Abstract Book of EASIAM 2018
Plenary Talk (23, June)
Room 213 (Building No.2)
Title: Modeling by networks
Speaker: Martin Golubitsky (Ohio State University)
Abstract: Networks consist of nodes and unidirectional arrows and encode systems of diﬀerential equations. The arrows indicate which nodes are coupled towhich. The nodes and arrows can be annotated to indicate which nodes are identical and which kinds of coupling are identical. We suggest that these types ofannotated networks should be thought of as modeling assumptions. Mathematically we ask the question: What kinds of dynamics are typical in the class of
diﬀerential equations that are associated to a given network? This project on
network dynamics is joint work with Ian Stewart.
What distinguishes a coupled network of diﬀerential equations from a large systemof diﬀerential equations in the desire to keep track of the output from each node
individually. It is then possible to compare signals from diﬀerent nodes (synchrony)and to keep track of singularities in individual nodes. This talk will illustrate theseideas with two biological applications: binocular rivalry (based on symmetry in
Wilson networks) and homeostasis (in gene regulatory networks).
In binocular rivalry subjects are shown two diﬀerent images - one to the right eye
and one to the left eye. Typically, the subject perceives alternation between thetwo images. Sometimes, subjects perceive a more complicated set of images. We
show how network modeling can help predict the set of possible percepts.
Homeostasis occurs when a certain feature of a family of equilibria remains approximately constant as an input parameter is varied. Again, we show how networkmodeling leads naturally to homeostasis and particularly to the possible existenceof housekeeper genes in gene regulatory networks. Housekeeper genes are ones
that keep the production of their associated proteins approximately constant andsuch genes are thought to be important in the regulation of gene networks.

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Invited Talks
Room 213 (Building No.2)
Invited Talk 1 (23, June)
Title: Topological data analysis and materials science
Speaker: Yasuaki Hiraoka (Kyoto University / RIKEN)
Abstract: Topological data analysis is an emerging concept in applied mathematics in which we characterize shape of data using topological methods. In
particular, the persistent homology and its persistence diagrams are nowadays applied to a wide variety of scientiﬁc and engineering problems including materials
science, life science and social networks etc. In my talk, I will give a survey ofthese concepts both from mathematics and applications in materials science. Iftime is allowed, I will show several future challenges which deepen the mathematical understandings and possible applications.
Invited Talk 2 (23, June)
Title: Rare event studies in systems biology
Speaker: Tiejun Li (Peking University)
Abstract: As a research ﬁeld to overcome the multiple time scale issues in noise
induced transitions, the rare event study has attracted much attention in recent
years. In this talk, I will review some recent results on the rare event study in systems biology done by our group. In theoretical aspect, I will introduce the so-called
two-scale large deviation theory for chemical kinetic processes motivated by biophysical study for genetic switches. This will induce to diﬀerent mean ﬁeld limits
and diﬀerent chemical Langevin approximations when the involved two timescales
has diﬀerent magnitudes. I will also mention the mathematical theory of energy
landscapes widely used in systems biology. In the application aspect, I will mention the rare events and energy landscape constructions for genetic switch, budding
yeast cell cycle and S-phase checkpoint activation mechanism related to the mostrecent biological experiments.
Invited Talk 3 (24, June)
Title: Phase retrieval algorithms with random masks
Speaker: Pengwen Chen (National Chung Hsing University)
Abstract: Phase retrieval aims to recover one unknown vector from its magnitude measurements, e.g., coherent diﬀractive imaging, where phase information is2

missing. The recovery of phase information can be formulated as one minimization
problem subject to a non convex high-dimensional torus set. In theory, uniquenessof solutions can be obtained under random masks. The introduction of randommasks actually breaks the symmetry of Fourier matrices and creates spectral gap
for the local convergence of many phase retrieval algorithms, including alternative
projection methods and Fourier Douglas-Rachford algorithms. The spectral gap isrelated to the local convergence rate. In this talk, we shall present the convergence
analysis as well as their ﬁxed point property.
On the other hand, these alternative algorithms still could fail to generate theglobal solution eﬀectively. To alleviate the stagnation of possible local solutions,
we propose one null vector method as an initialization method for phase retrieval
algorithms The method is motivated by the following observation: Gaussian random vectors in high dimensional space are always nearly orthogonal to each other.
According to magnitude data, we can construct one sub-matrix assembled fromthe sensing vectors nearly orthogonal to the unknown vector. One candidate forthe initialization vector is given by the singular vector of the sub-matrix corresponding to the least singular value. Thanks to isometric Fourier matrices, thisvector coincides with the dominant singular vector of the complement sub-matrix.
Empirical studies (non-ptychography and ptychography) indicate that its incredible closeness to the unknown vector, compared with other existing methods. In
this talk, we present one nonasymptotic error bound in the case of random complex Gaussian matrices, which sheds some light on its superior performance in the
Fourier coherent diﬀractive case with random masks.
If time permitted, we shall discuss the performance of these iterative algorithms
on ptychographic Fresnel data. We shall show that the spectral gap becomes verysmall for certain Fresnel numbers. Actually, we do observe poor reconstruction
occurs under these Fresnel numbers.
This is a joint work with Albert Fannjiang(UC Davis) and Giren Liu (NCKU).
Invited Talk 4 (24, June)
Title: Highly eﬃcient semismooth Newton augmented Lagrangian methods forsolving lasso problems
Speaker: Kim Chuan Toh (National University of Singapore)
Abstract: In this talk, we present a fast and robust algorithmic framework SSNAL
for solving large scale lasso problems. SSNAL is a semismooth Newton based augmented Lagrangian algorithmic framework. We show that for lasso problems, boththe primal and dual iteration sequences generated by SSNAL possess a remarkably fast linear convergence rate, which can even be superlinear asymptotically.
We also conduct variational analysis to analyse the second order sparsity structure3

of the underlying problems and proposed eﬃcient numerical techniques to exploit
the structure in our algorithm. Numerical comparison between our approach andstate-of-the-art solvers on real data sets are presented to demonstrate the higheﬃciency and robustness of our proposed algorithm in solving diﬃcult large scalelasso problems. For example, for a problem with over 4 million features and 16000
samples, SSNAL can solve it in 20 seconds, while the best alternative solver took
2400 seconds. This talk is based on joint work with Professor Sun Defeng and Dr
Li Xudong.

Room AD (Building No.6)
Invited Talk 5 (25, June)
Title: Solving multi-linear systems with M-tensors
Speaker: Yimin Wei (Fudan University)
Abstract: This talk is concerned with solving some structured multi-linear systems, especially focusing on the equations whose coeﬃcient tensors are M-tensors,
or called M-equations for short. We prove that a nonsingular M-equation witha positive right-hand side always has a unique positive solution. Several iterativealgorithms are proposed for solving multi-linear nonsingular M-equations, generalizing the classical iterative methods and the Newton method for linear systems.
Furthermore, we apply the M-equations to some nonlinear diﬀerential equations
and the inverse iteration for spectral radii of nonnegative tensors.
Invited Talk 6 (25, June)
Title: New analysis on Galerkin FEMs for nonlinear parabolic PDEs
Speaker: Weiwei Sun (City University of Hong Kong)
Abstract: Linearized (semi)-implicit schemes are the most commonly-used approximations in numerical solution of nonlinear parabolic equations since at eachtime step, the schemes only require the solution of a linear system. However thetime step restriction condition of schemes is always a key issue in analysis andcomputation. For many nonlinear parabolic systems, error analysis of Galerkin
type ﬁnite element methods with linearized semi-implicit schemes in the time direction is established mostly under certain time step condition τ ≤ hα for some
α > 0. Such a time-step condition may result in the use of a very small time stepand extremely time-consuming in practical computations. The problem becomes
more serious when a non-uniform mesh or adaptive meshing is used.
In this talk, we introduce a new approach to unconditional error analysis of linearized semi-implicit Galerkin FEMs for a large class of nonlinear parabolic PDEs.
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Student Paper Prize Session (23, June)
Room 213 (Building No.2)
Title: Unconditional convergence in maximum-norm of a second-order linearized
scheme for a time-fractional Burgers-type equation
Speaker: Pin Lyu (University of Macau)
Abstract: We study linearized ﬁnite diﬀerence scheme for a time-fractional Burgerstype equation in this paper. A linearized scheme with second-order accuracy intime and space is proposed. The advantage of the scheme is that iterative method
is not required for ﬁnding the approximated solutions. Nonlinearity involving
derivatives causes diﬃculties in analysis. By reﬁned estimates of our previous
study, we show that the scheme unconditionally converges with second-order inmaximum-norm. The theoretical results are justiﬁed by numerical tests.

Title: Point vortex dynamics on a toroidal surface
Speaker: Yuuki Shimizu (Kyoto University)
Abstract: Vortex interactions play an important role in the qualitatively understanding of two-dimensional incompressible and inviscid ﬂow evolutions. We
assume the vorticity is discretized with a ﬁnite number of delta-functions, called
point vortices. Thanks to the invariance of the vorticity along the orbit of a ﬂuid
particle, the evolution of the vorticity can be described in terms of interactions
between those point vortices. The dynamics of point vortices has been investigatedwell in unbounded planes with boundaries as well as on a sphere owing to their
physical relevance. On the other hand, it is of a theoretical interest to investigate
how geometric structures of curved surfaces gives rise to diﬀerent vortex interactions that are not observed in vortex dynamics in the plane and on the sphere. In
this talk, we compare the dynamics of point vortices on a toroidal surface with thatin a plane, on a sphere and a hyperbolic disc and consider how is the dynamics
aﬀected by shapes of these surfaces. This is a joint work with Takashi Sakajo at
Kyoto University.

Title: An eﬃcient iterative thresholding method for image segmentation
Speaker: Dong Wang (The Hong Kong University of Science and Technology)
Abstract: We proposed an eﬃcient iterative thresholding method for multi-phase
image segmenta-tion. The algorithm is based on minimizing piecewise constant5

Mumford-Shah functional in which the contour length (or perimeter) is approximated by a non-local multi-phase energy. The minimization problem is solved
by an iterative method. Each iteration consists of computing simple convolutions
followed by a thresholding step. The algorithm is easy to implement and has the
optimal complexity O(N logN ) per iteration. We also show that the iterative algorithm has the total energy decaying property. We present some numerical results
to show the eﬃciency of our method.

