2024 Japan-Taiwan Joint Workshop on
Numerical Analysis and Inverse Problems

Date: 9 (Sat.), 10 (Sun.), March, 2024
Venue: Room 111 (総合講義室 2) at Faculty of Engineering
Integrated Research Bldg. Kyoto University.

Program
9 (Sat.), March
10:50-11:00 Opening AddressJenn-Nan Wang and Hiroshi Fujiwara(Chair : Masato Kimura; Kanazawa University)
11:00-11:40 Jenn-Nan Wang (王 振男)
Bayesian approach to the inverse medium scattering problem
11:40-12:20 Hiroshi Fujiwara (藤原 宏志)
Numerical feasibility of optical tomography by the stationary radiative transport equation(Chair : Satoshi Taguchi; Kyoto University)
14:00-14:40 Masato Kimura (木村 正人)
Energy dissipation structure of the Zener-type viscoelastic wave equation and its applications
14:40-15:20 Shih-Hsien Yu (尤 釋賢)
Navier-Stokes equations in gas dynamics: Greens function, singularity, and well-posedness(Chair : Jenn-Nan Wang; National Taiwan University)
15:40-16:20 I-Kun Chen (陳 逸昆)
Geometric effects on $W^{1,p}$ regularity of the stationary linearized Boltzmann equation
16:20-17:00 Tomoyuki Miyaji (宮路 智行)
Traveling waves and the Neimark--Sacker bifurcation with discrete rotational symmetry

17:00-18:00Discussion

10 (Sun.), March(Chair : Daisuke Kawagoe; Kyoto University)
10:50-11:10 Masaki Imagawa (今川 真城)
On a boundary value problem of a first-order PDE and its regularized problems

11:10-11:30 Kousei Takahashi (高橋 幸聖)
Numerical computation for transonic potential flow with particular solutions of Chaplygin's
equation
11:30-11:50 Takuma Tomita (富田卓磨)
Numerical analysis of a rarefied gas flow past a circular disk: peculiar behavior near the sharpedge
11:50-12:00 Group Photo(Chair : Hiroshi Fujiwara; Kyoto University)
13:30-14:10 Hiroshi Isakari (飯盛 浩司)
Towards a topology optimisation of wave devices with wide working bandwidth
14:10-14:50 Yu-Chen Shu (舒 宇宸)
Numerical investigations to time-fractional Allen-Cahn equation(Chair : I-Kun Chen; National Taiwan University)
15:10-15:50 Pu-Zhao Kow (邱 普照)
Some free boundary methods and their application to inverse scattering problems
15:50-16:30 Hiroshi Takase (髙瀬 裕志)
Inverse source problems for wave equations with inverse square potentials
16:30-16:40 Closing Address

Yuusuke Iso

Organizers
Jenn-Nan Wang (National Taiwan University)
Yuusuke Iso (Kyoto University)

王 振男 (臺灣大學)

磯 祐介 (京都大学)

Hitoshi Imai (Doshisha University)

今井 仁司 (同志社大学)

Hiroshi Fujiwara (Kyoto University)

藤原 宏志 (京都大学)

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Bayesian approach to the inverse medium scattering
problemJenn-Nan Wang1
1. Institute of Applied Mathematical Sciences, National Taiwan University,
Taipei, 11529, Taiwan
E-mail:jnwang@math.ntu.edu.tw
Abstract
In this talk, I plan to discuss the inverse scattering problem of determining an
unknown refractive index from the far-field measurements using the Bayesian approach.
Our aim is to prove the Bernstein-von Mises theorem for this inverse scattering problem
when the unknown coefficient is in a class of piecewise constant functions. We also
provide numerical simulations based on the MCMC algorithm combining with a Gibbstype sampling and MALA to verify the Bernstein-von Mises theorem. This talk is basedon joint works with Takashi Furuya.

