudi, Sagrada Familia: https://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia[7] Gateway arch: https://www.gatewayarch.com/[8] Casa Mila: https://en.wikipedia.org/wiki/Casa_Mil%C3%A0105

Reeb graphs of smooth functions with prescribed preimages
Naoki Kitazawa
Institute of Mathematics for Industry, Kyushu University.
n-kitazawa@imi.kyushu-u.ac.jp

Abstract
This poster is on realization of graphs as Reeb graphs of smooth functions of several good classes with prescribed preimages. The Reeb space of a (smooth) map c is the space of all connectedcomponents of preimages of a (smooth) map. For smooth functions of a wide class, the Reeb space is a graph and it is called the Reeb graph. Reeb spaces (graphs) are important in singularity
theory of differentiable maps, its applications to geometry of manifolds and some applied mathematics.

Preliminaries.

4↑

Existing studies on Main Problem and Main Theorem.

 Construction of smooth functions on closed surfaces (see [5] and [2]).

 Manifolds are spaces which are locally regarded as the Euclidean space of a fixed

dimension and have local coordinates.
 Differentiable (smooth) manifolds are manifolds for differential calculus.

Smooth maps and singular points.
f : M → N : a smooth map between smooth manifolds.
p ∈ M is a singular point of f : at p(the rank of the differential df p)< min{dim M, dim N } and f (p) is a singular value.
Euclidean spaces, unit spheres and unit disks.
Rk : the k-dim. Euclidean space and for p ∈ Rk (R1 is denoted by R).
||p|| : the distance between p ∈ Rk and the origin 0 where the underlying metric is thestandard metric.
S k := {p ∈ Rk+1 | ||x|| = 1} : the k-dim. unit sphere.
Dk := {p ∈ Rk | ||x|| ≤ 1} : the k-dim. unit disk.

A function on a closed orientable surface of genus 2 and its Reeb graph.
 Construction of Morse functions s.t. connected components of preimages contain no sin-gular points are copies of a unit sphere ([3]).
Main Theorem ([1]). l : a non-negative integer valued function on E of the graph
G := (V, E).
g : G → R: a continuous function which is injective on each edge.
→ ∃M : a 3-dim. closed, connected and orientable manifold.
∃f : M → R : a smooth function satisfying the following.

Graph.
G := (V, E) : a graph s.t.
 V : the vertex set.

E ̸= ∅ : the edge set.

 It may be a multigraph. It is with no loops and finite.

1. Wf and G are isomorphic as graphs (we identify them suitably).

2. qf −1(a) is a closed and orientable surface of genus l(e) (a ∈ Int e).

→ A 1-dim. compact polyhedron.

2

3. g(a) = f¯(a) (a ∈ V : f¯ ◦ qf = f ).

Morse(-Bott) functions.









Definition 1. f : M → R is a Morse function: a smooth function s.t. at each singum−i(p)i(p)
lar point p it is represented by (x1, · · · , xm) → Σj=1 xj 2 − Σj=1xm−i(p)+j 2) + f (p)for some integer 0 ≤ i(p) ≤ m and suitable coordinates.

→ i(p) is chosen uniquely. Singular points appear discretely.

4. f is a Morse function, or if not, f is a function which is locally a Morse-Bottfunction around each singular point except finitely many singular points.

↑Some local functions.

A Morse-Bott function: a smooth function locally represented as the composition of asubmersion with a Morse function around each singular point.

↑



⤴

1

 (First) A Morse function around a vertex v of degree 1 adjacent to an edge e satisfyingl(e) = 0.

 (Second) A Morse function around a vertex v where r does not have a local extremum

3

Reeb spaces, Reeb graphs and realization of graphs.

For a map between spaces f : M → N · · ·p1∼f p2 : p1, p2 ∈ M are in a same connected component of f −1(p) for some p ∈ N .
→ ∼f is an eq. rel. on M .
Definition 2. Wf := M/∼f is the Reeb space of f .



qf : M → Wf : the quotient map.
Some important properties.