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Parallel Sessions
Room AD (Building No.6)
Parallel Session 1 (23, June)
Title: Convergence and stability of the MAC scheme for Stokes/Darcy coupling
problems based on ﬁnite diﬀerence methods
Speaker: Ming-Cheng Shiue (National Chiao Tung University)
Author(s): Ming-Cheng Shiue, Kian Chuan Ong and Ming-Chih Lai
Abstract: In this talk, to begin with, Stokes/Darcy coupling ﬂows which arise
from physical models such as biology, engineering and geophysical ﬂuid dynamics
are considered. In the literature, there are many numerical methods related toﬁnite element methods applied to solve this coupling problem. Unlike ﬁnite element
methods, due to that there is lack of natural variational formulation, in general,
the analysis of the scheme based on ﬁnite diﬀerence methods becomes complicate.
In this work, the MAC scheme for this coupling problem based on ﬁnite diﬀerence
methods is used. Convergence and stability of the scheme will be presented. The
second part will present the development of numerical schemes for Navier-Stokes
and Darcy coupling problems based on projection methods. Numerical simulations
demonstrate the results which match the case of only Navier-Stokes equations that
also are computed using the projection method. These are joint research works
with Ming-Chih Lai and Kian Chuan Ong.
Keywords: Stokes-Darcy, Mac scheme, ﬁnite diﬀerence, convergence, stability

Title: Numerical and mathematical analysis for the blow-up curve of solutions to
1-dimensional nonlinear wave equations
Speaker: Takiko Sasaki (Meiji University)
Author(s): Takiko Sasaki, Tetsuya Ishiwata
Abstract: We study a blow-up curve for the one dimensional wave equation∂t2 u − ∂x2 u = 2p |∂t u|p−1 ∂t u with the Dirichlet boundary condition. The purposeof this talk is to show that the blow-up curve T satisﬁes that T ′ (x) → −1 as
x → 0 + 0 (1) under the suitable initial conditions. To prove the result, weconvert the equation into a ﬁrst order system, and then present some numericalinvestigations of the blow-up curves. From the numerical results, we were able to
conﬁrm (1) holds numerically. Moreover, under some assumptions, we were alsoable to conﬁrm (1) holds mathematically.
Keywords: second-order hyperbolic equations, blow-up

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Title: High-order compact schemes for fractional diﬀerential equations with mixed
derivatives
Speaker: Chenyang Shi (University of Macau)
Author(s): Seakweng Vong , Chenyang Shi, Pin Lyu
Abstract: In this article, we consider two-dimensional fractional subdiﬀusion
equations with mixed derivatives. A high-order compact scheme is proposed tosolve the problem. We establish a suﬃcient condition and show that the scheme
converges with fourth order in space and second order in time under this condition.
Keywords: fractional diﬀerential equation, high-order compact scheme, mixed
derivatives

Title: Numerical analysis for blow-up problems of stochastic diﬀerential equations.
Speaker: YoungChol Yang (Shibaura Institute of Technology)
Author(s): Tetsuya Ishiwata, YoungChol Yang
Abstract: We consider a blow-up problem of stochastic diﬀerential equation(SDE) dX(t) = aX(t)p dt + X q dW (t), and its numerical approximation. P. Groisman et al show that this equation almost surely have a blow-up solution undercertain conditions. Its solution and blow up time are derived as stochastic processes. In this talk, we will add a few constraints and express the solution explicitly
in the form of Brownian motion by using Ito’s formula. Moreover, we show thatthe blow-up time of solution matches the ﬁrst hitting time of the Brownian motion to a speciﬁc value M , which depends only on parameter p and initial value
X0 . This result makes analyzing the blow-up time simple. We also propose a newnumerical method and show some numerical examples.
Keywords: blow-up, stochastic diﬀerential equation, numerical approximation

Parallel Session 2 (23, June)
Title: Research and development of STEM curriculum in high school
Speaker: Yi-Wen Su (University of Taipei)
Author(s): Yi-Wen Su, Chinghsiang Yu, Ting-mei Yu, ChunTsung Hsueh, ChiaYen Tsai
Abstract: Science, Technology, Engineering, and Mathematics (STEM) Education has attracted the attention of the education community in response to changes
in the times and education reforms. Designing STEM courses so that students canuse what they have learned is therefore a topic that concerns teachers in all levels. The purpose of this study is to explore how to develop and practice STEM
8

courses in the high school. The researchers are teacher educator and high school
teachers. The course developed by this research is IQ light, which is a course module, is coordinated between the Mathematics and Living Technology teachers. The
main design concept is that mathematics teachers use magnetic constructs to allow
students to explore the structure of polyhedrons. The principles of basic polyhedrons, such as Plato’s polyhedron and Archimedes’ polyhedron, are introduced to
guide students how to think about polyhedrons using mathematical modeling andto make polyhedron lamps. Then, the living technology teacher introduced the
Sketch Up 3D modeling software, which allowed students to use this software topresent the concepts and clarify the structure of polyhedrons. This software wasused to draw the polyhedral unit shape, and together with a laser cutting machine
to combine a polyhedron ball lamp and LED light bar, connecting a USB wireby welding to produce an IQ light ﬁnished product. The ﬁnished products, completed by the students, were exhibited in the Maker Faire Taipei 2017 and werealso warmly received by the public. This study explores the eﬀectiveness of teacher
cooperation and students’ learning in the development of STEM curriculum, andputs forward relevant reviews and recommendations.
Keywords: STEM, IQ light, high school

Title: Global synchronization of coupled memristor-based neural networks withtime delays
Speaker: Jui-Pin Tseng (National Chengchi University)
Author(s): Jui-Pin Tseng
Abstract: This talk presents an approach to establish the global synchronizationof coupled multiple memristor-based neural networks with delays. The coupled
networks that are considered can incorporate both the internal delay in each individual network and the transmission delay across diﬀerent networks. The coupling
scheme, which consists of a nonlinear term and a sign term, is rather general. Basedon an iterative approach, the respective global synchronization criteria, which depend on whether a transmission delay exists, are derived respectively.
Keywords: synchronization, memristor, neural network, delay

Title: Machine-learning prediction of ﬂuid variables from data using reservoir
computation
Speaker: Kengo Nakai (The University of Tokyo)
Author(s): Kengo Nakai, Yoshitaka Saiki
Abstract: We predict both microscopic and macroscopic variables of a chaotic
ﬂuid ﬂow using reservoir computing. In our procedure of the prediction, we assume no prior knowledge of physical model describing a ﬂuid ﬂow except that its
9behavior is complex but deterministic. We present two ways of prediction of thecomplex behavior; the ﬁrst called partial-prediction requires continued knowledgeof partial time-series data during the prediction as well as past time-series data,
while the second called full-prediction requires only past time-series data as training data. For the ﬁrst case, we are able to predict long-time motion of microscopic
ﬂuid variables. For the second case, we show that the reservoir dynamics constructed from only past data of energy functions can predict the future behavior
of energy functions and reproduce the energy spectrum. This implies that theobtained reservoir system constructed without the knowledge of microscopic datais equivalent to the dynamical system describing macroscopic behavior of energyfunctions.
Keywords: machine-learning, ﬂuid ﬂow

Parallel Session 3 (24, June)
Title: Regularity of solutions to the stationary transport equation with the incoming boundary data
Speaker: Daisuke Kawagoe (Inha University)
Author(s): Daisuke Kawagoe
Abstract: We consider a boundary value problem of the stationary transport
equation in a bounded convex domain with the incoming boundary condition, andwe obtain two results on regularity of its solutions. The ﬁrst result is to describe
discontinuity of the solution which arises from discontinuous points of the inoming
boundary condition, and we show the exponential decay of a jump of the solution
on a discontinuous point. The decay gives an idea for solving an inverse problem
to determine an unknown coeﬃcient from boundary measurements of the solution
to the boundary value problem. The second one is to give a precise estimate in a
two dimensional case. We obtain a W 1,p estimate of the solution for 1 ≤ p < pm ,where pm is a real number depending only on the shape of the domain.
Keywords:

Title: Regularity of solutions for non-uniform nonlinear elliptic equations in aheterogeneous medium
Speaker: Li-Ming Yeh (National Chiao Tung University)
Author(s): Li-Ming Yeh
Abstract: Uniform estimate for the solutions of non-uniform nonlinear elliptic
equations in a heterogeneous medium is considered. The heterogeneous medium
consists of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let ϵ ∈ (0, 1] denote the10

size ratio of the matrix blocks to the whole domain and let ω 2 ∈ (0, 1] denotethe permeability ratio of the disconnected matrix block subset to the connected
sub-region. The C 0,α norm for α ∈ (0, 1) for the nonlinear elliptic solutions in thehigh permeability sub-region are shown to be bounded uniformly in ω, ϵ. However,
the C 0,α norm of the nonlinear elliptic solutions in the low permeability subset can
only be bounded uniformly in ϵ. Relations between the sources and the variationof the solutions in the low permeability subset are also presented in this talk.
Keywords: Non-uniform nonlinear elliptic equations, heterogeneous medium,
permeability ratio

Title: Hopf-pitchfork bifurcation in a reaction-diﬀusion system with an integral
term.
Speaker: Shunsuke Kobayashi (Meiji university)
Author(s): Shunsuke Kobayashi
Abstract: We study the bifurcation structure of chaotic solution in a two-component
reaction-diﬀusion system with an integral term in one spatial dimension. The system has the triply degenerate point at the trivial solution. Around the triply
degenerate point, a three-dimensional dynamical system on the center manifold isobtained. The reduced system can be transformed into normal form for the HopfPitchfork bifurcation. The truncated normal form can possess the invariant tori
and the heteroclinic loop. Furthermore, the system under the non S 1 -symmetricperturbation may possess a heteroclinic cycle. Numerical results for the system
are presented and found to be convincing.
Keywords: reaction-diﬀusion system, Hopf-pitchfork bifurcation, chaos