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Numerical Feasibility of Optical Tomography
by the Stationary Radiative Transport Equation
Hiroshi Fujiwara1
1. Graduate School of Informatics, Kyoto University,
Kyoto, 606-8501 Japan
E-mail:fujiwara@acs.i.kyoto-u.ac.jp
Abstract
We study reconstruction of optical properties of biological tissue by using nearinfrared light which is scattered and absorbed during propagation. In this talk we
use the stationary radiative transport equation (RTE) as a mathematical model for
NIR-light propagation, and show numerical reconstruction of attenuated coefficientsin RTE from simulated data. The discontinuity of the solution to RTE induced by
the boundary condition plays an important role in the proposed approach. We also
introduce a numerical scheme for the boundary value problem of RTE by the piecewise
constant approximation in order to detect discontinuities of the solution efficiently.
This talk is based on a collaboration with I.-K. Chen (National Taiwan University),
D. Kawagoe (Kyoto University), and N. Oishi (Kyoto University).

References[1] I.-K. Chen, H. Fujiwara, and D. Kawagoe, Tomography from scattered signals obeying
the stationary radiative transport equation, Mathematics for Industry 37 (2023) pp.27–
46

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Energy dissipation structure of the Zener-type
viscoelastic wave equation and its applications
Masato Kimura 1
1. Faculty of Mathematics and Physics, Kanazawa University,
Kanazawa, 920-1192 Japan
E-mail:mkimura@se.kanazawa-u.ac.jp
Abstract
We consider a Zener-type viscoelastic wave equation in a bounded domain, which
consists from n Maxwell-type viscoelastic components connected in parallel and a
dumping term. Based on the observation that a physically natural energy dissipation
equality holds for a smooth solution, we develop the following mathematical analysis for
the Zener-type viscoelastic wave equation: 1) A time discretized model is constructed
and its energy dissipation equality is proved. 2) Existence of a weak solution is shownby a compactness argument for the time discretized solution as the time increment
tends to zero. 3) The energy dissipation equality is shown for weak solutions and theuniqueness of a weak solution follows from it. 4) The exponential decay property as
time tends to infinity is proved even for the case without the dumping term.
This research is an extension of previous studies[1, 2, 3], and is a joint work with
J.-T. Li (Kanazawa U.), Y. Liu (Kyoto U.), G. Nakamura (Hokkaido U.), H. Notsu(Kanazawa U.), and Y. Ueda (Kobe U.).

References[1] M. Kimura, H. Notsu, Y. Tanaka, and H. Yamamoto: The gradient flow structure of an
extended Maxwell viscoelastic model and a structure-preserving finite element scheme.
Journal of Scientific Computing, Vol.78 (2019) pp.1111-1131. doi:10.1007/s10915-0180799-2[2] J.-T. Li: Mathematical analysis of the Zener-type viscoelastic wave equation. Divisionof Mathematical and Physical Science, Graduate School of Natural Science and Technology, Kanazawa University, Master Thesis (2023 February).
[3] H. Yamamoto: Energy gradient flow structure of the Zener viscoelastic model and its
finite element analysis. Master theis, Division of Mathematical and Physical Sciences,
Graduate School of Natural Science and Technology, Kanazawa University (February2019) 36 pages (in Japanese).

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Navier-Stokes Equations in Gas Dynamics: Greens
Function, Singularity, and Well-Posedness
Tai-Ping Liu 1,2 and Shih-Hsien Yu 2
1. Department of Mathematics, Stanford University,
Stanford, CA 94305, USA
E-mail:liu@math.stanford.edu

2. Institute of Mathematics, Academia Sinica,
Taipei, 11529, Taiwan
E-mail:matysh@gate.sinica.edu.tw
Abstract
We are interested in the well-posedness theory and the propagation of singularity
in the weak solutions for the initial value problem. Our approach is to convert thedifferential equations into integral equations on the level of weak solutions. This depends on exact analysis of the associated linear equations and their Green’s functions.
We carry out our approach for the Navier-Stokes equations in gas dynamics. Localin time as well as time- asymptotic behaviors of weak solutions, and the continuous
dependence of the solutions on their initial data are established.

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Geometric effects on W 1,p regularity of the stationary
linearized Boltzmann equation
I-Kun Chen2 , Chun-Hsiung Hsia2 , Daisuke Kawagoe1 , Jhe-Kuan Su 2
1. Kyoto University,
Kyoto, 606-8501 Japan

2. National Taiwan University,
Taipei, 11529, Taiwan
Abstract
We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in W 1,pspaces for 1 ≤ p < 2 for small domains. In contrast, if we further assume the positivityof the Gaussian curvature on the boundary, we prove the existence of solutions in W 1,pspaces for 1 ≤ p < 3 provided that the diameter of the domain is small enough. In
both cases, we provide counterexamples in the hard sphere model; a bounded convex
domain with a flat boundary for p = 2, and a small ball for p = 3.