∃!f¯ s.t. f = f¯ ◦ qf .

s.t. for any edge e ∋ v, l(e) = 0.
⇒ This is due to [3]. Our Main Theorem extends this to the case l(e) > 0 may hold.

 (Third) A smooth map into the plane (onto the disk surrounded by the dotted blue

circle). We need this to obtain a smooth function around a vertex of degree 1 adjacent toan edge e satisfying l(e) > 0.
⇒ The straight lines show the singular values of the map into the plane. This map islocally regarded as the product map of a Morse function and the identity map on a line.
⇒ Circles and ”l(e) copies · · · ” are for preimages of the corresponding points in the plane.
⇒ We compose the map with a function in the first figure to obtain a desired function.

 Reeb spaces often inherit invariants for the manifolds such as homology groups.
 They are often graphs (a Reeb graph: the graph Wf in Fact 1).
 In applications of mathematics such as visualizations, they are strong tools.

Fact 1 ([4]). f : a smooth function with finitely many singular values on a closed
manifold.
→ Wf is a graph with the vertex set V := {p ∈ Wf | qf −1(p) has at least one
singular point.}.



↑

5


Main Problem (A realization). For a given graph G = (V, E), can we construct
a smooth function f of a certain good class s.t. the Reeb graph Wf is isomorphic to
G (with prescribed preimages)?

















Problem 1. Can we obtain higher dimensional variants of our Main Theorem?

⇒ [4] has a result where connected components of preimages containing no singular points
are compact (closed) manifolds of general dimensions. However, different from our study,
explicit types of singular points are not studied.

Problem 2. Find useful applications to visualizations, data analysis etc.

⇒ Our problem may give new methods in function fittings. For example, in suitable situations, datasets may be regarded as the union of (finitely many) prescribed preimages in our
problem and our solution to Main Problem may give rise to a suitable fitting function.

6

Acknowledgement.

This work is supported by JSPS KAKENHI Grant Number JP17H06128 (Principal Investigator: Osamu Saeki).

References

A Morse function on the unit sphere S m−1 ⊂ Rm and its Reeb graph and preimages
for m ≥ 2: isolated dots and ”S m−1” are for preimages (left). A Morse-Bott function
on a torus and its Reeb graph and preimages (right).
Future work.

[1] N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely
many singular values, accepted for publication in Topol. Methods in Nonlinear Anal. after a refereeing process, arxiv:1902.08841.
[2] Y. Masumoto and O. Saeki, A smooth function on a manifold with given Reeb graph, Kyushu J. Math. 65 (2011), 75–84.

[3] L. P. Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold. Topol. Methods in Nonlinear
Anal. 52 (2) (2018), 749–762, arxiv:1805.06727.
[4] O. Saeki, Reeb spaces of smooth functions on manifolds, Intermational Mathematics Research Notices, maa301,
https://doi.org/10.1093/imrn/maa301, arxiv:2006.01689.



[5] V. Sharko, About Kronrod-Reeb grpah of a function on a manifold, Methods of Functional Analysis and Topology 12 (2006),
389–396.

106

Strategic Delegation in Bilateral Environmental Agreements
under Heterogeneity
Qian Li

Institute of Mathematics for Industry, Kyushu University, Japanq-li@imi.kyushu-u.ac.jp

1. Introduction
Transboundary pollutant problems can be solved via international cooperation, of which a common way is forming bilateral environmental agreements(BEAs). Former studies modelling the formation of BEAs regard each country as a single player, without considering the domestic politics. In this study,
we construct a political-economy model of strategic environmental policymaking with heterogeneous countries. In both the developed and developing
countries, domestic households that have heterogeneous environmental preferences vote for the government separately and then these governments
negotiate about each country’s individual abatement levels and transfers between them.

2. The model setting

3. Cooperative environmental policies

There are two countries, of which one is developing country and another is developed country.
In each of them, there are households who havedifferent environmental preferences, denoted as
θih . In the former country (denoted as 1), eachhousehold’s payoff function is

Cooperative environmental policies mean that the elected governments decide individual abatementlevels and transfers in order to maximize their aggregate payoff.