Parallel Session 4 (24, June)
Title: Asymptotic behavior of numerical solutions for the Allen-Cahn equation
with coarse meshes
Speaker: Tomoya Kemmochi (Nagoya University)
Author(s): Tomoya Kemmochi
Abstract: The Allen-Cahn equation is a semilinear parabolic equation with asmall parameter ε > 0, which is the simplest equation of phase ﬁeld models. Most
parts of the domain is separated into two parts. The solution is approximately one
in the one part and minus one in the other. The parameter ε describes the ”width”
of the interface in a sense, that is, the measure of the part where the solution isapproximately zero.
We are interested in numerical computation of the Allen-Cahn equation. Since thesolution has a sharp interface with the width O(ε), the mesh size, which is denoted11

by h, should be smaller than ε. However, of course, when h becomes smaller, thenthe computational cost becomes much higher. Therefore, it is important to choose
appropriate mesh size h in view of both accuracy and cost.
In this talk, we consider semi-discretization in space for the Allen-Cahn equationby the ﬁnite diﬀerence method with coarse meshes. When the mesh size h is bigger
than ε, the numerical computation fails as expected. Indeed, the interface does not
move. We will demonstrate why this phenomenon occurs, which gives a necessary
condition for the mesh size for proper computation.
Keywords: Allen-Cahn equation, numerical analysis, ﬁnite diﬀerence method

Title: Improved error estimates for ﬁnite element solutions of parabolic Dirichletboundary control problems
Speaker: Buyang Li (Hong Kong Polytechnic University)
Author(s): Wei Gong, Buyang Li
Abstract: The parabolic Dirichlet boundary control problem and its ﬁnite element discretization are considered in convex polygonal and polyhedral domains.
We improve the existing results on the regularity of the solutions by establishingand utilizing the maximal Lp -regularity of parabolic equations under inhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, we prove O(h1 − 1/q − ε) convergence for the semi-discrete ﬁnite element
solutions for some q > 2, with q depending on the maximal interior angle at thecorners and edges of the domain and ε being a positive number that can be arbitrarily small.
Keywords: Dirichlet boundary control, parabolic equation, ﬁnite element method,
error estimate

Title: An asymptotic-based adaptive ﬁnite element method for Kohn-Sham equation
Speaker: Yedan Shen (University of Macau)
Author(s): Yedan Shen, Yang Kuang, Guanghui Hu
Abstract: In this paper, we propose an a posteriori error estimation for KohnSham equation by coarsening mesh. The error estimation is derived from the difference of the total energies on two meshes, respectively. With the help of Hardy’s
inequality, it is shown that our analysis works smoothly for the all-electron model
for a given electronic structure. Numerical tests show that such an a posteriori
error estimation works very well in the proposed h-adaptive ﬁnite element method
framework. In addition, a Poisson problem is proposed to approximate the wavefunctions on the coarsened mesh for improving the eﬃciency of the algorithm. The
eﬀectiveness of this improvement is analyzed and numerically tested.
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Keywords: electronic structure calculation, Kohn-Sham density functional theory, adaptive ﬁnite element method, ground state energy, coarsening mesh
Title: Finite element analysis for a generalized Robin boundary value problem ina smooth domain
Speaker: Takahito Kashiwabara (The University of Tokyo)
Author(s): Takahito Kashiwabara
Abstract: We consider the Poisson equation in a domain Ω, and also the Poisson
equation on the boundary, which involves the Laplace-Beltrami operator. The two
PDEs are then coupled through the Neumann operator (i.e., the normal derivativeon the boundary). This can be viewed as a generalized Robin boundary value
problem, which is related with reduced-order models to examine ﬂuid-structureinteraction. In this talk, a linear ﬁnite element approximation to this problem,
assuming that Ω is a smooth domain, is studied. The main diﬃculty is that weneed to take into account the polyhedral approximation of the curved boundary,in particular, the error caused by the approximation of the Laplace-Beltrami operator. We prove the optimal rate of convergence O(h) in the H 1 -norms in thedomain and on the boundary, under a regularity assumption on the exact solution.
Keywords: ﬁnite element method, error estimate, Laplace-Beltrami operator
Parallel Session 5 (24, June)
Title: Unique solvability of discontinuous ODE system for conﬁned particles motion
Speaker: Zhenxing Yang (Kanazawa University)
Author(s): Zhenxing Yang, Masato Kimura, Patrick van Meurs
Abstract: We consider a system of discontinuous ordinary diﬀerential equations
which represents general motion of particles in a ﬁxed region. Since a classicalsolution may not exist, we deﬁne a mild solution to the discontinuous ODE system and show the existence and uniqueness of the solution. As an application,
we consider a gradient ﬂow of an interactive particle energy of particles conﬁned
in a bounded domain. Furthermore, we show several numerical examples of thisconﬁned gradient ﬂow system and observe how particles attach and detach fromthe boundary.
Keywords: discontinuous ODE system, particle, mild solution
Title: Structure-preverving methods for PDEs via Green-Gauss formulae on Voronoi
cells
Speaker: Daisuke Furihata (Osaka University)
13Author(s): Daisuke Furihata
Abstract: In these decades, we have developed a few numerical methods to inherit
essential properties from the original target problems, such as PDEs/ODEs. They
are commonly called structure-preserving methods, and we indicated they are efﬁcient, reliable and durable via some numerical computations on one-dimensional
problems. However, if we would like to introduce reference points located arbitrarily in two- or three-dimensional regions, it is hard to design some structurepreserving methods. The reason is that we should deﬁne some ﬁnite diﬀerence/volume
operators to discretize diﬀerential operators, however, in general, it is also severe
problems to ﬁnd/design some general Gauss-Green formulae based on them. Thisdiﬃculty often prevents to design structure-preserving methods since Green-Gauss
formulae are an essential key for variational calculation included in the process ofthe design.
In these years, we have found that there exist some rigorous discrete Gauss-Green
formulae among ﬁnite diﬀerence/volume operators based on the Voronoi-Delaunay
triangulations. Furthermore, we can apply them to design some structure-preserving
numerical methods for some PDE problems and run numerical computations.
In this talk, we would like to indicate those ﬁnite diﬀerence/volume operators,
Green-Gauss formulae and their proofs in detail and the obtained discrete variational derivative methods, which is one of the structure-preserving methods for
PDEs, based on Voronoi cells.
Keywords: structure-preserving methods, Voronoi cell, discrete varitational derivative method

Title: 3D Poisson-Boltzmann computation of KcsA potassium channel
Speaker: Tzyy-Leng Horng (Feng Chia University)
Author(s): Tzyy-Leng Horng
Abstract: Ion channels are pore-forming trans-membrane proteins that allow
ions to enter/leave cell. There are many important cell functions involving ion
channel, e.g., establishing and regulating action potential in neurons and myocytes. The average time for an ion passing through ion channel is in the orderof ms, which is infeasible for molecular dynamics simulation so far. Continuum
model like Poisson-Boltzmann equation (PB) and Poisson-Nernst-Planck (PNP)
equations are popular to describe ion channel in equilibrium and non-equilibrium
situations. KcsA potassium channel is chosen to be studied here, since it is oneof few ion channels having X-ray crystallographic structure. 3D PB and PNP
simulations of KcsA channel have been a challenging task, since (1) geometry iscomplicated especially the narrow ﬁlter part requiring high resolution when generating meshes; (2) mathematical models are complicated since there are various
versions of modiﬁed PB/PNP to choose; (3) physics is complicated such as distri14

butions of dielectric constant and diﬀusion coeﬃcient, necessity to employ stericeﬀect or not and solvation energy should be included or not. Here, a PDB 3F7Y
KcsA structure with ﬁlter part replaced by that of PDB 1K4C is used as the structure for simulation. Unlike all other KcsA PDB structures, this synthetic structure
guarantees that the channel is open. PB and modiﬁed PB equations are ﬁrst extended to be pseudo-time-dependent with the steady-state solution being our onlyinterest. The numerical framework adopted here to solve these time-dependentequations for electric potential is method of lines (MOL). Governing equation isﬁrst semi-discretized in space by 2nd order ﬁnite volume method under Cartesian
grids with the edge value to cope with interface condition. This semi-discretizedsystem forms a system of ordinary diﬀerential algebraic equations (ODAE) thatcan be further integrated by popular ODAE solvers. Mathematical models simulated here are (I) Classical PB, (II) modiﬁed PB with steric eﬀect described by
Bikerman model, and (III) modiﬁed PB as (II) with solvation energy includedin addition. From simulation results, we found potassium ion is unrealistically
crowded in the ﬁlter for model (I). For model (II), though potassium ion is no
more unrealistically crowded in ﬁlter due to the inclusion of steric eﬀect, there is
no room for water to be in the ﬁlter. Finally, model (III) delivers the most reasonable physical result among all three models by obtaining reasonable potassium
concentration under steric eﬀect and allowing water residence in the ﬁlter at thesame time.
Keywords: ion channel, Poisson-Boltzmann, ﬁnite-volume method, transmembrane protein

Parallel Session 6 (24, June)
Title: Computer-assisted proof of the existence of unimodal solutions to theProudman-Johnson equation
Speaker: Tomoyuki Miyaji (Meiji University)
Author(s): Tomoyuki Miyaji, Hisashi Okamoto
Abstract: We present a computer-assisted proof for the existence of unimodal solutions to the Proudman-Johnson equation, which gives some self-similar solution
to the 2D Navier-Stokes equations. We formulate the shooting method for the stationary Proudman-Johnson equation and solve it via the interval Newton method.
A direct application of this strategy, however, does not work due to numericalinstability of shooting method for a large Reynolds number. We overcome thisdiﬃculty by using both multiple-shooting method and multiple precision intervalarithmetic.
Keywords: the Proudman-Johnson equation, unimodal solutions, interval analysis, computer- assisted proof, multiple shooting method
15