References[1] Geometric effects on W 1,p regularity of the stationary linearized Boltzmann equation
arXiv preprint arXiv:2311.12387, 2023

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Traveling waves and the Neimark–Sacker bifurcation
with discrete rotational symmetry
Tomoyuki Miyaji1 and Kazuya Okamoto2 and Akiyasu Tomoeda3
1. Department of Mathematics, Kyoto University,
Kyoto, 606-8502 Japan
E-mail:miyaji.tomoyuki.4m@kyoto-u.ac.jp

2. Department of Pure and Applied Mathematics, Waseda University,
Tokyo, 169-8555, Japan
E-mail:poulkun67@gmail.com

3. Faculty of Informatics, Kansai University,
Osaka, 569-1095, Japan
E-mail:tomoeda@kansai-u.ac.jp
Abstract
The Neimark–Sucker bifurcation is a bifurcation of a ﬁxed point in discrete-time
dynamical systems, which leads to the appearance of a branch of invariant closed
curves. We consider this bifurcation for smooth maps with Zn -symmetry and pointout that a branch of traveling waves corresponding to invariant closed curves bifurcates.
We discuss a method of numerical continuation of the bifurcating traveling waves. We
also discuss its application to a system of delay-diﬀerence equations, a traﬃc ﬂow model
proposed by the authors.

References[1] P. Chossat and M. Golubitsky, Iterates of maps with symmetry, SIAM J. Math. Anal.
19 (1988) 1259–1270.
[2] M. Golubitsky, I. Stewart, and D.G. Schaeﬀer, Singularities and Groups in Bifurcation
Theory II, Springer, New York, 1988.
[3] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Fourth edition Springer Nature, Switzerland, 2023.
[4] R. Mazrooei-Sebdani and Z. Eskandari, Neimark-Sacker Bifurcation with Zn -Symmetry
and a Neural Application, Qual. Theory Dyn. Syst. 18 (2019) 931–946.
[5] K. Okamoto, T. Miyaji, and A. Tomoeda, Nonlinear delay diﬀerence equation withbistability as a new traﬃc ﬂow model, submitted.

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

On a boundary value problem of a first-order PDE andits regularized problems
Masaki Imagawa1
1. Graduate School of Informatics, Kyoto University,
Kyoto, 606-8501 Japan
E-mail:m imagawa@acs.i.kyoto-u.ac.jp
Abstract
In the study of PDEs, it is often useful to consider approximate equations for theequations of interest. In this talk, we will consider a bounday value preblems for firstorder PDE and the regularized elliptic problems that approximate it, which is obtainedby adding the Laplacian with small parameters to the original equation. We introduce
convergence estimates and numerical examples of approximate solutions. This talk isbased on joint work with Dr. Daisuke Kawagoe (Kyoto Univ.).

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Numerical computation for transonic potential flow withparticular solutions of Chaplygin’s equation
Kousei Takahashi 1
1. Graduate School of Informatics, Kyoto University,
Kyoto, 606-8501, Kyoto
E-mail:taka hijiri@acs.i.kyoto-u.ac.jp
Abstract
We introduce a numerical method based on the hodograph transformation for theplane transonic potential flow past a wing. The plane transonic potential flow is governed by Chaplygin’s equation in hodograph, which is a mixed type linear partial
differential equation. However, hodograph transformation generally suffers from its
non-univalency, e.g. limiting line singularity. We will overcome this difficulty by analytic continuation of its solutions on the corresponding Riemann surface.

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Numerical analysis of a rarefied gas flow past a circular
disk: peculiar behavior near the sharp edge
Takuma Tomita1
1. Graduate School of Informatics, Kyoto University,
Kyoto, 606-8501, Kyoto
E-mail: tomita.takuma.83s@st.kyoto-u.ac.jp
Abstract
In the framework of rarefied gas dynamics, a flow around a body induces a nonuniform temperature field, even if the body has a uniform temperature. This effect iscalled thermal polarization and one of the most significant differences from ordinary
fluid dynamics, in which flow and temperature fields are determined independently.
From this point of view, we study a uniform flow past a circular disk numerically.
The boundary value problem is composed of the Bhatnagar-Gross-Krook model of the
Boltzmann equation and the diffuse reflection boundary condition. To solve this BVP,
we construct a numerical scheme based on the method of characteristics, and performed large-scale computing by using a supercomputer to obtain the detailed profilesof velocity distribution function. From the numerical result, it is revealed that thermal
polarization is localized near the edge of the disk in the continuum limit. In the talk,after describing the method of numerical analysis, we present numerical results withemphasis on the peculiar behavior near the edge.

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Towards a topology optimisation of wave devices withwide working bandwidth
Hiroshi Isakari1
1. Department of Science and Technology, Keio University,
Yokohama, 223-0061 Japan
E-mail: isakari@sd.keio.ac.jp
Abstract
Topology optimisation [1], originally proposed in the ﬁeld of mechanics aiming toachieve light and stiﬀ structural members, is now expanding its application ﬁelds tovarious engineering ﬁelds such as heat, ﬂuid, and wave device designs [2]. Usually,
the existing topology optimisations for wave devices assume a “monochrome” incident
excitation oscillating with a single frequency. As a result, the performance of thederived conﬁguration can be sensitive to the variation in the excitation frequency. Since
the frequency can deviate from the assumed value at the design stage in a practicalsituation, it is important to provide wave devices with robust performance against
the frequency perturbation. In the presentation, we shall introduce our recent eﬀort
to realise such a robust topology optimisation. Our method [3, 4] rely on the fast
frequency sweep based on the boundary integral method [5], and the level-set method
utilising the topological derivative [6].

References[1] M P Bendsøe and N Kikuchi 1988 Generating optimal topologies in structural design using a homogenization method Computer Methods in applied Mechanics and Engineering
71(2) 197-224[2] H Isakari K Kuriyama S Harada T Yamada T Takahashi and T Matsumoto 2014 A
topology optimisation for three-dimensional acoustics with the level set method and thefast multipole boundary element method Mechanical Engineering Journal 1(4) CM0039[3] J Qin H Isakari K Taji T Takahashi and T Matsumoto 2021 A robust topology optimization for enlarging working bandwidth of acoustic devices International Journal for
Numerical Methods in Engineering 122(11) 2694-2711[4] Y Honshuku and H Isakari 2022 A topology optimisation of acoustic devices based on
the frequency response estimation with the Padé approximation Applied Mathematical
Modelling 110 819-840[5] J C Nédélec 2001 Acoustic and electromagnetic equations: integral representations forharmonic problems 144 New York: Springer.
[6] S Amstutz and H Andrä 2006 A new algorithm for topology optimization using a levelset method. Journal of Computational Physics 216(2) 573-588

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Numerical Investigations to Time-Fractional Allen-Cahn
Equation
Bing-Ze Lu1 and Yu-Chen Shu 1,2
1. Department of Mathematics, National Cheng Kung University,
Tainan, 70101 , Taiwan
E-mail:l18081028@gs.ncku.edu.tw

2. Department of Mathematics, National Cheng Kung University,
Tainan, 70171, Taiwan
E-mail:ycshu@mail.ncku.edu.tw
Abstract
In this talk, we present a dissipative functional for the time-fractional Allen-Cahn
equation and examine the properties analytically and numerically. The time differential
operator here is realized by a Caputo-type differential operator. In order to answer the
open problem, the dissipative property of classical energy, the dissipative functional isderived from the formulation of classical energy and Caputo differential operator. We
show the functional is unconditional dissipation, and the decay rate is shown to follow
tα−1 . In addition, we provide the error analysis of the numerical solution. The analysis
depends on the regularity of the solution and the implicit-explicit notion inherited in
the convex splitting method. We show the convergence order is of tα regardless ofthe chosen convex splitting method, and the numerical solutions obey the maximum
principle.

References[1] Q. Du, J. Yang, and Z. Zhou, 2020 Time-fractional allen–cahn equations: analysis andnumerical methods, Journal of Scientific Computing, 85 , pp. 1–30
[2] T. Tang, H. Yu, and T. Zhou, 2019 On energy dissipation theory and numerical stability
for time-fractional phase-field equations, SIAM Journal on Scientific Computing, 41,
pp. A3757–A3778[3] H.-l. Liao, T. Tang, and T. Zhou, 2021 An energy stable and maximum bound preserving
scheme with variable time steps for time fractional allen–cahn equation, SIAM Journal
on Scientific Computing, 43, pp. A3503–A3526.636

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Some free boundary methods and their application to
inverse scattering problems
Pu-Zhao Kow1 , Simon Larson2 , Mikko Salo3 and Henrik Shahgholian4
1. Department of Mathematical Sciences, National Chengchi University,
No. 64, Sec. 2, ZhiNan Rd., Wenshan District, Taipei 116302, Taiwan
E-mail:pzkow@g.nccu.edu.tw

2. Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg,
SE-41296 Göteborg, Sweden
E-mail:larsons@chalmers.se

3. Department of Mathematics and Statistics, University of Jyväskylä,FI-40014 Jyväskylä, FinlandE-mail:mikko.j.salo@jyu.fi

4. Department of Mathematics, KTH Royal Institute of Technology,
SE-10044 Stockholm, Sweden
E-mail:henriksh@kth.se
Abstract
In this talk, we consider the inverse scattering problem of acoustic waves, which
can be modeled by the Helmholtz equation. We will focus on the non-scattering phenomena. This topic has been presented via Zoom two years ago (https://www-an.
acs.i.kyoto-u.ac.jp/tj2022/). Now I will elaborate this talk according to some of
my recent works.
Some of the non-scattering domain can be constructed using partial balayage [1]or the minimizing problem (with free boundary) [2]. We also discuss the possible
scattering and non-scattering behavior of an anisotropic and inhomogeneous Lipschitz
medium, by connecting the anisotropic non-scattering problem to a Bernuolli type freeboundary problem [3].

References[1] P.-Z. Kow, S. Larson, M. Salo, and H. Shahgholian, Quadrature and nonscattering domains for the helmholtz equation, Potential Anal., 60 (2024), pp. 387–424.
MR4696043, Zbl:7798456, doi:10.1007/s11118-022-10054-5. The results in the appendix
are well-known, and the proofs can found at arXiv:2204.13934.
[2] P.-Z. Kow, M. Salo, and H. Shahgholian, A minimization problem with freeboundary and its application to inverse scattering problems, arXiv preprint, (2023).
2303.12605. To appear in Interfaces Free Bound.
[3], On scattering behavior of corner domains with anisotropic inhomogeneities, arXiv
preprint, (2023). 2309.11213.

2024 Taiwan-Japan Joint Workshop on Numerical Analysis and Inverse Problems
Kyoto University, Kyoto
March 9–10, 2024

Inverse source problems for wave equations
with inverse square potentials
Hiroshi Takase1
1. Institute of Mathematics for Industry, Kyushu University,
Fukuoka, 819-0395 Japan
E-mail:htakase@imi.kyushu-u.ac.jp
Abstract
We will show the global Lipschitz stability estimate for an inverse source problem of
the one-dimensional wave equation with inverse square potential with the singularity
inside a domain. The key tool is the global Carleman estimate applicable to equations
with singular potentials. Because of the singularity of the potentials, conventional
perturbation methods for bounded potentials cannot be applied, and a new weightfunction must be chosen that resolves this difficulty. Also, the regularity of the solution
expected from the general theory of well-posedness currently only allows us to prove
this estimate in one dimension.

References[1] A. Enciso, A. Shao, and B. Vergara. Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials. J. Eur. Math.
Soc., 23(10):3459–3495, 2021.
[2] J. Vancostenoble. Lipschitz stability in inverse source problems for singular parabolic
equations. Comm. Partial Differential Equations, 36(8):1287–1317, 2011.
[3] J. Vancostenoble and E. Zuazua. Hardy inequalities, observability, and control for
the wave and Schrödinger equations with singular potential. SIAM J. Math. Anal.,
41(4):1508–1532, 2009.