π1h = θ1h (x1 + x2 ) − γcx1 2 ,(1)while in the later (denoted as 2), it becomes
π2h = θ2h (x1 + x2 ) − cx2 2 .

(2)0 < γ < 1 suggests that each household in developing country has lower abatement cost thandeveloped one, for the same abatement level.
Moreover, θ1m < θ2m means that the medianvoter in country 1 has a lower monetary valuation over the improvement of environmental
quality than that in country 2. The formation of
the agreement can be represented using a threestage game, the framework of which is represented as follows:

After solving the game via backwards induction, the equilibrial individual abatement levels suggestθ g +θ gθ g +θ gthat x∗1c = 12cγ 2 > x∗2c = 12c 2 , where subscript c represents the cooperative case. In addition,
the elected governments’ environmental preferences are that θ1g∗ < θ1m and θ2g∗ < θ2m . These confirm
the results in [1] and [2] that households have incentives to delegate less green government whencooperate, in the asymmetric case.

4. Non-cooperative environmental policies
Under the non-cooperative environmental policies, each government decides the abatement level
individually in order to maximize its own payoff, which can be shown as follows.

6. References[1] Buchholz, W., Haupt, A. Peters, W.
(2005). International environmental agreements
and strategic voting. Scandinavian Journal of
Economics. 107, 175–195.
[2] Harstad, B. (2008). Do side payments help?
Collective decisions and strategic delegation.
Journal of the European Economic Association,
6(2/3), 468–477.
[3] Siqueira, K. (2003). International externalities, strategic interaction, and domestic politics.
Journal of Environmental Economics and Management, 45, 674–691.

The equilibrial results show that households will delegate sincerely. The abatement level of each
θ1m
θ2m
and x∗2I = 2c
, where I means the non-cooperative case.
country is x∗1I = 2cγ

5. Conclusions
We prove that cooperation is effective in improving the abatement levels.
1. Under cooperative environmental policies, the governments being elected in both the developing
and the developed country have lower environmental preferences than the median household.
The developing country always abate more than the developed one.
2. Under both the cooperative and non-cooperative environmental policies, the government being
elected in the developed country has a higher environmental preference than the developingone.
3. Cooperation brings higher total abatement level and is always effective since it brings higher
aggregate payoff, which is in contrast with [3].

107

108

The insights from our model has the potential to be applied toother new suburbs to assist with the planning phase of suburb
development.

•

Figure 2 Percentage price change from housing features

Our model is able to predict house prices in the new suburb of
Hobsonville with reasonable accuracy.

•• Understand the impact of individual housing features on sale pricein a new suburb.

• Validate the robustness of our proposed model by making
predictions for a new suburb.

The proposed method of modelling sale prices relative to a‘standard’ house effectively captures the non-linear effects ofindividual housing features and price change over time.

 

Figure 4 Box plot of prediction residuals against sale year

Figure 3 Scatter plot of predicted price against actual price

We use our ridge regression model fitted on Fairview Heights, Oteha, andStonefields to make house price predictions for sales in the Hobsonvillesuburb. Root mean square error (RMSE) is $129,779 after backtransformation. The box plot of residuals against sale years does not show
any clear trend of residual changes across time (see Fig. 3). Residuals are
relatively uniform across sale years. This implies that the non-linear house
price trend over time is effectively captured by our model. There is one clear
outlier with a residual just above one million in magnitude. This outlier
property is a townhouse sold in 2018 and with a floor area of 400m2 . This is
more than twice the median floor area of other townhouses, but the actual
sale price is unexpectedly low for unknown reasons. Overall, approximately
91% of our predictions are within $200,000 of the true sale price.



•We conduct a case study on modelling house sales in three new
Auckland suburbs-Fairview Heights, Oteha, and Stonefields. The
estimated percentage change in price for individual housing featurescompared to a standard house are calculated from the fitted
coefficients of housing features. A house with only one bedroom isestimated to have a price approximately 37.25% lower than the priceof a standard house with three bedrooms. A house with four
bedrooms is estimated to be 10.43% higher in price than the standard
house, but the increase in price for each added bedroom plateaus
above four bedrooms. This indicates that most buyers are not satisfied
with single-bedroom houses, but five bedrooms or above can become
excessive. Additionally, houses with at least one free-standing garage
are estimated to have prices 5.01% higher than those who do not.



The chosen formula for our model is:• Impacts of individual housing features on sale price relative to thestandard house• Price of a standard house over time

This study proposes separating the effects of price change over timefrom the effects of individual housing features by decomposing the
problem into two components: 

This study focuses on analyzing newly developed suburbs in Auckland
to understand housing feature impacts on sale price in these new
suburbs and provide interpretable. Defining a reliable point ofreference for our model is critical for conveying the results to nonexperts in machine learning, such as housing developers. Our chosenpoint of reference is the price of a standard house in the new suburbs
over time. Modelling housing feature impacts relative to a standard
house will create a unique solution to our regression problem that iseasy to interpret. The main objectives of this study are to:  

Figure 1 Map of Greater Auckland

Shortage of affordable housing is a growing crisis all around the world.
Surveys conducted by Gallup‘s World Poll from 2015 to 2017 across
140 different countries indicate that on average 27% of a country’s
population cannot afford adequate housing. Countries are looking for
new housing solutions to create sustainable and affordable housing
conditions for middle-class families. One new solution is the massdevelopment of entirely new suburbs that support a local community.
Auckland Council has presented a plan to develop outer suburbs suchas Fairview Heights and Hobsonville and expand the current city into
`Greater Auckland‘. These new suburbs have not been adequately
studied because this is a relatively new approach to property
development. Modelling housing feature impacts in the new suburbs
will be increasingly relevant to future developers.

 

Christina Yin-Chieh Lin (UOA), Andreas W. Kempa-Liehr (UOA), Andrew Mason (UOA)
Department of Engineering Science
University of Auckland

Modelling Housing Feature Impacts on Sale Price in Newly Developed Suburbs

109

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MI レクチャーノートシリーズ刊行にあたり
本レクチャーノートシリーズは、文部科学省 21 世紀 COE プログラム「機能数理学の構築と展開」（H.15-19 年度）において作成した COE Lecture Notesの続刊であり、文部科学省大学院教育改革支援プログラム「産業界が求める数学博士と新修士養成」（H19-21 年度）および、同グローバル COE プログラ
ム「マス・フォア・インダストリ教育研究拠点」（H.20-24 年度）において行われた講義の講義録として出版されてきた。平成 23 年 4 月のマス・フォア・
インダストリ研究所（IMI）設立と平成 25 年 4 月の IMI の文部科学省共同利用・共同研究拠点として「産業数学の先進的・基礎的共同研究拠点」の認定を受け、今後、レクチャーノートは、マス・フォア・インダストリに関わる国内外の研究者による講義の講義録、会議録等として出版し、マス・フォア・インダ
ストリの本格的な展開に資するものとする。
平成 30 年 10 月
マス・フォア・インダストリ研究所
所長

佐伯修

Proceedings of Forum “Math-for-Industry” 2021
-Mathematics for Digital Economy発
編

行
集

発

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印

刷

2022年３月28日
Osamu Saeki, Ho Tu Bao, Shizuo Kaji, Kenji Kajiwara, Nguyen Ha Nam, Ta Hai Tung,
Melanie Roberts, Masato Wakayama, Le Minh Ha, Philip Broadbridge
九州大学マス・フォア・インダストリ研究所
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白井 朋之
若山 正人
脇
隼人
Philip Broadbridge
山本 昌宏

Study Group Workshop 2015 Abstract, Lecture & Report
156pages

MI Lecture Note Vol.67

Institute of Mathematics
for Industry, Kyushu
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IMI-La Trobe Joint Conference“Mathematics for Materials Science and Processing”
66pages

February 5, 2016

MI Lecture Note Vol.68

古庄
小谷
新甫

英和
久寿
洋史

結び目と Grothendieck-Teichmüller 群
116pages

February 22, 2016

MI Lecture Note Vol.69

土橋
鍛治

宜典
静雄

Symposium MEIS2016:
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74pagesNovember 5, 2015

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MI Lecture Note Vol.70

Institute of Mathematics
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MI Lecture Note Vol.71

小磯
二宮
山本

Study Group Workshop 2016 Abstract, Lecture & Report

November 21, 2016

深幸
嘉行
昌宏

143pagesシリーズ既刊
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Author／Editor
朝雄
泉
文生

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Published

MI Lecture Note Vol.72

新井
小嶋
廣島

MI Lecture Note Vol.73

穴田 啓晃
Kirill Morozov
須賀 祐治
奥村 伸也
櫻井 幸一

Secret Sharing for Dependability, Usability and Security of Network
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March 15, 2017

MI Lecture Note Vol.74

QUISPEL, G. Reinout W.
BADER, Philipp
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TAGAMI, Daisuke

IMI-La Trobe Joint Conference
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March 31, 2017

MI Lecture Note Vol.75

手塚
田上
山本

Study Group Workshop 2017 Abstract, Lecture & Report

MI Lecture Note Vol.76

宇田川誠一

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August 4, 2017

MI Lecture Note Vol.77

松谷 茂樹
佐伯
修
中川 淳一
田上 大助
上坂 正晃
Pierluigi Cesana
濵田 裕康

平成29年度 九州大学マス・フォア・インダストリ研究所
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December 20, 2017

MI Lecture Note Vol.78

瀧澤 重志
小林 和博
佐藤憲一郎
斎藤
努
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神山 直之

平成29年度 九州大学マス・フォア・インダストリ研究所
プロジェクト研究 研究集会（I）
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February 26, 2018

MI Lecture Note Vol.79

神山
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February 28, 2018

MI Lecture Note Vol.80

Kirill Morozov
Hiroaki Anada
Yuji Suga

IMI Workshop of the Joint Research Projects
Cryptographic Technologies for Securing Network Storage
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March 30, 2018

MI Lecture Note Vol.81

Tsuyoshi Takagi
Masato Wakayama
Keisuke Tanaka
Noboru Kunihiro
Kazufumi Kimoto
Yasuhiko Ikematsu

IMI Workshop of the Joint Research Projects
International Symposium on Mathematics, Quantum Theory,
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September 25, 2019

MI Lecture Note Vol.82

池森

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新型コロナウイルス感染症にかかわる諸問題の数理
145pages

March 22, 2021

集
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直之
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俊文

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133pages

January 27, 2017

71pages

118pages

October 20, 2017

シリーズ既刊
Issue

Author／Editor

Title

Published

MI Lecture Note Vol.83

早川健太郎
軸丸 芳揮
横須賀洋平
可香谷 隆
林
和希
堺
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July 28, 2021

MI Lecture Note Vol.84

Taketoshi Kawabe
Yoshihiro Mizoguchi
Junichi Kako
Masakazu Mukai
Yuji Yasui

SICE-JSAE-AIMaP Tutorial
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December 27, 2021

MI Lecture Note Vol.85

Hiroaki Anada
Yasuhiko Ikematsu
Koji Nuida
Satsuya Ohata
Yuntao Wang

IMI Workshop of the Joint Usage Research Projects
February 9, 2022
Exploring Mathematical and Practical Principles of Secure Computationand Secret Sharing 114pages

MI Lecture Note Vol.86

濱田 直希
穴井 宏和
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千葉 一永
佐藤 寛之
能島 裕介
加葉田雄太朗
一木 俊助
早野 健太
佐伯
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2020年度採択分 九州大学マス・フォア・インダストリ研究所
共同利用研究集会
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110pages

February 22, 2022

九州大学マス・フォア・インダストリ研究所

九州大学大学院 数理学府
URL http://www.imi.kyushu-u.ac.jp/