Title: Numerical solution of the Ostrovsky equation over variable topography
Speaker: Azwani Alias (Universiti Malaysia Terengganu)
Author(s): Azwani Alias, Nik Nur Amiza Nik ismail
Abstract: In the real world phenomena, internal nonlinear waves are often propagating through a non-uniform background medium known as variable topography.
The eﬀect of variable topography on the propagation of oceanic internal solitary
waves in the absence of background rotation is relatively well studied and understood by variable coeﬃcient Korteweg-de Vries type equations. However, theworks that collaborate with background rotation and various variable topography,
known as variable coeﬃcient Ostrovsky equations is less understood. The aim of
this research is to study the eﬀects of various variable topography on the propagation of the internal nonlinear waves in the presence of rotation eﬀect. Numericalsimulation such as Pseudo-Spectra method will be used to analyze these eﬀects.
The intention is to get deeper understanding of the propagation of nonlinear wave
packets with various variable topography.
Keywords: Ostrovsky equation, variable topography, pseudo spectral method

Title: Numerical computation of the radiative transport equation on nonconformal meshes
Speaker: Hiroshi Fujiwara (Kyoto University)
Author(s): Hiroshi Fujiwara
Abstract: In this talk, we propose the use of non-conformal meshes for numerical computations of the stationary radiative transport equation (RTE). RTE isan integro-diﬀerential equation, and is used in medical engineering ﬁelds as amathematical model of near-infrared light propagation in biological tissue. Our
numerical scheme is basically based on the discontinuous Galerkin method, andthus regularity and conformal meshes are assumed from the viewpoint of error
analysis. However, in medical application, optical ﬁbers are used to emit light,
which is rather smaller than a target domain, and adaptive meshes have eﬃciency
from quantitative viewpoint. In the presentation, we shall show error analysis anddiscuss comparisons with adaptive conformal mesh case in 2 dimensions.
Keywords: numerical analysis, radiative transport equation, nonconformal meshes

Parallel Session 7 (25, June)
Title: Improving numerical stability and analyzing backward error for heavily
damped quadratic eigenvalue problem
Speaker: Hongjia Chen (University of Tsukuba)
16Author(s): Hongjia Chen, Akira Imakura,Tetsuya Sakurai
Abstract: In this talk, we propose a method with Betcke’s technique to improve
the numerical stability for heavily damped quadratic eigenvalue problem (QEP).
Heavily damped QEP is a special type of QEP that the norm of damping matrix islarger than the norm of other coeﬃcient matrices. A classical approach for solving
QEP is via a linearization form. We usually use QZ method to compute eigenpairs of the linearization form, and ﬁnally recover the eigenpairs of QEP from theeigenpairs of the linearization form. However, it can suﬀer from numerical instability when the norms of coeﬃcient matrices vary widely. In this study, we have
two objectives. The ﬁrst objective is to improve numerical stability of computing
eigenpairs in heavily damped QEP. To improve numerical stability, we combine
the linearization form with Betcke’s technique. The second objective is to explain
why the use of Betcke’s balancing technique improves the numerical stability ofcomputing eigenpairs for the heavily damped QEP. To achieve the second objective, we investigate the backward error of the proposed method. The numericalexperiments show that the proposed method improves numerical stability of computing eigenpairs for the heavily damped QEP.
Keywords: balancing technique, heavily damped QEP, backward error

Title: Relaxed MILU(0) Factorization with Block Red-Black Ordering for GPU
parallelization of PIC simulation
Speaker: Akemi Shioya (The University of Electro-Communications)
Author(s): Akemi Shioya, Yusaku Yamamoto
Abstract: The combination of the block red-black ordering and the relaxed modiﬁed incomplete LU factorization without ﬁll-ins (MILU(0)) provides a stable andhighly parallelizable preconditioner for Krylov subspace methods. Introducing
relaxation prevents zero pivots produced by the combination of the MILU(0) preconditioner with the red-block ordering, while keeping the beneﬁt of reducing thenumber of iteration and synchronization points. In this strategy, the number of
iterations decreases as the block size becomes larger. To take advantage of the preconditioner and the computational power of GPU, we consider an implementationin which large block sizes and performant data access patterns are compatible.
The eﬀects of improvement are investigated by solving Poisson’s equation that
accounts for the majority of computation time in PIC simulation for 3D process
plasma apparatus. In this simulation, electric ﬁelds are repeatedly solved to calculate the behavior of sample particles at each time step. To reduce the overheadof ordering, we put the setting of index data that does not change with the time
step out of the time loop. We illustrate that the computation time in GPU environments becomes small when the block size satisﬁes both conditions for good
convergence and suﬃcient parallelism.
17

Keywords: linear system, modiﬁed incomplete factorizations, block red-black ordering, parallel processing

Title: The AAA algorithm for rational approximation
Speaker: Yuji Nakatsukasa (National Institute of Informatics)
Author(s): Yuji Nakatsukasa, Olivier Sete, Lloyd N. Trefethen
Abstract: Rational approximation can outperform polynomial approximation bya landslide when there are singularities in or near the domain of approximation.
However, its use has been limited relative to polynomials, primarily due to illconditioning (in addition to spurious poles). In this work we show that the conditioning, and the overall utility of rational functions, can be improved dramatically by an adaptive barycentric representation for rational functions, wherein one
chooses a basis depending not only on the domain but also the function to beapproximated.
We introduce a new algorithm, called AAA (triple A, standing for “adaptive
Antoulas-Anderson”) for rational approximation on a real or complex domain.
Even on a disk or interval the algorithm may outperform existing methods, andon more complicated domains it is especially competitive. The core ideas are (1)representation of the rational approximant in barycentric form with interpolation
at certain support points and (2) greedy selection of the support points (which
determines the basis) to avoid exponential instabilities.
Keywords: rational approximation, barycentric representation, stability

18

Room 61 (Building No.6)
Parallel Session 1 (23, June)
Title: PDE models for pricing Asian options
Speaker: Chi-Tien Lin (Providence University)
Author(s): C. Brown, J. C. Handley, CT Lin and K. J. Palmer
Abstract: In this talk, we consider ﬁxed and ﬂoating strike European style Asiancall and put options. For such options, there is no convenient closed-form formula for the prices. Previously, Rogers and Shi, Vecer, and Dubois and Lelievre
have derived partial diﬀerential equations with one state variable, with the stock
price as numeraire, for the option prices. In this talk, we derive a whole familyof partial diﬀerential equations, each with one state variable with the stock priceas numeraire, from which Asian options can be priced. Any one of these partial
diﬀerential equations can be transformed into any other. This family includes four
partial diﬀerential equations which have a particularly simple form including thethree found by Rogers and Shi, Vecer, and Dubois and Lelievre. Recently, Vecer
derive a new PDE using the average asset as numeraire. We perform numericalcomparisons of the ﬁve partial diﬀerential equations by Crank-Nicolson method
and conclude, as expected, that Vecer’s equations and that of Dubois and Lelievre
do better when the volatility is low but that with higher volatilities the performance of all ﬁve equations is similar. Vecer’s equations are unique in possessing acertain martingale property and they perform numerically well or better than theothers.
Keywords: Asian options, partial diﬀerential equations, one state variable, Crank–
Nicolson

Title: State space model of realized volatility under the existence of dependent
market microstructure noise
Speaker: Toru Yano (Shibaura Institute of Technology)
Author(s): Tetsuya Ishiwata, Toru Yano
Abstract: Volatility means the degree of variation of a stock price which is important in ﬁnance. However, volatility is unobservable and is estimated from observable data such as stock price. Realized Volatility (RV) is an estimator of thevolatility calculated using high-frequency observed prices. RV has lately attracted
considerable attention of econometrics and mathematical ﬁnance. However, it
is known that high-frequency data includes observation errors called market microstructure noise (MN), and the RV calculated from this is also aﬀected by MN.
So, we can not get true volatility and we get noise contaminated RV (NCRV).
Nagakura, Watanabe [2009] proposed a state space model that resolves NCRV
19

into true volatility and inﬂuence of MN. With this model, NCRV can be used as
observation data to estimate and predict the true volatility excluding the inﬂuenceof MN on a model basis. Although they assume independence for MN, empiricalstudies of Hansen, Lunde [2006] point out that MN has correlation with true stock
price and autocorrelation. In this study, we assume these to MN and attempt todevelop a model of Nagakura, Watanabe [2009]. In this talk, we introduce derived
model and perform model comparison by applying the models to actual data.
Keywords: ﬁnance, data science, state space model, time series analysis, realized
volatility, microstructure noise

Title: Stochastic polynomial chaos expansion method for a ﬁnancial market modelwith uncertain volatility
Speaker: Yin-Tzer Shih (National Chung Hsing University)
Author(s): Yu-Tuan Lin, Yin-Tzer Shih
Abstract: This paper presents an arbitrary polynomial chaos expansion citeOladyfor solving a ﬁnancial market model with transactions costs and uncertain volatility. The nonlinear Black–Scholes equation provides the pricing of the European
call option for the real market stock price. The stochastic Galerkin method with
employing historical volatilities is used to approximate the raw data of volatilities
distribution in log–normal random when implementing the arbitrary polynomialchaos expansion. The study illustrates that for the arbitrary polynomial chaos
expansion oﬀer a more accurate statistical estimation describing the uncertainty
for the Black–Scholes model compared to Monte Carlo simulations and the generalized polynomial chaos expansion. Arbitrary polynomial chaos expansion canalso be used for the model with unknown distributions of volatilities in short term.
Keywords: Black–Scholes formula, option pricing, polynomial chaos, uncertainty
random stochastic

Title: A fast preconditioned iterative method for two-dimensional options pricing
under fractional diﬀerential models
Speaker: Xu Chen (University of Macau)
Author(s): Xu Chen, Deng Ding, Siu-Long Lei, Wenfei Wang
Abstract: In recent years, fractional partial diﬀerential equation (FPDE) has
been widely applied in options pricing problem, which better explain many important empirical facts of ﬁnancial markets. However, the vast majority of theliterature focus on pricing the single asset option under the FPDE framework. In
this paper, a two-dimensional FPDE governing the valuation of rainbow options
whose two underlying assets follow independent exponential Levy processes is established, and its boundary conditions are determined by one-dimensional FPDEs.
20

An unconditionally stable and second-order accurate ﬁnite diﬀerence scheme is
proposed to discretize the established two-dimensional model. Given the block
Toeplitz with Toeplitz block structure of the coeﬃcient matrix, a fast Krylov
subspace method based on fast Fourier transform can be utilized for solving theresulting linear system with O(N log N ) computational complexity per iteration,
where N is the matrix size. Furthermore, a block circulant with circulant block
preconditioner is proposed to accelerate the convergence of the iterative methodwith theoretical analysis. In addition, a preconditioned Krylov subspace method
is also proposed for solving the boundary conditions numerically with theoreticalanalysis of convergence rate. Numerical examples are given to demonstrate theaccuracy and eﬃciency of our proposed fast preconditioned iterative methods.
Keywords: two-dimensional fractional partial diﬀerential equation, rainbow options pricing, ﬁnite moment log stable model, preconditioner

Parallel Session 2 (23, June)
Title: A numerical study of the Chebyshev collocation method for solving higherorder partial diﬀerential equations
Speaker: Somruthai Apornsaengsawang (Mahidol University)
Author(s): Somruthai Apornsaengsawang and Farida Chamchod
Abstract: This present study is devoted to investigating numerical solutions of thehigher-order partial diﬀerential equations by the Chebyshev collocation method.
At the collocation points, the problems are transformed into the matrix equationsof linear algebraic equations with the unknown Chebyshev coeﬃcients. Approximations of the solutions are then interpolated. To demonstrate eﬃciency andaccuracy of the method, we include some examples and interpret their numericalresults and errors. Our results verify the high accuracy of the method.
Keywords: Chebyshev collocation method, higher-order partial diﬀerential equations, interpolation

Title: An investigation of the hyperbolic telegraph equation with time delay by
the Haar wavelet method
Speaker: Patipol Saengduean (Mahidol University)
Author(s): Patipol Saengduean, Farida Chamchod
Abstract: In this present study, the Haar wavelet method was implemented toapproximate a solution of the hyperbolic telegraph equation. Due to mathematical
simplicity of the method, it can be eﬃciently adjusted to tackle with time delay inthe equation. This technique converts the problem to the algebraic equations ofHaar coeﬃcients that can be easily solved and used to interpolate a solution. To
21

analyze the eﬃciency and accuracy of such method, we provided some examples
and compared our numerical results with exact solutions or solutions obtained
from the ﬁnite diﬀerence method.
Keywords: Haar wavelet method, telegraph equation, time delay, ﬁnite diﬀerence
methodTitle: A mixed H 1 -conforming ﬁnite element method for Maxwell’s equations
with non-H 1 solution
Speaker: Suh-Yuh Yang (National Central University)
Author(s): Huo-Yuan Duan, Roger C. E. Tan, Suh-Yuh Yang*, Cheng-Shu You
Abstract: We propose and analyze a mixed H1-conforming ﬁnite element method
for solving Maxwell’s equations in terms of electric ﬁeld and Lagrange multiplier,
where the multiplier is introduced accounting for the divergence constraint. We
mainly focus on the case that the physical domain is non-convex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell’sequations to be a non-H 1 very weak function and thus causes many numericaldiﬃculties. A pair of H 1 -conforming CP2-P1 elements for electric ﬁeld and multiplier is studied and its stability and error bounds are also derived. Numericalexperiments for source problems as well as eigenvalue problems are presented toillustrate the high performance of the proposed method.
Keywords: Maxwell’s equations, non-H 1 solution, H 1 -conforming ﬁnite element
method, stabilized ﬁnite element method

Parallel Session 3 (24, June)
Title: Classiﬁcation and evolution of bifurcation curves for the one-dimensional
perturbed Gelfand equation with Dirichlet-Neumann boundary conditions
Speaker: Shin-Hwa Wang (National Tsing Hua University)
Author(s): Yu-Hao Liang and Shin-Hwa Wang
Abstract: We study the classiﬁcation and evolution of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand equation with DirichletNeumann boundary conditions given by( au ){ ′′u (x) + λ exp a+u= 0; 0 < x < 1,′u(0) = 0, u (1) = −c < 0,where 4 ≤ a < a1 , for some a1 ≈ 4.107. We prove that, for 4 ≤ a < a1 , there exist
two nonnegative c0 , (= c0 (a)) < c1 , (= c1 (a)) satisfying c0 > 0 for 4 ≤ a < a∗ withsome a∗ ≈ 4.069, and c0 = 0 for a∗ ≤ a < a1 , such that, on the (λ, ∥u∥∞ )-plane, (i)when 0 < c < c0 , the bifurcation curve is strictly increasing; (ii) when c = c0 , the22bifurcation curve is monotone increasing; (iii) when c0 < c < c1 , the bifurcation
curve is S-shaped; (iv) when c ≥ c1 , the bifurcation curve is subset-shaped. This
work is a continuation of Liang and Wang (J. Diﬀerential Equations 260 (2016)
8358–8387) where authors studied this problem for a ≥ a1 , and our results partially prove a conjecture on this problem for 4 ≤ a < a1 in Liang and Wang (J.
Diﬀerential Equations 260 (2016) 8358–8387).
Keywords: multiplicity, positive solution, perturbed Gelfand equation, S-shapedbifurcation curve, ⊂-shaped bifurcation curve, time map
Title: Finite-time singularity for ODEs from the viewpoint of dynamical systems
Speaker: Kaname Matsue (Kyushu University)
Author(s): Kaname Matsue
Abstract: A geometric treatment of ﬁnite-time singularities for autonomous ordinary diﬀerential equations (ODEs) including their asymptotic behavior is concerned. In particular, blow-up behavior of solutions is the one of our main concerns. We show that dynamics on center-stable manifolds of “invariant sets at
inﬁnity” with appropriate time-scale desingularizations as well as blowing-up of
singularities characterize dynamics of blow-up solutions as well as their rigorous
blow-up rates not only of so-called “Type-I” but also of other types. This approach not only characterize monotonous blow-up behavior but also oscillatory
blow-ups in terms of stable manifolds of “periodic orbits at inﬁnity” for example.
Moreover, this mechanism has the similar nature to other ﬁnite-time singularities
such as ﬁnite traveling waves including extinction, compacton waves and quenching behavior. Consequently, the above geometric treatment reveals a common
mechanism among ﬁnite-time singularities in (ordinary) diﬀerential equations.
Keywords: ﬁnite-time singularities (blow-up, extinction, compacton, quenching),
asymptotic behavior, compactiﬁcations, desingularizations

Title: Rigorous numerics of ﬁnite-time singularity for ODEs
Speaker: Kaname Matsue (Kyushu University)
Author(s): Kaname Matsue
Abstract: This talk aims at providing rigorous numerical computation procedurefor ﬁnite-time singularities in dynamical systems based on interval arithmetic.
Combination of time-scale desingularization, computable quantitative estimationof asymptotic behavior and standard integration procedure of ordinary diﬀerential
equations give us validations of true trajectories of dynamical systems involvingﬁnite-time singularities. Our focus includes ﬁnite-time extinction, traveling wave
solutions with half-line or compact support, including several quantitative information such as extinction times and size of supports for compactons. Such validated
23

solutions lead to a plenty of composite wave solutions for degenerate parabolic
equations, for example, with concrete information of proﬁles and evolutions. Finally note that blow-up solutions are also inside our focus, and we have several
rigorous validation results of blow-up solutions.
Keywords: rigorous numerics, ﬁnite traveling waves, blow-up solutions, quantitative estimates in ﬁnite-time singularities

Parallel Session 4 (24, June)
Title: Error analysis of Crouzeix–Raviart and Raviart–Thomas ﬁnite element
methods
Speaker: Takuya Tsuchiya (Ehime University)
Author(s): Kenta Kobayashi, Takuya Tsuchiya
Abstract: We discuss the error analysis of the lowest degree Crouzeix–Raviart
and Raviart–Thomas ﬁnite element methods applied to a two-dimensional Poisson equation. To obtain error estimations, we use the techniques developed byBabuska–Aziz and the authors. We present error estimates in terms of the circumradius and the diameter of triangles in which the constants are independentof the geometric properties of the triangulations. Numerical experiments conﬁrm
the theoretical results obtained.
Keywords: Crouzeix–Raviart and Raviart–Thomas ﬁnite element methods, error
estimation, triangulations, circumradias

Title: On the IAS scheme for strongly coupled systems of singular perturbation
equations
Speaker: Po-Wen Hsieh (National Chung Hsing University)
Author(s): Po-Wen Hsieh, Suh-Yuh Yang, Cheng-Shu You
Abstract: The Il’in-Allen-Southwell (IAS) scheme is a robust ﬁnite diﬀerence
scheme for treating scalar singular perturbation convection-diﬀusion equation whose
solution may display strong boundary layer behavior. In this talk, we will apply
this scheme to 1-D strongly coupled systems. By employing an alternating direction technique, the scheme can be extended to 2-D cases. From the numericalresults, we can observe that when the perturbation parameter ε is small enough,the developed IAS scheme is ﬁrst order convergent in the discrete maximum norm
uniformly in ε on uniform meshes.
Keywords: strongly coupled system, singularly perturbed convection-diﬀusion
equation, boundary and interior layers

24Title: Remarks on the computation of blow-up problems
Speaker: Chien-Hong Cho (National Chung Cheng University)
Author(s): Chien-Hong Cho
Abstract: In many evolution equations, the solution may become unbounded ina ﬁnite time. Such a phenomenon is often called blow-up and the ﬁnite time iscalled the blow-up time. We consider two numerical methods which are often used
for the computation of such problems, one is to deﬁne the temporal increment
adaptively while another is to compute with uniform time mesh. It was known
that even the numerical solution and the numerical blow-up time converge to theexact ones, the asymptotic behaviors, such as the blow-up rate and blow-up set,
for the numerical solution do not always coincide with those of the exact solution.
We would like to explore this problem in more detail. Moreover, we would also
like to investigate in this talk a problem as to whether a higher order scheme gives
a better approximation for blow-up solutions. Several examples will be reported
for these problems.
Keywords: semi-linear heat equation, blow-up, blow-up rate, numerical blow-up
rate

Title: A numerical estimating method of blow-up rates for nonlinear evolution
equations with a scale invariance
Speaker: Takeo Ushijima (Tokyo University of Science)
Author(s): Koichi Anada, Tetsuya Ishiwata, Takeo Ushijima
Abstract: Many nonlinear evolution equations possess solutions which blow up
in ﬁnite time. It is one of the most interesting issues concerning blow-up phenomena how the solutions develop singularities. To determine the blow-up rate is amajor problem in such a direction. Depending on the equations, blow-up rates
take the forms of various types such as simple power type and logarithmic type,and even more complex ones. There are vast amounts of literature deeply considering how solutions blow up. To our best knowledge, however, only a few studies
have investigated numerical methods for blow-up rates. In this talk, we propose
a simple but eﬀective numerical method which estimates blow-up rates for a classof nonlinear evolution equations. Here, we consider the class of equations which
satisfy a scaling invariance. Thanks to this scaling invariance, we adopt the rescaling algorithm and construct a sequence whose behavior leads to the blow-up rate.
Applying the method to several nonlinear equations, we examine the eﬀectivenessof the method. We show that our method is applicable to not only the simple
power type blow-up rates but also more complex ones.
Keywords: nonlinear equation, blow-up rate, scale invariance, numerical method,
rescaling algorithm

25

Parallel Session 5 (24, June)
Title: Discontinuous Galerkin method for an N -dimensional spherically symmetric Poisson equation
Speaker: Yuki Chiba (The University of Tokyo)
Author(s): Yuki Chiba, Norikazu Saito
Abstract: In the theory of PDE, it is known several critical phenomena withcritical value related to space dimension N . If we can make numerical method
for nonlinear PDEs in higher dimension, it is useful for studying critical phenomena through experimental consideration. So, we consider a spherically symmetric
Poisson equation in N -dimensional ball as a model problem. This equation can be
rewritten as a one-dimensional equation with singularity near origin. The previous
study proposed ﬁnite element method using weight function to reduce singularity. One approach to use weight function xN −1 showed optimal weighted L2 error
estimate. However, if space dimension N is too large, there is a disadvantage ofincreasing error near origin because of ﬂatness of weight function. The anotherapproach to use weight function x showed optimal pointwise error estimate. In
this paper, we see the PDE as singularly perturbed convection-diﬀusion equation
through later approach, and apply discontinuous Galerkin method to it. We show
some estimates and oﬀer some numerical results.
Keywords: numerical analysis, discontinuous Galerkin method

Title: High order accuracy of self-gravitational force calculation on inﬁnitely thingaseous disks
Speaker: Chien-Chang Yen (Fu Jen Catholic University)
Author(s): Chien-Chang Yen, Rui-Zhu Wu
Abstract: In this talk, we focused on the self-gravitational force calculation ofinitesimally thin and bounded distribution of surface density in 3-D space. The
method we proposed is to expand the surface density function based on B-spline
function, and calculated the B-spline function and kernel integration by integral
formula directly. The numerical simulations demonstrate the proposed method ishigh order accuracy and fast calculation.
Keywords: direction method, B-spline, self-gravitational force calculation

Title: Convergence analysis of a conservative ﬁnite diﬀerence scheme for the modiﬁed Hunter–Saxton equation
Speaker: Shun Sato (The University of Tokyo)
Author(s): Shun Sato
Abstract: The modiﬁed Hunter–Saxton equation models the propagation of short26

capillary-gravity waves. As it involves a mixed derivative, its initial value problem
on the periodic domain is much more complicated than the standard evolutionary
equations. In particular, the convergence analysis of the numerical methods forthe class of equations with a mixed derivative is yet to be investigated, and there
are only one result for the simplest case in the literature. In this talk, we provethe convergence of a conservative ﬁnite diﬀerence scheme for the modiﬁed Hunter–
Saxton equation. There, in order to overcome the diﬃculty appeared due to amixed derivative, the mass conservation plays a crucial role. Since the mass conservation is a common property of equations with a mixed derivative, our strategy
is expected to be a standard method for convergence analysis of the other equations.
Keywords: modiﬁed Hunter–Saxton equation, conservative numerical methods,
convergence

Parallel Session 6 (24, June)
Title: A penalty method for the Stokes-Darcy problem and the discontinuous
Galerkin approximation
Speaker: Guanyu Zhou (Tokyo University of Science)
Author(s): Guanyu Zhou, Takahito Kashiwabara, Issei Oikawa, Eric Chung
Abstract: For the Stokes-Darcy problem with a curved interface, it requires additional techniques for the numerical implementation of the interface boundary condition. To tackle this problem, we introduce a penalty method to treat the interface boundary condition which is easy to implement using discontinuous Galerkinmethod. The optimal error estimates of the penalty method and the ﬁnite element
discretization are investigated. The numerical experiments are carried out which
comﬁrms the theoretical results.
Keywords: Stokes-Darcy problem, discontinuous Galerkin method, penalty method,
error estimates

Title: On Lyapunov functions constructed by veriﬁed computation
Speaker: Nobito Yamamoto (The University of Electro-Communications)
Author(s): Nobito Yamamoto, Masao Nakamura, Gen Terasaka, Kouki Nitta
Abstract: Constructing Lyapunov functions by veriﬁed computation is discussed.
In the previous works by some of the authors, numerical veriﬁcation methods havebeen established to construct Lyapunov functions of quadratic form within a certain neighborhood of hyperbolic equilibria of given dynamical systems described
by ODEs. However it is impossible generally to ﬁnd any Lyapunov function ofquadratic form for a non-hyperbolic equilibrium. Our approach to solve the diﬃ27

culty is adopting some sort of projections to ﬁnd so called quasi-Lyapunov functions, and we have derived special conditions which enable us to construct quasiLyapunov functions of quadratic form within a neighborhood of non-hyperbolic
equilibria. In our talk, following an introduction of numerical veriﬁcation methods
to construct Lyapunov functions for hyperbolic equilibria, our approach to nonhyperbolic cases will be described together with some numerical examples.
Keywords: veriﬁed computation, dynamical system, ODEs, Lyapunov function

Title: Eﬃcient numerical methods for stochastic optimal control problems for
PDEs
Speaker: Hyung-Chun Lee (Ajou University)
Author(s): Hyung-Chun Lee
Abstract: In this talk, we consider an optimal control problem for an elliptic partial diﬀerential equation and Stokes equations with random inputs. To determine
an applicable deterministic control f (x), we consider the several cases which wecompare for eﬃciency and feasibility. We prove the existence of optimal states,
adjoint states and optimality conditions for each cases. We also derive the optimality systems for the four cases. The optimality system is then discretized bya standard ﬁnite element method for physical space and sparse grid collocation
method and/or Monte Carlo method for probability space, respectively. The numerical experiments are performed for their eﬃciency and feasibility.
Keywords: stochastic optimal control, Monte Carlo, sparse grid collocation, ﬁnite element28

Room 62 (Building No.6)
Parallel Session 1 (23, June)
Title: A framework of explicit eigenvalue bounds for self-adjoint partial diﬀerential operators
Speaker: Xuefeng Liu (Niigata University)
Author(s): Xuefeng Liu
Abstract: For the purpose of high-precision explicit eigenvalue bounds for selfadjoint diﬀerential operators deﬁned over complicated domains, a general framework based on ﬁnite element methods is proposed.
First, rough but rigorous eigenvalue bounds can be obtained by using lower order ﬁnite element method along with a fundamental theorem in em X. Liu, A
framework of veriﬁed eigenvalue bounds for self-adjoint diﬀerential operators, Applied Mathematics and Computation, 267, pp.341-355, 2015. For example, for the
Laplace operator, the eigenvalues can be easily bounded by applying CrouzeixRaviart ﬁnite elements. Explicit eigenvalue bounds for the following eigenvalue
problems have been successfully obtained: the Laplacian, the Biharmonic operator, the Stokes operator, the Steklov eigenvalue problems.
Second, to obtain high-precision eigenvalue bounds, Lehmann-Goerisch’s theorem
along with high-order ﬁnite element methods is adopted in bounding eigenvalues.
By further adopting the interval arithmetic, the explicit eigenvalue bounds fromnumerical computations can be mathematically correct. As a computer-assisted
proof, the veriﬁed eigenvalue bounds have been used to investigate the solution
existence of semi-linear elliptic diﬀerential equations.
Keywords: eigenvalue problem, lower bound, diﬀerential operator, ﬁnite element
methodTitle: On the semigroup property for some structured iterations
Speaker: Chiang Chunyueh (National Formosa University)
Author(s): Chun-Yueh Chiang , Matthew M. Lin
Abstract: Matrix equations are encountered in many applications of appliedmathematics and engineering problems. Problems in determining the solutionsof a matrix equation are closely related to a wide range of challenging scientiﬁcareas. Traditional approaches for ﬁnding numerical solution are based on ﬁxed
point iterations and the speed of the convergence is usually linear.
Recently, we built up a semigroup property for some binary matrix operations; toconstruct such a type of iterations for solving several diﬀerent matrix equations,while the speed of convergence can be R-superlinearly with any order r is given.
In this work, we want to generalize this property to analyze a special instance of29some structured iterations arising from a class of matrix equations. We show that
this property can not only ﬁnd out the solution, but also construct an iterative
approach which converges to the solution with any desired order. Some examples
are shown to demonstrate the robustness of our method. (A joint work with Prof.
Matthew M. Lin)
Keywords: accelerated iterative method, matrix equation, semigroup property

Title: Solving two generalized nonlinear matrix equations
Speaker: Peter Chang-Yi Weng (Academia Sinica)
Author(s): Peter Chang-Yi Weng
Abstract: In this paper, we consider the numerical solutions of two generalized
nonlinear matrix equations. Newton’s method is applied to compute one of generalized nonlinear matrix equations and we get the generalized Stein equation, thenwe adapt the generalized Smith method to ﬁnd the maximal Hermitian positivedeﬁnite solution. Furthermore, we consider the properties of the solution for thegeneralized nonlinear matrix equation. The other generalized nonlinear matrix
equation is computed by Newton’s method to ﬁnd the minimal Hermitian positivedeﬁnite solution. Finally, two numerical examples are presented to illustrate theeﬀectiveness of the theoretical results and the convergence behaviour of the considered methods.
Keywords: generalized nonlinear matrix equations, Newton’s method, maximaland minimal Hermitian positive deﬁnite solutions, generalized Smith method, perturbation analysis

Title: Designing robust control using bilinear matrix inequalities
Speaker: Roberd Saragih (Institut Teknologi Bandung)
Author(s): Roberd Saragih
Abstract: The robust control has been applied to many areas since it can cover
the problem having the disturbance, perturbation, and uncertainty. The diﬃculty
associated with the robust control, however, is in solving the state dependent algebraic Ricatti equations (SDARE). To overcome this diﬃculty, we present the bilinear matrix inequalities (BMI) method for solving the SDARE. The BMI problem
are not convex and have multiple local solutions and it is not solvable in polynomial time. The easiest method to implement for solving the BMI problem is thealternative minimization algorithm, this method is based on the iterative schemesof the alternation between analysis and synthesis via the linear matrix inequalities. It will transform the BMI problem to the LMIs problem, which can be solved
easily via LMI solvers. However, this algorithm might converge very slowly, andeven stop at the non stationary point. The choice of the initial value is important30

for the convergence, so that an acceptable solution could be achieved.
Keywords: robust control, SDARE, bilinear matrix inequalities, linear matrix
inequalities

Parallel Session 2 (23, June)
Title: Integer low-rank approximation and its applications
Speaker: Min Hsiung Lin (National Cheng Kung University)
Author(s): Lin, Min-Hsiung
Abstract: The low-rank approximation of integer matrices has received attention
recently due to its capacity of naturally representing parts of integer data sets.
Diﬀerent from the general low-rank approximations, the integer approximation isnaturally discrete, therefore, the conventional techniques for matrix approximation, such as SVD and non-negative matrix approximation, are inappropriate andunable to solve this problem. To the best of our knowledge, a numerical method
for ﬁnding a low-rank integer approximation of an integer matrix has not been proposed in the literature earlier. In this talk, we want to propose a method to obtain
the integer low-rank approximation of integer matrices. This method consists ofrecursively ﬁnding integer solutions of integer least square problems. Applications
on the real world problems will be given.
Keywords: data mining, matrix factorization, integer least squares problem, clustering, association rule, pattern extraction

Title: Finding the largest eigenpair of an irreducible nonnegative tensor
Speaker: Liu, Ching-Sung (National University of Kaohsiung)
Author(s): C. H. Guo, and W. W. Lin
Abstract: In this talk, we will introduce a Newton–Noda iteration (NNI) forcomputing the Perron pair of a weakly irreducible nonnegative mth-order tensor
A, by combining the idea of Newton’s method with the idea of the Noda iteration. The method requires the selection of a positive parameter θk in the kth
iteration, and produces a scalar sequence approximating the spectral radius of A
and a positive vector sequence approximating the Perron vector. We propose ahalving procedure to determine the parameters θk , starting with θk = 1 for eachk, such that the scalar sequence is monotonically decreasing. Convergence of thissequence to the spectral radius of A (and convergence of the vector sequence tothe Perron vector) is guaranteed for any initial positive unit vector, as long as thesequence θk so chosen is bounded below by a positive constant. In this case, wealways have θk = 1 near convergence and the convergence is quadratic. Very often,
the halving procedure will return θk = 1 (i.e., no halving is actually used) for each31

k. If the tensor is semisymmetric, m ≥ 4, and θk = 1, then the computational
work in the kth iteration of NNI is roughly the same as that for one iteration of
the Ng–Qi–Zhou algorithm, which is linearly convergent for the smaller class of
weakly primitive tensors.
Keywords: nonnegative tensor, M -matrix, nonnegative matrix, positivity preserving, quadratic convergence, Perron vector, Perron root
Title: A method to calculate SVD of sparse and tall-skinny matrices
Speaker: Hiroshi Murakami (Tokyo Metropolitan University)
Author(s): Hiroshi Murakami
Abstract: For a sparse and very tall-skinny matrix A of large size, we calculate
the singular value decomposition A =: QΣW T by using the classical method. We
calculate C := AT A using the sparsity of the √matrix A, and make the eigenvalueT
decomposition of C =: W DW and set Σ := D.
Even if the original matrix A is very large, by the use of the sparsity it could be
stored. However, the matrix Q will be dense in general, and it could be diﬃcult
to store the entire matrix elements of Q.
We can avoid to store Q in dense form. Small size matrices W and D can beconstructed without change of A as above procedure, then Q can be represented
implicitly by using the relation Q = AW Σ† , here † on the shoulder denotes thegeneralized inverse.
From the implicit representation of Q, we can easily calculate singular vectors(columns of Q) on demand. To keep the memory requirement low, instead tomake and store all vectors at once, we generate only a small number of vectors at
a time on demand.
Keywords: tall-skinny, SVD, sparse matrix
Parallel Session 3 (24, June)
Title: High-performance computing for 3D Maxwell equations with fourteen Bravais lattices
Speaker: Tsung-Ming Huang (National Taiwan Normal University)
Author(s): Tsung-Ming Huang, Tiexiang Li, Wei-De Li, Jia-Wei Lin, Wen-Wei
Lin
Abstract: The numerical simulation of the band structure of three-dimensional
photonic crystals leads to large-scale generalized eigenvalue problems (GEPs). Dueto a high dimensional subspace associated with the eigenvalue zeros, it is very challenging to solve the GEP. In this paper, we focus on developing a high-performancecomputing method to solve GEP for all fourteen Bravais lattices. For each lattice, we derive the explicit matrix form of the discrete double-curl operator by32

using Yee’s scheme and classify all the matrices into four general types. The
eigen-decompositions of these four general matrices are then derived. Based onthese eigen-decompositions, the nullspace-free method is applied to exclude thezero eigenvalues from the associated generalized eigenvalue problem. Applying
these theoretical results, a high-performance computing package FAME (Fast Algorithm for Maxwell’s Equations) with GPU acceleration is proposed to ﬁnd thetarget eigenpairs for any lattices. Numerical results illustrate that FAME successfully solves each of a set of 5.184 million dimension eigenvalue problems within 18
to 50 seconds on a workstation with NVIDIA Tesla P100 GPUs.
Keywords: Maxwell’s equation, three-dimensional photonic crystals, generalized
eigenvalue problems, Bravais lattices, eigen-decomposition, nullspace-free method,
FAME, GPU

Title: An eﬃcient and stable solver for Kohn–Sham equation
Speaker: Yang Kuang (University of Macau)
Author(s): Yang Kuang, Guanghui Hu
Abstract: We present an eﬃcient and stable solver for Kohn–Sham equation
which can help to speed up the convergence of self-consistent ﬁled (SCF) iteration, and even eﬀectively avoid the failure of SCF iteration. We ﬁnd that forelectronic structure calculations the imaginary time propagation (ITP) method to
obtain the ground state will always give a convergent result. However, the wavefunctions should be propagated long time enough to get an accurate result whichleads to the ineﬃciency of ITP. In this work, we take advantage of the convergenceof ITP to generate a qualiﬁed initial guess for SCF iteration. Numerical examples
show that with the ITP-generated initial guess, the SCF iteration is faster andstronger than it with random initial guess.
Keywords:

Title: Application of an energy-preserving integrator to quantum-mechanicalwavepacket dynamics
Speaker: Yusaku Yamamoto (The University of Electro-Communications)
Author(s): Tsubasa Sakai, Shuhei Kudo, Hiroto Imachi, Yuto Miyatake, Takeo
Hoshi and Yusaku Yamamoto
Abstract: Simulation of quantum-mechanical wavepacket dynamics plays a crucial role for the design of organic electronic devices. Here, the governing equationis the nonlinear time-dependent Schrodinger equation and the potential has both arandom component due to structural disorder and a delta function-like component
due to impurity atoms. To solve such an equation stably and accurately, we use
the BM4 method, which is a parallelizable 4th order energy-preserving continuous
33

stage Runge-Kutta method recently proposed by Miyatake and Butcher (Y. Miyatake and J. C. Butcher, SIAM J. Numer. Anal., 54(3), 1993-2013). We applied
the method to a two-dimensional time-dependent nonlinear Schrodinger equation
with a delta function-like potential discretized on a regular grid with up to 640
x 640 mesh points. This system is a prototypical model of a two-dimensional
disordered organic material, in which each mesh point corresponds to an organic
molecule such as pentacene (C22H14) and the potential corresponds to a defect.
Numerical tests show that the BM4 method can solve the equation stably despite
the existence of the delta function-like potential and preserves the total energy
to 14-digit accuracy throughout the simulation. We also parallelized the method
using its inherent parallelism and achieved up to 2.8 times speedup with 3 computational nodes.
Keywords: quantum simulation, Runge-Kutta methods, structure-preserving methods, parallel computing

Parallel Session 4 (24, June)
Title: A direct-forcing immersed boundary projection method for ﬂuid-solid interaction problems
Speaker: Cheng-Shu You (National Central University)
Author(s): Tzyy-Leng Horng, Po-Wen Hsieh, Suh-Yuh Yang, Cheng-Shu You
Abstract: We introduce a simple direct-forcing immersed boundary projection
method in conjunction with a prediction-correction process for simulating the dynamics of ﬂuid-solid interaction problems. The immersed solid object can be stationary or moving in the ﬂuid with a prescribed velocity. This method is mainly
based on the introduction of a virtual force which is distributed only on the immersed solid bodies and appended to the ﬂuid momentum equations to accommodate the internal boundary conditions at the immersed solid boundaries. Morespeciﬁcally, we ﬁrst predict the virtual force on the immersed solid domain by using the diﬀerence between the prescribed solid velocity and the computed velocity,which is obtained by applying the Choi-Moin projection scheme to the incompressible Navier-Stokes equations on the entire domain including the portion occupiedby the solid bodies. The predicted virtual force is then added to the ﬂuid momentum equations as an additional forcing term and we employ the same projection
scheme again to correct the velocity ﬁeld, pressure and virtual force. Numericalexperiments of several benchmark problems are performed to illustrate the simplicity and high performance of the proposed prediction-correction approach.
Keywords:

34

Title: Generation of good point conﬁgurations by global optimization for interpolation in reproducing kernel Hilbert spaces
Speaker: Ken’ichiro Tanaka (The University of Tokyo)
Author(s): Ken’ichiro Tanaka
Abstract: The accuracy of approximation methods for functions depends crucially on the location of data points. In approximation theory, there are some
popular points such as Chebyshev points, Fekete points, Leja points, Padua points,
etc. They are mainly aimed at keeping the Lebesgue constant under control andknown to be good also for kernel-based approximation.
For kernel-based approximation methods, there are other approaches based on theerror bound of the interpolation in a reproducing kernel Hilbert space. In fact,
such a bound is provided by the power function, which depends on the data points
and the kernel of the space. For example, DeMarchi et al. (2005) propose a greedy
algorithm for placing points one by one, and Iske (2000) uses a similar approach.
In this work, we propose a global optimization approach for generating good point
conﬁgurations for for interpolation in reproducing kernel Hilbert spaces. We provide an approximation of the power function based on Mercer’s theorem and obtainan optimization problem to determine a good point conﬁguration. Then, we adopt
a convex relaxation of the problem to obtain its good approximate solution, which
provides a good point conﬁguration. By some numerical experiments, we can observe the good property of the point conﬁguration.
Keywords: reproducing kernel Hilbert space, interpolation, point conﬁguration,
power function, global optimization
Title: Optimal estimation for the Fujino-Morley interpolation error constants
Speaker: Shih-Kang, Liao (Niigata University, JP & National Cheng Kung University, TW)
Author(s): Shih-Kang Liao, Xuefeng Liu
Abstract: Let K be a triangle with the largest edge length as h. The vertices of
K are denoted by O, A and B and the edges by e1 , e2 , e3 . Given u ∈ H 2 (K), theFujino-Morley interpolation ΠFM maps u to a quadratic polynomial that satisﬁes(ΠFM u − u)(P ) = 0, P = O, A, B;∫∂(ΠFM u − u)mds = 0, i = 1, 2, 3.
∂n
ei

(1)
(2)

In this talk, we consider the Fujino-Morley interpolation error constant C0 and C1 ,which satisfy∥ΠFM u − u∥ ≤ C0 ∥ΠFM u − u∥2 ,∥ΠFM u − u∥1 ≤ C1 ∥ΠFM u − u∥2 .
35

In [Numer. Math., bf 126(1):33–51, may 2014.], em Carsten Carstensen and em
Dietmar Gallistl provided rough bounds of constants C0 and C1 . In our research,
the problem of constant estimation is transformed to the eigenvalue problem forcertain Biharmonic diﬀerential operator, which is further solved by applying the
eigenvalue estimation method developed by em X. Liu[Appl. Math. Comput.,bf
267:341-355, 2015.]. Particularlly, for triangle elements with longest edge length
less than 1, the optimal estimation for the constants is obtained as follows,
0.0735 ≤ C0 ≤ 0.0736,

0.1886 ≤ C1 ≤ 0.1888.

Keywords:
Title: Approximate inversion method for time-fractional subdiﬀusion equations
Speaker: Seakweng Vong (University of Macau)
Author(s): Seakweng Vong
Abstract: When ﬁnite-diﬀerence method is applied to time-fractional subdiﬀusion
equations, it usually leads to a large-scale linear system with a block lower triangular Toeplitz coeﬃcient matrix. In this talk, the approximate inversion method
is employed to solve this system. A suﬃcient condition is proved to guarantee
the high accuracy of the approximate inversion method for solving the block lower
triangular Toeplitz systems, which are easy to verify in practice and have a wide
range of applications. Numerical experiments are presented to verify the validityof theoretical results. This talk is partially supported by the grant MYRG201700098-FST from University of Macau
Keywords: time-fractional subdiﬀusion equations, Approximate inversion method
Parallel Session 5 (24, June)
Title: Fast solution algorithms for exponentially tempered fractional diﬀusion
equations
Speaker: Siu-Long Lei (University of Macau)
Author(s): Siu-Long Lei, Daoying Fan, Xu Chen
Abstract: In this talk, a fast-iterative method and a fast-direct method is proposed for solving one-dimensional and two-dimensional tempered fractional diﬀusion equations with constant coeﬃcients. The proposed iterative method is accelerated by circulant preconditioning which is shown to converge superlinearly while
the proposed direct method is based on circulant and skew-circulant representation for Toeplitz matrix inversion. Numerical examples are provided to illustrate
the eﬀectiveness and eﬃciency of the proposed methods.
36

Keywords: Toeplitz matrix, circulant preconditioner, fast Fourier transform,
tempered fractional diﬀusion equations, circulant and skew-circulant representation of Toeplitz inversion

Title: Validated computation for the matrix principal logarithm
Speaker: Shinya Miyajima (Iwate University)
Author(s): Shinya Miyajima
Abstract: A matrix X is a logarithm of a matrix A if the exponential of X coincides with A. Any nonsingular matrix has inﬁnitely many logarithms, but the onethat is most useful in practice is the principal logarithm. For a matrix with noeigenvalues on the closed negative real axis, the principal logarithm is the globallyunique one whose eigenvalues have imaginary parts lying in the interval (−π, π).
Numerical algorithms for the logarithm have been developed.
The work presented in this talk addresses the problem of veriﬁed computations
for the principal logarithm, speciﬁcally, numerically computing an interval matrix
which is guaranteed to contain the logarithm. The pioneering work seems to be
the VERSOFT routine VERMATFUN. This routine is applicable not only to alogarithm but also to other matrix functions, requires quadruple complexity, anddoes not say that the logarithm contained in the computed interval is principal.
The purpose of this talk is to propose two iterative veriﬁcation algorithms for the
principal logarithm. The ﬁrst algorithm is based on a numerical spectral decomposition and requires only cubic complexity per iteration. The second algorithm isbased on a numerical Jordan decomposition and applicable even for defective matrices. Numerical results show the eﬀectiveness and robustness of the algorithms.
Keywords: matrix logarithm, principal logarithm, veriﬁed numerical computation

Title: Computation of the matrix fractional power based on the double exponential formula
Speaker: Fuminori Tatsuoka (Nagoya University)
Author(s): Fuminori Tatsuoka, Tomohiro Sogabe, Yuto Miyatake, Shao-Liang
Zhang
Abstract: The matrix fractional power has attracted attention because it arises
in many research ﬁelds, such as lattice QCD calculations, fractional partial diﬀerential equations, and quantum communication channel.
We consider computation of the matrix fractional power by numerical integration.
The matrix fractional power can be represented as an improper integral of a slowly
decaying function over the half inﬁnite interval. Conventional approaches are basedon some variable transformations (e.g. [J. R. Cardoso, ETNA, 2012]) in order to37use Gaussian quadrature. However, the convergence of the approaches could beslow for an ill-conditioned matrix or a speciﬁc exponent. Therefore, these methods
may not be the best choice in that situation.
When considering an improper integrals with one variable, the double exponential(DE) formula is known to be one of the best choices. This motivates us to consider
the DE formula for the matrix fractional power. In this talk, we propose an algorithm for computing the matrix fractional power based on the DE formula. Since
the transformed interval is inﬁnite, we estimate the truncation error and give a
promising ﬁnite interval. Some numerical examples show that the proposed algorithm works well even for the cases the convergence of the conventional approaches
is slow.
Keywords: matrix fractional power, matrix function, numerical integration, theDE formula

Parallel Session 6 (24, June)
Title: Fast validated computation for the minimal nonnegative solution of thenonsymmetric algebraic Riccati equation
Speaker: Shinya Miyajima (Iwate University)
Author(s): Shinya Miyajima
Abstract: Consider the nonsymmetric algebraic Riccati matrix equation (NARE)
associated with M-matrix. The NARE of this type appears in ﬂuid queues models
and transport equations. The solution of practical interest in these applications
is the minimal nonnegative solution. Conditions for the existence of the solution
and numerical algorithms for the solution are extensively studied.
The work presented in this talk addresses the problem of veriﬁed computation
for the minimal nonnegative solution, speciﬁcally, computing an interval matrix
which is guaranteed to contain the solution. While there are well-establishedveriﬁcation algorithms for continuous and discrete time algebraic Riccati equations,less attention has been paid to the NARE. To the author’s best knowledge, [S.
Miyajima, Fast veriﬁed computation for solutions of algebraic Riccati equations
arising in transport theory, Numer. Linear Algebra Appl., 24, 1-12, 2017] is the
only literature which mentions the veriﬁcation algorithm for the NARE. On theother hand, this paper treats a special case only.
The purpose of this talk is to propose an iterative veriﬁcation algorithm for theminimal nonnegative solution. This algorithm requires only cubic complexity periteration, and moreover veriﬁes that the solution contained in the interval is unique.
Numerical results show properties of the algorithm.
Keywords: nonsymmetric algebraic Riccati equation, minimal nonnegative solution, veriﬁed numerical computation38

Title: Validated computation for the Hermitian positive deﬁnite solution of the
conjugate discrete-time algebraic Riccati equation
Speaker: Shinya Miyajima (Iwate University)
Author(s): Shinya Miyajima
Abstract: In this talk, we consider the conjugate discrete-time algebraic Riccati
matrix equation (CDARE). Investigating the CDARE is a parallel study of a conjugate nonlinear matrix equation, which has an application in the study of modern
quantum theory. The solution of interest in the CDARE is the Hermitian positivedeