マス・フォア・インダストリ研究 No.24

Fiber Topology Meets
Applications 2

Editors :Daisuke Sakurai
Osamu Saeki
Shigeo Takahashi
Hamish Carr
Takahiro Yamamoto
Naoki Hamada

Institute of Mathematics for Industry
Kyushu University

About the Mathematics for Industry Research
The Mathematics for Industry Research was founded on the occasion of the certificationof the Institute of Mathematics for Industry (IMI), established in April 2011, as a MEXT
Joint Usage/Research Center – the Joint Research Center for Advanced andFundamental Mathematics for Industry – by the Ministry of Education, Culture, Sports,
Science and Technology (MEXT) in April 2013. This series publishes mainly proceedingsof workshops and conferences on Mathematics for Industry (MfI). Each volume includes
surveys and reviews of MfI from new viewpoints as well as up-to-date research studies
to support the development of MfI.
October 2018
Osamu Saeki
Director
Institute of Mathematics for Industry

Fiber Topology Meets Applications 2
Mathematics for Industry Research No.24, Institute of Mathematics for Industry, Kyushu University
ISSN 2188-286X
Editors: Daisuke Sakurai, Osamu Saeki, Shigeo Takahashi, Hamish Carr, Takahiro Yamamoto,
Naoki Hamada
Date of issue: 15 March 2022
Publisher:
Institute of Mathematics for Industry, 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/
Printed by
Kijima Printing, Inc.
Shirogane 2-9-6, Chuo-ku, Fukuoka, 810-0012, Japan
TEL +81-(0)92-531-7102 FAX +81-(0)92-524-4411

Fiber Topology Meets Applications 2

Editors: Daisuke Sakurai
Osamu Saeki
Shigeo Takahashi
Hamish Carr
Takahiro Yamamoto
Naoki Hamada

Daisuke Sakurai
Kyushu University
February, 2022

Introduction
In diﬀerential topology, the study of the topology of Þbers, known as Þber topology, extends theMorse theory, of functions, to that of maps. This establishes a thorough exploration of topologicaltransitions of inverse images. The topological analysis of the level set, from which Þber topologybased data analysis originates, has been proven powerful for understanding todayÕs data whose sizeand complexity are overwhelming. We have worked on establishing data analysis utilizing Þbertopology with regard to theory and computation, and now are on the stage to co-design applications
and further developments for them.
Continuing the eﬀort in the previous meeting, we propose to gather researchers from academia
and the industry to discuss the future directions. The participants included mathematics,
computation, and applications. For this, participants bring themes from broad spectra ofpractitioners including those from mathematics, topological analysis in visualization, the gameindustry and environmental sciences.
Our collaboration for application in the game industry started by designing and automating gamemusic developments for mobile devices. We have since expanded the aim to into general
optimization problems for mobile games. A key problem is to understand multi-objective
optimization, for which Þber topology can contribute by designing and understanding benchmarks.
Another application is environmental science, for which we discuss integrating Þber topology into
HPC simulations.
All these pose a challenge to state-of-the-art algorithms. The current, rather loose, mathematics
found in the computer science literature meets a more rigorous treatment by mathematicians and wewill discuss the direction we can take.

Acknowledgement
This forum was co-sponsored by Kyushu University Institute of Mathematics for Industry underFY2022 IMI Joint Usage/Research Program and Pan-Omics Data-Driven Research Innovation Centerof the same university.

－i－Contents
プログラム����������������������������� iii
Fiber Topology Meets Applications: This Year

Daisuke Sakurai (Kyushu University) ���������������������

On-line Diagnostics in Climate Simulations -Status and Perspectives

Bastian Kern and Hiroshi Yamashita (German Aerospace Center (DLR))��������

1

5

Explicitly Multimodal Benchmarks for Multi-Objective Optimizations

Reiya Hagiwara (Kyushu University), Ryosuke Ota (Kyushu University), Naoki Hamada (KLab Inc.), Takahiro Yamamoto (Tokyo Gakugei University), Daisuke Sakurai (Kyushu University) ��������������������� 23

The Centroids of Isosurface Components for All Isovalues

Akito Fujii, Daisuke Sakurai, Kenji Ono (Kyushu University)������������ 29

Thinking multi-objective optimization problem topologically

Likun Liu, Kenji Ono and Daisuke Sakurai (Kyushu University)����������� 35

Topological invarinats of smooth map germs of manifolds with boundary

Takahiro Yamamoto (Tokyo Gakugei University)����������������� 41

Reeb Diagram and Visualization of Monodromy

Osamu Saeki (Kyushu University)����������������������� 54－ii－

九州大学 IMI 共同利用・一般研究-研究集会(II) 公開プログラム

Fiber Topology Meets Applications 2
日

場

時： 2021年12月1日（水）17：00 〜 18：15
2021年12月2日（木）17：00 〜 18：15
2021年12月3日（金）17：00 〜 18：40
所： 九州大学 伊都キャンパス ウエスト1号館 D棟4階
IMIコンファレンスルーム（W1-D-414）（対面＋オンラインのハイブリッド形式）
研究代表者： 櫻井 大督（九州大学情報基盤研究開発センター附属
汎オミクス計測・計算科学センター）※プログラムは都合により変更になる場合がありますので予めご了承ください。
最新情報はホームページをご覧ください。

＜プログラム＞
12 月 1 日（水）17:00-17:25
Fiber Topology Meets Applications: This Year
Daisuke Sakurai (Kyushu University)
17:25-17:50
Fibers, Covers, Tessellation & Partial Orders: Different Approaches toAnalysing Multi-Variate Data
Hamish Carr (University of Leeds)
17:50-18:15
On-line Diagnostics in climate simulations - Status and perspectives
Bastian Kern and Hiroshi Yamashita (German Aerospace Center)－iii－

12 月 2 日（木）17:00-17:25
Explicitly Multimodal Benchmarks for Multi-Objective Optimizations
Reiya Hagiwara (Kyushu University), Ryosuke Ota (Kyushu University), Naoki Hamada(KLab Inc.), Takahiro Yamamoto (Tokyo Gakugei University), Daisuke Sakurai (Kyushu
University)

17:25-17:50
The Centroids of Isosurface Components for All Isovalues
Akito Fujii, Daisuke Sakurai and Kenji Ono (Kyushu University)
17:50-18:15
Thinking about multi-objective optimization problems topologically
Likun Liu, Daisuke Sakurai and Kenji Ono (Kyushu University)

12 月 3 日（金）17:00-17:25
Topological invariants of smooth map germs of manifolds with boundary
Takahiro Yamamoto (Tokyo Gakugei University)
17:25-17:50
Reeb diagram and visualization of monodromy
Osamu Saeki (Institute of Mathematics for Industry, Kyushu University)
17:50-18:15
Combinatorics Meets Topology at Piecewise Linear Reeb Spaces
Peter Hristov (University of Leeds)
18:15-18:40
Concluding Remarks
Daisuke Sakurai (Kyushu University)－iv－

Fiber Topology Meets Applications:This Year
Daisuke Sakurai
2021-12-01

1This Forum•

Forum for Þber topology & Reeb spaces(Generalized contour topology & Reeb graphs)•

From math, compsci & applications•

Share recent progress in retreat-style

2－1－

My Talk (25min Total)

1. Introduction to the forum
2. Progress on our Reeb space computation

3

Topology(Squeeze
Horizontally)

HeightReeb Graph:
CompactiÞes Landscape

Layout Coordinate

f : R 2 →Height

Contour Tree
4－2－

Reeb Space:
CompactiÞes Multi-Value Landscape
R3

Topology(Squeeze)

f

c2
c2

c1
Minimum c1, c2

c1

Reeb space off
5

Talks fromTarget Applications•

Machine learning••

Multi-objective optimization••

Hyper-parameter tuning (Likun on Thu)
Benchmarking (Reiya on Thu)

Atmospheric sciences•

HPC simulations (Bastian & Hiroshi today)•

Multi-objective optimization (Dec 15 @ IMI)
6

－3－

Talks from
Topological Analysis•

Review of related data analyses (Hamish today)•••

JCN, Mapper, etc.

ClassiÞcation of Reeb spaces for PL maps
(Petar on Fri)
Centroids of isosurfaces (Akito on Thu)

7

Talks from Math••Monodromy (twists of Þbers)(Saeki-sensei on Fri)A study of topological invariants withvisualization(Takahiro on Fri)

8

－4－

DLR.de • Chart 1> On-line diagnostics > Kern & Yamashita, DLR-PA • Fiber Topology Meets Application 2 > 01.12.2021

On-line Diagnostics in Climate Simulations Status and Perspectives
Bastian Kern and Hiroshi Yamashita
Institute of Atmospheric Physics – Earth System Modelling
German Aerospace Center (DLR)
Oberpfaffenhofen

DLR.de • Chart 2> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Overview• Introduction• Climate modelling• Challenges of high data volume• On­line Diagnostics (examples)• Exploiting the native time­step of the simulation• Get results, that are not possible off­line• In­situ analysis• Perspectives• Upcoming project at DLR• Summary / Discussion－5－DLR.de • Chart 3> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Introduction

DLR.de • Chart 4> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021(Numerical) Modelling in a nutshell• There is no second Earth (to experiment with)a)a)

NOAA/NASAhttps://www.nasa.gov/image­feature/new­weather­satellite­sends­first­images­of­earthb)b)Wikimedia, user Hellerick
https://commons.wikimedia.org/wiki/
File:Division_of_the_Earth_into_Gauss­Krueger_zones_­_Globe.svg－6－

DLR.de • Chart 5> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021(Numerical) Modelling in a nutshell• There is no second Earth (to experiment with)• A mathematical model based on physical equations
 coupled system of (non­linear) partial differential equations
 solved numerically on a “supercomputer”

https://www.dkrz.de/about/media/galerie/Media­DKRZ/hlre­3

DLR.de • Chart 6> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Climate projections with EMAC
• Model simulations with 2 PByte output• Contribution to Chemistry Climate Initiative (CCMI)

Total ozone column (DU), Antarctic, October• Data for WMO Ozone Assessment andIntergovernmental Panel on Climate Change (IPCC)

Dohmse et al., ACP, 2018－7－

DLR.de • Chart 7> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Limits of hardware• Storage hierarchy

https://en.wikipedia.org/wiki/Memory_hierarchy

DLR.de • Chart 8> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

How does the I/O gap impact a typical workflow?
1) simulation
2) data output

memory3) post­processing(subset)3b) post­
processing(reduced)disk(scratch/work)archive (tape)4) archiving3a) post­
processing－8－

DLR.de • Chart 9> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

How does the I/O gap impact a typical workflow?
1) simulation
2) data output

memory5a) post­
processing

DLR.de • Chart 105) post­processing(subset)3) archiving4) get data(subset)disk(scratch/work)5b) post­
processing(reduced)archive (tape)6) archiving(reduced)> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

How does the I/O gap impact a typical workflow?
1) simulation
2) data output

memory5a) post­
processing5) post­processing(subset)3) archiving

disk(scratch/work)5b) post­
processing(reduced)4) get data(subset)6) archiving

Bottleneck 1: ~400 GB/s

Bottleneck 2: ~20 GB/s

－9－archive (tape)DLR.de • Chart 11> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

An idealised on­line diagnostic workflow1) simulation
2) data output(reduced)memorydisk(scratch/work)3) Archiving(reduced)archive (tape)1a) online­
diagnostics

DLR.de • Chart 12> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

What are on­line diagnostics?
• Calculate something from the model state during the simulation• Write out the result• Use the result as input in the model system• What are off­line diagnostics?
• “The post­processing step”…

Why do we use on­line diagnostics?
• Use results of calculations in the model system (e.g. as input in parameterisations)• Get ready­to­use results, for evaluation, comparison between model and observation, …
• Benefit from the native resolution of the host model (temporal and spatial)• Use model information, which are not available from output• Reduce output data volume－10－

DLR.de • Chart 13> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Modular Earth Submodel System (MESSy)
EMAC = ECHAM/MESSy Atmospheric Chemistry

COSMO

ECHAM

CESM1

CLaMS

ICON

MECO(n)Modular Earth Submodel System Infrastructure
SWITCH CHANNELTRACERTIMER

IMPORT

…

…

GRID

…

…

…

> 20 partner institutes
Framework to couple
scientific codes tonumerical weather
prediction and climate
modelsDLR.de • Chart 14

…

…

...

...

> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Make use of native temporal model resolution
Sampling along satellite swaths

－11－

MESSy
infrastructure

https://www.messy­
interface.org/

DLR.de • Chart 15> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

SORBIT ­ Sampling along sun­synchronous satellite ORBITs
Sun­synchronous orbit• Equator crossing at approx. the same local time each day(and night)• Consistent scientific observations• Relatively constant angle between the Sun and the Earth’s
surface

Ground paths of the multiple orbital revolutions during one
day for a near­polar orbiting satellite.
Three consecutive orbits of a sun­synchronous satellite with anequatorial crossing time of 1:30 pm.

https://en.wikipedia.org/wiki/Sun­synchronous_orbit
https://earthobservatory.nasa.gov/features/OrbitsCatalog/page2.php
http://tornado.sfsu.edu/geosciences/classes/m407_707/Monteverdi/Satellite/PolarOrbiter/Polar_Orbits.html

DLR.de • Chart 16> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

SORBIT
O3 @ 50hPa, μmol mol­1
15 Jan 2006 16 UTC, orbit 20276

daily
SORBIT
output ENVISAT orbit

15 Jan 2006, orbit 20276 ­ 20282

0 6 12 18 24 LT→

at output time
60°N
output time ±Δt

0°5h modeloutput30°S← latitude

30°N

60°S

Jöckel et al., GMD, 2010
doi:10.5194/gmd­3­717­2010－12－

DLR.de • Chart 17> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

SORBIT
ENVISAT orbit

NO @ 50hPa, μmol mol­1
15 Jan 2006 16 UTC, orbit 20276

15 Jan 2006, orbit 20276 ­ 20282

0 6 12 18 24 LT→

at output time
60°N
output time ±Δt

0°
30°S

← latitude

30°N

60°S

Jöckel et al., GMD, 2010
doi:10.5194/gmd­3­717­2010

DLR.de • Chart 18> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

SORBITNO @ 50 hPa, monthly average (Jan 2006)
5h model output

SORBIT

difference

80°N
40°N
0°
40°S

80°S
180°W

0°

180°E 180°W
0.0

0.5
nmol mol­10°1.1

180°E 180°W
­0.38

0°
0
nmol mol­1

180°E
0.38

Jöckel et al., GMD, 2010
doi:10.5194/gmd­3­717­2010－13－

DLR.de • Chart 19> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

SORBIT
Total O3 column, monthly average (Jan 2006)
5h model output

SORBIT

difference

80°N
40°N
0°
40°S

80°S
180°W

0°

180°E 180°W

0°180°E 180°W­6

280 360 440 520
DU

0°
­3

0
DU

180°E

3

6

Jöckel et al., GMD, 2010
doi:10.5194/gmd­3­717­2010

DLR.de • Chart 20> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Make use of native temporal model resolution – 2
Sampling in space and time－14－

DLR.de • Chart 21> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

S4D – Sampling in 4 dimensions
spatio­temporalinterpolation

08 Sep 2006 22:00 – 240 hPa
model output

nearest grid
point

10­18

10­17

10­16

10­15

10­14

S4D sampling at modeltime step (Δt=12 min)10­13

OH in mol mol­1

DLR.de • Chart 22

Jöckel et al., GMD, 2010
doi:10.5194/gmd­3­717­2010

spatio­temporalinterpolation
nearest gridpoint
sampling at
model time step> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

S4D – Sampling in 4 dimensions

8 Sep 2006 23:00 – 205 hPa

nearestinterpolation

model output

10­18

9 Sep 2006 00:45 – 205 hPa

10­17

10­16

10­15

OH in mol mol­1

－15－

10­14

10­13

Jöckel et al., GMD, 2010
doi:10.5194/gmd­3­717­2010DLR.de • Chart 23> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Get results, that are not possible from off­line data
Tagging of chemical emissions

DLR.de • Chart 24> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

TAGGING “Classic” ­ Differences“reference run” minus “perturbed run”,“ sensitivity study”

 “Tagging”

marking emissions and tracking

N

N
O

N

O

O

O

O
O

O

O
O

O

O
O
Grewe et al., GMD, 2017
doi:10.5194/gmd­10­2615­2017－16－

DLR.de • Chart 25> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

TAGGING
Contribution of road traffic emissions to O3• Ozone decreases because ozone from road traffic
decreases (12%)• Ozone net production rates increase  ozone fromother sectors increases(­10%)• Net Ozone change = 2%

DLR.de • Chart 26> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Current implementation in our model system－17－

DLR.de • Chart 27> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Interfaces• Climate/weather model

COSMO

CESM1

CLaMS

ICON

MECO(n)Modular Earth Submodel System Infrastructure• Submodels

SWITCH CHANNEL TRACER TIMER IMPORT GRID…

• Implementation: Fortran

…

…

...

...

…

…

…

MESSy
infrastructure• The Modular Earth Submodel
System (MESSy)

ECHAM

…

Jöckel et al., ACP, 2005, doi:10.5194/acp­5­433­2005
Jöckel et al., ACP, 2006, doi:10.5194/acp­6­5067­2006
Jöckel et al., GMD, 2010, doi:10.5194/gmd­3­717­2010

DLR.de • Chart 28> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

MESSy
infrastructure

MESSy
submodels“communication” between
individual components
through mutual access of
distributed objects via
standard interfaces“dynamics”

Smemory­/ T
object­D
manage­ .
ment
I
flow­
N
controlTE
input/
R
outputF
A
C…
E“physics”
expandability without
intervention in othercomponents“chemistry”“diagnostics”
●
●

...

－18－

scalable development
coexistence of alternatives→ “community”­ansatz

DLR.de • Chart 29> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Monitor and analyse simulations in­situ

DLR.de • Chart 30> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Online Analysis and Visualisation• Paraview / Catalyst

Simulation
Catalyst
AdaptorParaview

https://www.paraview.org/in­situ－19－

DLR.de • Chart 31> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Online Analysis and Visualisation• Catalyst Adaptor for MPAS Ocean

diagnosing model setup errors
In­situ visualisation

DLR.de • Chart 32

Ayachit et al., ISAV2015, 2015
doi:10.1145/2828612.2828624
https://www.paraview.org/in­situ> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Perspectives－20－

DLR.de • Chart 33> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

DLR Project “VisPlore” (VISual & exPLORE)• 6 DLR Institutes• 3 years, starts 01/2022• Standardised interface in simulation codes of different disciplines (CFD, climate simulation, EO, …)• Provide metadata and data fields• Paraview/Catalyst• Interactive exploration• Python­based (inter­disciplinary) toolset• Web­based User­portal

DLR.de • Chart 34> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

DLR Project “VisPlore” (VISual & exPLORE)
Service Portal
register data

Metadata

Webservices☺
Interaction & Streaming

HPC system
Pre­ and Postprocessing

Simulation
Simulation
Simulation
Paraview / Catalyst

In­Situ

Paraview
Server

ViracochaRemoteVisualisation
CosmoScout
VR
Paraview
ClientI/Oadapted from VisPlore project plan－21－

DLR.de • Chart 35> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Summary• Need for On­line diagnostics and examples• Current Implementation via (Fortran) “Submodels”• Future plan of common interface for different disciplines• Common tool­set of algorithms (Python)• Discussion:• Will such an interface support implementation of novel algorithms?
• Will it enhance opportunity to apply new tools from other disciplines to gain new insights?
• Whole history stored off­line vs. high­frequent, but limited number of timesteps on­line?

DLR.de • Chart 36> On­line diagnostics > Kern & Yamashita, DLR­PA • Fiber Topology Meets Application 2 > 01.12.2021

Thank
you!

－22－

Explicitly Multimodal Benchmarks forMulti-Objective Optimizations
Reiya Hagiwara (Kyushu University), Ryosuke Ota (Kyushu University),
Naoki Hamada (KLab Inc.), Takahiro Yamamoto (Tokyo Gakugei University),
Daisuke Sakurai (Kyushu University)
2021-11-29

Multi-Objective Optimization•

Multiple objectives f : ℝn → ℝm•

An “optimal” x:•

Pareto: if fi(x) > fi(x′) then fj(x) < fj(x′)•

Locally Pareto•

Objective 2“Optimal”: Pareto front

Image of f

Improve f

Objective 1

Pareto optim. in neighborhood of x
[https://www.sciencedirect.com/topics/engineering/pareto-front]

2

－23－

Multimodality
in Multi-Objective Optimization?
Single Objective Optimization

Multi-Objective Optimization？

OptimalLocal optim.

[wikimedia.org]

Multimodality:• Depth / breadth of valleys• Connectivity etc.

Multimodality:• ?

3Multi-Mordality According to
Kerschke et al. (’16)
contour：Obj. 1

Multimodality (MM):••：Obj. 2）

Solvers: feel MM if••

Can trace local Pareto to Þnd global
ones
Unsolved:•

Kerschke, Pascal, Hao Wang, Mike Preuss, Christian Grimme, André Deutz, Heike
Trautmann, and Michael Emmerich. "Towards analyzing multimodality of continuous
multiobjective landscapes." In International Conference on Parallel Problem Solving fromNature, pp. 962-972. Springer, Cham, 2016.

Connectivity of (local) Pareto（•

MM at regular (not a local Pareto) x?

•

Explicit design of MM in benchmarks
4－24－

Reeb Space and Multi-Modality[Sakurai et al., 16]

Range

DomainReeb space•

Sheets = modes?

•

UI for singular Þbers
(2016)
Sheets = Multi-Modality?

5

Starting Collaboration
N Hamada

Model Multi-Modality→ Design Benchmark Problems（Work In Progress）

Multi-Obj
Optim.

Vis

SakuraiSingularity
Theory•Jan 2021: rough ideas•

At present: formalization

T Yamamoto
R Hagiwara
R Ota
6

－25－

Our Model of MM•

Admissible solution at x：•

Local Pareto component reachable by “optimizing” fi(x) (ie no sacriÞce of any objective)
DomainPoint xAdmissible at
Admissible at
Admissible at: {━, ━}
: {━}
: {━}

Hierarchy of feasibility
Local Pareto solutions•

Multimodality: Characteristics of admissible solutions
7

Hierarchy and Multi-Modality

Multi-modality
in small scaleMulti-modality
in large scale[Weber et al., 07]•

Can consider multiple scales•

Eg: evaluation of solvers:•

Sensitivity to scales•

Escape from small scalefeatures?

Also important in Vis

8

－26－

Designing Our Benchmark Problem

Polynomial maps：difÞcult to control•

Existing benchmarks: multi-modality unknown

singularity globally• 1990s: No standard problem• Researchers made their own problems• Early 2000: ZDT [Zitzler+ 2000]→ PL-maps

3• De facto standard problem set for 2 objectives• Late 2000: DTLZ [Deb+ 2001], WFG [Huband+ 2006]

2

ℝ →ℝ

•

• Extended ZDT for n objectives• 2010s: Inverted-DTLZ [Lain+ 2013], Minus-WFG [Ishibuchi+ 2017]• Good methods for DTLZ/WFG don’t work for negative costs

9

Primitives of Our Benchmark
f1 = x 3 + zx + y 2
f2 = z3 modesz=0Discretization
Non-linear transformationz = 0.5

Benchmark problem

1 modez=1zPrimitives:••

Local multi-obj
optimization problem

…

y
x

10

－27－

Forming Benchmark
From Primitives: Domain: Local optima (deÞnite folds): indeÞnite folds

11

Outlook

Benchmarks for•

R3 → R2
2
2• R → R••

Latest manuscript in arxiv.org

12

－28－

The Centroids of Isosurface
Components for All Isovalues
Akito Fujii
Daisuke Sakurai
Kenji Ono
(Kyushu University)

1

Analyzing multiple fields
Eg analysis with meteorologists:
Find relative positions of isosurface components
from 3 scalar fields
Cloud

VortexJet stream

2

－29－

Subjectivity in isosurfacing
Vorticity

Hypothesis (Vortex
between cloud & jet)

0.019

Stands not

0.018

Stands0.017

Stands not
3

Biases in visualization-based analysis! Validity of hypotheses is biased by user inputs! Multiple fields! Multiple isovalues! Analyze with which isovalues user specified hypotheses on
relative positions stand (future research idea)

4

－30－

Preparation (presented today):compute centroid positions for all isovaluesin scalar field! Compute the centroids of isosurface components! Use it in the future for analyzing relative positions in multiple
fields5Computing geometrical information of
isosurfaces! Bajaj et al. 1997

! Computed some values for each isosurface as a whole! length, 2D area, 3D volume etc.

! Carr et al. 2010

! Computed geometrical values for isosurface components! For contour trees! Ours! Compute positions for manifold domains! Easy implementation! Insert few lines to existing Reeb graph implementation (of [Parsa 2012])
6

－31－

!

The centroid network! Composed of! Vertex set: centroids at topological events in
isosurface! Edge set: trace of centroids! Gaps in the network occur wherevercontours split or merge! Topology: from Reeb graph! Encode centroid positions on the graph

Yellow lines are the network

7

Computing centroid in the triangle! Split a triangle into two small triangles by the line ! " #$%! Compute centroids for each small triangle for parallelization
"!"&

"!"&

"!"#

#!"&

"!"#

"!$%

Trace of centroid bychanging isovalues

－32－

#!"&

"!$%
8

Preimage graph: encodes contour
connectivity !" #! $ % $ " #" &

!

$

"-

"+

"(

",

"*
")

[Parsa!2012]! Nodes"&!&" ' &#&" ' &#&$ ' (&#&%(' (&&&%(! Arcs " ) &!&#&" ' &#&"&$ ' (&&&#&%(*! Keep the transition of connectedcomponents of contours by changing +"'9

Alteration to pseudo code
CreateNetwork(mesh) vertices are sorted in the increasing order.
Preimage_graph is the graph which has vertices but no edges.
Network is the graph which has no vertices and edges.
Triangles = []# initializationfor vertex in mesh:
G = 0, S = 0
Lc = LowerComps(vertex, Preiamge_graph, Triangles)
if Lc != []:ComputeCentroid(vertex, Lc, Triangles)

Subroutines of creating theReeb graph [Parsa, 2012]

UpdatePreimage(vertex, Preimage_graph, Triangles)
Uc = UpperComps(vertex, Preimage_graph, Triangles)
if Uc != []:ComputeCentroid(vertex, Uc, Triangles)

UpdateNetwork(Lc, Uc, Network)
Return Network
10

－33－

UpdatePreimage(vertex, Preimage_graph, Triangles)

Compute and update preimage graph

Triangles.add(t)[Parsa!2012]11

Current situation! I’m implementing now! VTK + Python

12

－34－

Thinking multi-objective
optimization problem
topologicallyStarting from hyperparameters
Likun Liu, Kenji Ono, Daisuke Sakurai(Kyushu University)

Overviewu

Theory basisuSimplicial problemuData abstractionuSpace partitioninguProposed MethoduCurrent progress

2

－35－

Simplicial problemuDefinitionuuu

A multi-objective optimization problem is 𝐶𝐶 ! simplicial if the Pareto set
and the Pareto front are 𝐶𝐶 ! diffeomorphic to a simplex and, under the
𝐶𝐶 ! diffeomorphisms, each face of the simplex corresponds to the Pareto
set and the Pareto front of a subproblem, where 0 ≤ 𝑟𝑟 ≤ ∞𝐶𝐶 ! is a topology space that met certain criteria

Diffeomorphic: The mapping is bidirectional and is r-th orderdifferentiable3

Simplicial Optimization Problems[1]: model parameters trained: interpolate model without trainingu

For simplicial problems, training results can be approximatedu

No re-training requiredu

The determination of the simplicial problem[1]
K. Kobayashi, N. Hamada, A. Sannai, A. Tanaka, K. Bannai, and M. Sugiyama, “Bézier Simplex Fitting: Describing
Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization,” Proc. AAAI Conf. Artif. Intell.,vol. 33, no. 01, pp. 2304–2313, 2019.

－36－

4

Determination – None Simplicial[2]uDefination
u
u
u
u

u

Let ∆#$"≔ 𝑤𝑤 ∈ 0,1 # ∑% 𝑤𝑤% = 1}

1) Exist Homeomorphic Mapping ∅: ∆#$"→ Χ ∗ 𝑓𝑓 s. t. ∅ ∆& = Χ ∗ 𝑓𝑓& (∀𝐼𝐼)
2) Restrict Mapping 𝑓𝑓|'∗ (" → ℝ# ∀𝐼𝐼 is embeded in 𝐶𝐶 )It’s difficult to prove if ∅ exist, 1) can be interpreted as:uuu

Let Χ ∗ 𝑓𝑓 be the pareto set of function 𝑓𝑓 = (𝑓𝑓", … , 𝑓𝑓# )Χ ∗ 𝑓𝑓& and ∆|&|$" is homeomorphic
𝑖𝑖𝑖𝑖𝑖𝑖Χ ∗ 𝑓𝑓& ∩ 𝑖𝑖𝑖𝑖𝑖𝑖Χ ∗ 𝑓𝑓+ = ∅If Homology group 𝐻𝐻∗ (𝐴𝐴) and 𝐻𝐻∗ (𝐵𝐵) are not isomorphism, tha space 𝐴𝐴 and 𝐵𝐵
are not homeomorphicuProve that exist homology 𝑞𝑞 𝑠𝑠. 𝑡𝑡. 𝐻𝐻, Χ ∗ 𝑓𝑓+≇ 𝐻𝐻, ∆|&|$"

5[2]

濱田 直希, “多目的最適化の解集合のトポロジーの検定法,” in 日本応用数理学会2020年年会, 2020, pp. 267–268.

Determination – Simplicial[2]uLet 𝑀𝑀 be a compact 𝐶𝐶 # manifoldsuuu

If 𝑀𝑀 and 𝜕𝜕𝑀𝑀 are both simply connected space
𝐻𝐻, 𝑀𝑀 ≅ 𝐻𝐻, ∆- ≅ F

𝕫𝕫 (𝑞𝑞 = 0)0 (𝑞𝑞 ≠ 0)

𝑀𝑀 and ∆- are Homeomorphic

6[2]

濱田 直希, “多目的最適化の解集合のトポロジーの検定法,” in 日本応用数理学会2020年年会, 2020, pp. 267–268.

－37－

Space partitioning
Data structure

Applicablesituations

Grade of N

Time complexity

Dynamically
updatable

Quad Tree

Scene
management
without height,
rendering

Nmuber ofobjects

𝑂𝑂(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛)

Yes

Large (Depend on area
size and number ofobjects)

Oct TreeScenemanagement,
rendering

Nmuber ofobjects

𝑂𝑂(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛)

Yes

Large (Depend on area
size and number ofobjects)

BVH TreePhysics,
rendering (almost
anything)

Number ofobjects

𝑂𝑂(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛)

Yes

Medium (Depend on
the number ofobjects)

BSP Tree

Editor, complex
scene

Number of planesNo

Large (Depend on thenumber of planes)

Accerated BSP
tree with BVH

Editor, complex
scene

Number of planes

𝑂𝑂(𝑛𝑛𝑙𝑙𝑙𝑙𝑙𝑙# 𝑛𝑛)

Partially Yes

Large (Depend on area
size and number ofobjects)

𝑂𝑂(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛)

Storage Required

7

Validation – Staring from MLuu

Generally, in algorithms, we have parameters that controlled the behavior of the algorithmu

SVM -> Kernelu

K-NN -> ku

Loose Oct/Quad Tree -> expansion factor k

The raise of the ML has made the selection of the hyperparameters importantuRunning time and storage costsuModel qualityu

Inference accuracyu*Model Architecture

8

－38－

Manual hyperparameter tuninguu

To set hyperparameters manually…
uType of hyperparametersuTraining erroruGeneralization error

When plotting generalization error regarding a hyperparameter

Model capacity:
The ability to represent
complex functions of themodel
Effective capacity:
The capacity that
actually used torepresent functions

9

Examplesuu

Number of hidden unit :uIncrease the hidden unit can increase the representational
capacity of the modelu

Increase the time and memory cost

Learning rate:uControls the effective capacity of the modelu

Have an optimal value for optimization problemu* If you only have time for one hyperparameter,
Go for Learning Rate

10

－39－

From Model-based methods
u The theory:u

The search for good hyperparameters can be cast as an optimization problemu

The decision variables are the hyperparametersu

The cost to be optimized is the validation set error that results from training using these
hyperparametersu MethodsuGradient-based methodsuBayesian Methods

11

Thanks forlistenging
12

－40－

Topological invarinats ofsmooth map germs ofmanifolds with boundary
Fiber

Topology

Meets

Applications

2

byTakahiro Yamamoto (Tokyo Gakugei University)
December 3, 2021

Contents
1. Introduction
A bit my self and back ground in Singularity theory

2. Invariants
α = σcusp − σdefB2

β = σ∂cusp + σB2

γ =#B2 mod 2

are introduced and they are calculated for some examples

3. Conclusions and Future works

In this talk, ∀ mfd.s and ∀ maps are smooth of classes C∞ unlessotherwise stated.
For a smooth map f : M → Rk , denote by S(f ) the set of singularpointsS(f ) = {p ∈ M | rankdfp < min{dim M, k}}.

－41－

Introduction

Introduction

1/23

Takahiro Yamamoto
INTEREST: To study TOPOLOGY and GEOMETRY of spaces M, inparticular manifolds, by using singularity of smooth maps
f : M → Rk , (dim M ≥ k).
KEY WORDS: Singularity of smooth maps, Stable Maps, Generic
Maps, Singular Fibers
FAVORITE FORMULAS: [Kamenosono -Y(’08)] For a stable map
f : Σg → S2 , we havecg = N + + (1 + I) − m(f ).
2[Saeki -Y(’06)] For a stable map f : M → R3 of a closed oriented
4-mfd, we haveσ(M) = ||III8 (f )||.

Introduction

2/23

－42－

Joint Works with Computer Sci.
(By O. Saeki, S. Takahashi, D. Sakurai, H. Wu, K. Kikuchi, H. Carr, D. Duke, Y)(1) The Impact of Applications on Mathematics, Proceedings of Forum of
Mathematics for Industry 2013, 51–65.
(2) IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS,
VOL. 22, NO. 1, 945–954, 2016.
! Contributed by classifying singular fibers of stable maps M3 → R2 of
cpt 3-mfds with boundary.
(D. Sakurai, Y)(3) Topological Methods in Data Analysis and Visualization VI, 181–196,
2021.
! Contributed by constructing the theory and calculating example byusing "jcnet".
(By R. Hagiwara, R. Ota, N. Hamada, D. Sakurai, Y)(4) Explicitly Multimodal Benchmarks for Multi-Objective Optimizations

Introduction

3/23

Back ground in Singularity thoery
Classifying mappings or map germs is an important andfundamental problem in Singularity theory.
! In order to classify mappings or map germs, we need INVARIANTS. It
implies that studying invariants are important.

For a given map germs, invariants which are calculated byperturbing the map germs stably are obtained in SOME cases:
Milnor (’68)
g : (Cn , 0) → (C, 0),
Fukuda-Ishikawa (’87) g : (R2 , 0) → (R2 , 0),..
.
Saeki (’07)
Y (’18, 19)

g : (Rn , 0) → (R2 , 0), andg : (Hn , 0) → (R2 , 0) ← Today!!

! It is NOT easy obtaining stable map germs by deforming {gt } a given
map germ g at the formula level, gt = g + tf for example.
Introduction

4/23

－43－

TodayFor map germs of 3-mfd.s with ∂ into R2 , there are three invariants
α = σcusp − σdefB2

β = σ∂cusp + σB2

γ =#B2 mod 2

calculated by perturbing the map germs stably.
In this talk, we show that for a given map germ, we can perturb the
map germ stably by using the fiber topology and its visualization
software. Then, the invariants are actually calculated by using theobtained stable map germs.
! By calculating α, β, γ in some cases, we estimate how strong
they are.

Introduction

5/23

Invariants

Invariants

6/23

－44－

Map germs (Hn, 0) → (R2, 0)
Denote by Hn the upper half subset of Rn , namely
Hn := {(x1 , . . . , xn−1 , y) ∈ Rn | y ≥ 0},
where {y = 0} corresponds to the bdry of our mfd, whose interior
is taken to be the part of Rn with y > 0.

Equivalence relation among map germs (Hn , 0) → (Rk , 0)g, g : (Hn , 0) → (Rk , 0): map germs.
g ∼ g : topologically B+ -equivalentdef⇔ ∃Φ : (Hn , 0) → (Hn , 0): homeo. preserving {y = 0}, {y > 0},∃ψ : (Rk , 0) → (Rk , 0): homeo. preserving orientation
s.t. g ◦ Φ = ψ ◦ g .

Invariants



7/23

Stable maps
f : M → Rk : smoothdef
f is stable ⇔ ∃N(f ) ⊂ C∞ (M, Rk ): a nbd of f s.t. ∀f  ∈ N(f ), f and f are equivalent, where f and f  are said equivalent if∃Φ : M → M: diffeo. preserving ∂, ∃ψ : Rk → Rk : diffeo. s.t. they
make the following diagram commutesΦM −−−−→ M




f
f
ψ

Rk −−−−→ Rk .

def

A map germ g : (Hn , 0) → Rk is stable ⇔ a representative U → Rk
of g is stable.
A stable map f : M3 → R2 is characterized by the following two(local and global) conditions.
Invariants

8/23

－45－

Stable maps M3 → R2 (Local cond. )iff

A C∞ map f : M3 → R2 is stable ⇔ f satisfies the conditions below:(1) For ∀p ∈ IntM, (f , p) is equiv. to
p is a reg. pt, or(x1 , x2 )(x1 , x2 , y) → (x1 , x22 ± y 2 )p: a def/indef fold pt,(x1 , x23 + x1 x2 + y 2 ) p: a cusp pt.
For ∀p ∈ ∂, (f , p) is equiv. to(x1 , x2 )(x , x 2 ± y)
1 2
(x1 , x2 , y) →(x1 , x23 + x1 x2 + y)(x , x 2 ± y 2 + x y)
1 2
1

p is a reg. pt of f |N(∂) , orp: a def/indef ∂fold pt,
p: a ∂cusp pt,
p: a def/indef B2 pt,

where IntM ⇔ {y > 0} and ∂ ⇔ {y = 0}.
Invariants

9/23

Stable maps M3 → R2 (Global cond. )iff

A C∞ map f : M3 → R2 is stable ⇔ f satisfies the conditions below:(1) . . .
(2) For ∀q ∈ f (S(f ) ∪ S(f |∂ )), (f (S(f ) ∪ S(f |∂ )), q) is equivalent to
one of the figures below:

1

2

3

where solid lines ⇔ f (S(f )) and dotted lines ⇔ f (S(f |∂ )).
! B2 is fold (or fold) if we ignore ∂ (resp. restrict f to ∂).
! For a stable map f : M3 → Q2 , the pre-image f −1 (q) of a reg value
consists of finite number of circles and arcs.

Invariants

10/23

－46－

Local degeneration of level sets
∅
(1)

(2)

(3)

(5)

(6)

∅
(4)

∅

(7)

(8)

Invariants

11/23

Fibers of smooth maps
For a C∞ map f : M → Q and q ∈ Q, the map germf : (M, f −1 (q)) → (Q, q)
is called the fiber over q ∈ Q.
Fibers fi : (Mi , fi−1 (qi )) → (Qi , qi ) (qi ∈ Qi ) are C0 -equiv if∃Ui ⊂ Qi : a nbd of qi ∈ Q, (i = 0, 1),∃ Φ : (f0−1 (U0 ), f0−1 (r0 )) → (f1−1 (U1 ), f1−1 (q1 )): homeo preserving ∂∃ ϕ : U0 → U1 : homeo with ϕ(q0 ) = q1
s.t. they make the following diagram commutative:Φ(f0−1 (U0 ), f0−1 (q0 )) −−−−→ (f1−1 (U1 ), f −1 (q1 ))




f0 
f 1
(U0 , q0 )

ϕ

−−−−→

Invariants(U1 , q1 ).

12/23

－47－

Fibers of M3 → Q2 of 3-mfd.s with ∂
K

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

B

L

K

0

211

1

2

3

4

5

6

7

8

9

a

Invariants

C

13/23

Invariants
THM 1 [Y’18] (n = 3)

g : (H3 , 0) → (R2 , 0): a "cone-like"map germ Then,for a stable map f : U → R2 obtained by perturbing a
representative U → R2 of g stably,(1) α(f ) = σcusp (f ) − σdefB2 (f ) (2) β(f ) = σ∂cusp (f ) + σB2 (f )(3) γ(f ) =#B2 (f ) mod 2
are topological B+ -invariants, where σ∗ = "the alg. # of
singularities of type ∗".

n2! For a generic map germ g : (R , 0) → (R , 0) and a generic hyperplane
H ⊂ Rn , a map germ g restrict to the upper half space of H is "conelike".

Invariants

14/23

－48－

Signs of cusps and ∂cusps, B2
f : M → R2 : a stable map of a cpt 3-mfd w/ ∂p: a cusp or a ∂cusp, a B2 point of f
Orient R2 in the counter-clockwise manner.
If the image around f (p) are in the left hand side of
indef

def

+1

indef

def

def

indef

-1

def

+1

indef

-1

+1

-1

then sign(p) = +1; otherwise, sign(p) = −1.
σcusp (or σ∂cusp , σdefB2 σB2 ) is the total of signs of cusps (resp.
∂cusps, definite B2 , B2 ) of f .

Invariants

15/23

Example 1: B3+ (well-known)! In the following, gi is perturbed by a software "jcnet"which draws thediscriminant (and the Reeb space) of gi . Furthermore "jcnet"can perturb agiven map germ by mouse interaction.

g1 = (x1 , x22 + y 3 + x1 y): B3+
G

indef B2PERTURB!!

PERTURB!!

def B2Cusp(α, β, γ)(g1 ) = (0, −1, 1) = (−1 − (−1), −1, 1)
Invariants

16/23

－49－

Example 2: B3− (well-known)
g2 = (x1 , x22 − y 3 + x1 y): B3−
G

PERTURB!!

Cusp

PERTURB!!

indef B2
def B2(α, β, γ)(g2 ) = (1, −1, 1) = (−(−1), −1, 1)!

It imples that (α, β, γ) distinguish B3+ and B3− singularities.

Invariants

17/23

Example 3: g3 = (x12 − x22 + y, 2x1x2)
g3 = (x1 +√−1x2 )2 + y = (x12 − x22 + y, 2x1 x2 )G

dcuspsPERTURB!!
indef B2(α, β, γ)(g3 ) = (0, (+1 · 2 − 1) + (−1), 1) = (0, 0, 1)!

It imples that (α, β, γ) distinguish B3+ and B3− , g3 .

Invariants

18/23

－50－

Example 4: g = (x − 3x x + y, 3x x − x )4g4 = (x1 +

√

3
1

2
1 2

2
1 2

3
2

−1x2 )3 + y = (x13 − 3x1 x22 + y, 3x12 x2 − x23 )G

dcuspsPERTURB!!

indef B2s(α, β, γ)(g4 ) = (0, (+1 · 2 − 1 · 2) + (+1 − 1), 0) = (0, 0, 0)!

It imples that (α, β, γ) distinguish B3+ and B3− , g3 , g4 .

Invariants

19/23

Example 5: g5 = (x12 − x22 + y 2, 2x1x2)
g5 = (x1 +√−1x2 )2 + y 2 = (x12 − x22 + y 2 , 2x1 x2 ) : (X 3 , 0) → (R2 , 0)G

dcuspsPERTURB!!
B!!

indef B2s(α, β, γ)(g5 ) = (0, (+1 · 1 − 1 · 4) + (+1 · 3 − 1 · 0), 1) = (0, 0, 1)! (α, β, γ)(g3 ) = (0, 0, 1). Actually, g3 and g5 are topologically B+
equivalent.

Invariants

20/23

－51－

Conclusions and Future Works

Conclusions and Future Works

21/23

Conclusion
In this talk, we show that invariants α, β, γ are actually calculated
via stable map germs obtained by using the fiber topology and its
visualization software.
Q Can the calculation process be automated by computer sci.?
Nemely,perturb stably

a map germ g −−−−−−−−→
calculate(α, β, γ)(g)In particular, if perturbing stably process are automated, then,
HIDDEN invariants might be discovered by studying a lot of
examples!!

Conclusions and Future Works

22/23

－52－

Future works
On the other hand, we have also obtained the invariants below:
THM 2 [Y’19] (n ≥ 2)

g : (Hn , 0) → (R2 , 0): a "cone-like"map germ Then,for a stable map f : U → R2 obtained by perturbing a
representative U → R2 of g stably,γ(f ) = #B2 (f ) mod 2
is a topological B+ -invariant.

! The invariant γ of a map germ g is calulated if we have asoftware like "jcnet"!!

Conclusions and Future Works

23/23

－53－

Reeb Diagram and Visualization of
Monodromy
Osamu Saeki(Kyushu University)

December 3, 2021
Fiber Topology Meets Applications 2
IMI, Kyushu University§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond§1. Fiber－54－

Setting§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

N n , M m : C ∞ manifolds,
f : Nn → MmnUsually, N is a bounded domain in Rn .
Typically M m = Rm , and in this casef = (f1 , f2 , . . . , fm )C ∞ map

multi-variate data

We assume f is generic (C ∞ stable, C 0 stable, finite codimension, etc.)
We are interested in the topology of ﬁbers f −1 (y), y ∈ M m .
Generically, dim f −1 (y) = n − m. We usually assume n ≥ m.

3 / 23

Example of ﬁbers§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

N

R

f

We can grasp global feature of data by chasing fibers (or level sets).
We have some singular ﬁbers (or critical level sets) where topologicaltransitions of ﬁbers occur.

4 / 23－55－

Singular points and Jacobi set§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

f : N n → M m (n ≥ m)

C ∞ mapDeﬁnition 1.1 For a point p ∈ N n , consider the diﬀerential
dfp : Tp N n → Tf (p) M m(linear map associated with the Jacobian matrix of f ).
Singular point is a point p ∈ N n with rank dfp < m.
The set of singular points
J(f ) = {p ∈ N n | rank dfp < m}is called the Jacobi set of f .
Generically, dim J(f ) = m − 1.
5 / 23

Jacobi set image§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Jacobi set image f (J(f )) divides the range M m into some regions.

Topology of fibers changes along the Jacobi set image.
Singular ﬁber is a fiber f −1 (y) with y ∈ f (J(f )).
It is important to know topological changes of fibers near a singular fiber.
6 / 23－56－

A formulation§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Mathematically, a ﬁber is, in fact, NOT just a subset in the domain N n , but
a MAP around a pre-image.
fi : Nin → Mim C ∞ maps, i = 0, 1
For points yi ∈ Mim , i = 0, 1, fibers over y0 and y1 are equivalent if∃commutative diagram∼=(f0−1 (U0 ), f0−1 (y0 )) −−−→ (f1−1 (U1 ), f1−1 (y1 ))
⏐
⏐
⏐
⏐f
f0 �
�1
(U0 , y0 )

∼
=

−−−→

(U1 , y1 )for some neighborhoods yi ∈ Ui ⊂ Mim , i = 0, 1.

In particular, equivalent fibers have the same topological transitions of
nearby fibers.
7 / 23

Classiﬁcation of singular ﬁbers§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

For certain dimensions, we can classify singular ﬁbers of generic maps.
Example 1.2 Classification results for (n, m) with n − m = 1.
For simplicity, we assume the domain N n is orientable.
We will ignore regular fiber components.
1. (n, m) = (2, 1) [Folklore]κ = 1 (codimension)

2. (n, m) = (3, 2) [Kushner–Levine–Porto, 1984]
κ=1

κ=2

8 / 23－57－

Singular ﬁbers for (n, m) = (4, 3)§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

3. (n, m) = (4, 3) [S.]
κ=1
κ=2

κ=3

4. (n, m) = (5, 4) [Yamamoto–S.]
9 / 23§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond§2. Reeb Space－58－

Reeb space§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

For a C ∞ map f : N n → M m , n > m, the space Rf obtained by contracting
each connected component of a fiber to a point is called the Reeb space off [Edelsbrunner–Harer–Patel, 2008].
Nn

Mm
f

f¯

qf

Rf

11 / 23

Local structures of Reeb spaces§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Classification of fibers⇒ Characterization of local structures of Reeb spaces
Example 2.1 1. (n, m) = (2, 1)

2. (n, m) = (3, 2) [Kushner-Levine-Porto, 1984]

3. (n, m) = (4, 3) [Hiratuka, 2001]
12 / 23－59－§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond§3. Case with n − m = 2 andBeyondRegular ﬁbers for n − m = 2§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

How about the case n − m = 2 ?
We assume the domain N n is orientable, compact, and w/o boundary.
Regular fibers are closed orientable surfaces.
ΣggΣg : closed orientable surface of genus g

14 / 23－60－

Reeb graph for (n, m) = (3, 1)§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Case with (n, m) = (3, 1).
Σg
Σg+1
S 2 = Σ0g
Σg1 +g2
Σg2

Σg1

g1

g2
15 / 23

Reeb space for (n, m) = (4, 2)§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Each regular “stratum” has its own label (genus of the corresponding regularfiber component). [Furuya, 1986]
g2

0

g3
g1
g1 = g2 + g3

g+2
g+1

g+1

g1

g

g3
g2

g+1

g1

g3

g2

g2
g1

g1

g1 + 1
g1 = g2 + g3gg

g2 + 1

g1 + 1
g1 = g2 + g3

0

g

g

g3
g4

g1 + g4
g2 = g3 + g4
g = g1 + g2

g+1
16 / 23－61－

0-th Homology§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Reeb space describes the connected components of ﬁbers and their“adjacencies”.

y1 y y2I
H0 (f −1 (y1 )) −−−→ H0 (f −1 (I)) ←−−− H0 (f −1 (y2 ))
⏐
⏐
⏐
⏐
⏐
⏐
∼
∼
∼
=�
=�
=�
H0 (f −1 (y))
⏐
∼⏐
=
�

Z⊕Z

Z

Z

17 / 23

Reeb diagram§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

This idea can be extended to homology groups of any dimension (orhomotopy groups, if you want).
Hi (f −1 (y1 )) → Hi (f −1 (y)) ← Hi (f −1 (y2 ))abαHi (a)Hi (α)

Hi (b)βγ
Hi (β)cdδHi (c)

Hi (δ)Hi (d)Hi (γ)=⇒ Notion of Reeb diagram.
18 / 23－62－

Category ?
§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

K1
H2
K2

H1
K4

G
H3

H4
K3
H1

K2

K1

G
H2

In a certain categorical formulation of a Reeb space, this can be considered
to be a functor.
Or any formulation using the notion of sheaves?
19 / 23

A formulation§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Reeb category
Object: each stratum (or simplex) of Rf
Morphism : τ ≺ σ ⇐⇒ τ ← σThen, Reeb diagram is a covariant functor from Reeb category to thecategory of groups.
Remark 3.1 This makes sense if each stratum is contractible.

20 / 23－63－

Monodromy§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

If a stratum is not contractible, we need to consider monodromy.
Suppose π : E → B is a C ∞ fiber bundle with fiber S.
MCG(S) = π0 (Diﬀ + (S)) mapping class group
Associated to π is the monodromy π1 (B, b0 ) → MCG(S).
This measures the “twist” of the fibers along a loop in the base B.

S

E
π
B

b0

21 / 23

Monodromy in Reeb diagram§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Given a generic map f : N n → M m , we can subdivide M m (or the Reeb
space Rf ) so that each stratum is contractible.
In this case, the monodromy is hidden in the Reeb diagram.

K1

H1

K2

H2
K3

H6

H3
K4

K6
H5

K5

H4

π1 (B, b0 ) → MCG(S) → Aut(H∗ (S))

Problem 3.2 Formulate all these, including monodromy! Sheaf? Category
theory? How to compute Reeb diagram and/or monodromy?
22 / 23－64－

Ending§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Thank you!

23 / 23－65－

「マス・フォア・インダストリ研究」シリーズ刊行にあたり
本シリーズは，平成 23 年 4 月に設立された九州大学マス・フォア・インダストリ研究所(IMI)が，平成 25 年 4 月に共同利用・共同研究拠点「産業数学の先進的・基礎的共同研究拠点」として，文部科学大臣より認定を受けたことにともない刊行するものである．本シ
リーズでは，主として，マス・フォア・インダストリに関する研究集会の会議録，共同研
究の成果報告等を出版する．各巻はマス・フォア・インダストリの最新の研究成果に加え，その新たな視点からのサーベイ及びレビューなども収録し，マス・フォア・インダストリの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Fiber Topology Meets Applications 2
マス・フォア・インダストリ研究 No.24, IMI, 九州大学
ISSN 2188-286X
発行日

2022 年 3 月 15 日

編

集

櫻井大督，佐伯発

行

九州大学マス・フォア・インダストリ研究所

修，高橋成雄，Hamish Carr，山本卓宏，濱田直希〒819-0395 福岡市西区元岡 744
九州大学数理・IMI 事務室
TEL 092-802-4402

FAX 092-802-4405

URL https://www.imi.kyushu-u.ac.jp/
印

刷

城島印刷株式会社〒810-0012 福岡市中央区白金 2 丁目 9 番 6 号
TEL 092-531-7102

FAX 092-524-4411

修シリーズ既刊
Issue

Author ／ Editor

Title

Published

マス・フォア・インダストリ
研究 No.1

穴田 啓晃
安田 貴徳
Xavier Dahan
櫻井 幸一

Functional Encryption as a Social Infrastructure and
Its Realization by Elliptic Curves and Lattices

26 February 2015

マス・フォア・インダストリ
研究 No.2

滝口
藤原

Collaboration Between Theory and Practice inInverse Problems

12 March 2015

マス・フォア・インダストリ
研究 No.3

筧

非線形数理モデルの諸相：連続，離散，超離散，その先Various aspects of nonlinear mathematical models: continuous, discrete, ultra-discrete, and beyond

24 March 2015

マス・フォア・インダストリ
研究 No.4

穴田
安田
櫻井
寺西

啓晃
貴徳
幸一
勇

Next-generation Cryptography for
Privacy Protection and Decentralized Control andMathematical Structures to Support Techniques

29 January 2016

マス・フォア・インダストリ
研究 No.5

藤原
滝口

宏志
孝志

Mathematical Backgrounds and Future Progress ofPractical Inverse Problems

1 March 2016

マス・フォア・インダストリ
研究 No.6

松谷
佐伯
中川
上坂
濵田

茂樹
修
淳一
正晃
裕康

結晶のらせん転位の数理

10 January 2017

マス・フォア・インダストリ
研究 No.7

滝口
藤原

孝志
宏志

Collaboration among mathematics, engineering andindustry on various problems in infrastructure andenvironment

1 March 2017

マス・フォア・インダストリ
研究 No.8

藤原
滝口

宏志
孝志

Practical inverse problems based on interdisciplinary
and industry-academia collaboration

20 February 2018

マス・フォア・インダストリ
研究 No.9

阿部
高島
縫田
安田

拓郎
克幸
光司
雅哉

代数的手法による数理暗号解析
Workshop on analysis of mathematical cryptography
via algebraic methods

1 March 2018

マス・フォア・インダストリ
研究 No.10

阿部
落合
高島
縫田
安田

拓郎
啓之
克幸
光司
雅哉

量子情報社会に向けた数理的アプローチ
Mathematical approach for quantum informationsociety

26 December 2018

孝志
宏志

三郎

（

）

IssueAuthor ／ Editor

Title

Published

マス・フォア・インダストリ
研究 No.11

松谷
佐伯
中川
濵田
上坂

茂樹
修
淳一
裕康
正晃

結晶転位の先進数理解析
Advanced Mathematical Investigation for Dislocations

7 January 2019

マス・フォア・インダストリ
研究 No.12

滝口孝志

Non-destructive inspection for concrete structuresand related topics

13 February 2019

マス・フォア・インダストリ
研究 No.13

宇波
長野
吉岡
田上
白井

耕一
智絵
秀和
大助
朋之

数理農学における時系列データのモデル化と解析
Modeling and Analysis of Time Series Data in MathAgro Sciences

28 February 2019

マス・フォア・インダストリ
研究 No.14

佐久間 弘文
大津 元一
小嶋 泉
福本 康秀
山本 昌宏
納谷 昌之

ドレスト光子に関する基礎的数理研究

18 March 2019

マス・フォア・インダストリ
研究 No.15

松谷
佐伯
中川
濵田
富安

結晶の界面，転位，構造の先進数理解析

2 December 2019

マス・フォア・インダストリ
研究 No.16

Takuro Abe
Yasuhiko Ikematsu
Koji Nuida
Yutaka Shikano
Katsuyuki Takashima
Masaya Yasuda

Quantum computation, post-quantum cryptography
and quantum codes

17 January 2020

マス・フォア・インダストリ
研究 No.17

河村
津曲
西澤
溝口

彰星
紀宏
弘毅
佳寛

代数・論理・幾何と情報科学―理論から実世界への展開

10 February 2020

マス・フォア・インダストリ
研究 No.18

Takashi Takiguchi

New technologies for non-destructive and noninvasive inspections and their applications

21 February 2020

マス・フォア・インダストリ
研究 No.19

Hirofumi Sakuma
Motoichi Ohtsu
Masayuki Naya
Izumi Ojima
Yasuhide Fukumoto

Basic mathematical studies on dressed photon
phenomena

19 March 2020

茂樹
修
淳一
裕康
亮子

マス・フォア・インダストリ
研究 No.20

松谷 茂樹
井上 和俊
加葉田雄太朗
佐伯
修
垂水 竜一
内藤 久資
中川 淳一
濵田 裕康

材料科学における幾何と代数 I

24 November 2020

マス・フォア・インダストリ
研究 No.21

Daisuke Sakurai
Shigeo Takahashi
Naoki Hamada
Osamu Saeki
Hamish Carr
Takahiro Yamamoto

Fiber Topology Meets Applications

10 March 2021

マス・フォア・インダストリ
研究 No.22

巴山
中丸
大垣

竜来
啓
真二

数理的生成手法による CG とデジタルファブリケー
ション

15 March 2021

松谷
井上
落合
佐伯
垂水
内藤
中川
濵田
松江

茂樹
和俊
啓之
修
竜一
久資
淳一
裕康
要

材料科学における幾何と代数 II

11 November 2021

マス・フォア・インダストリ
研究 No.23

Institute of Mathematics for Industry
Kyushu University

九州大学マス・フォア・インダストリ研究所〒819 - 0395 福岡市西区元岡 744

https://www.imi.kyushu-u.ac.jp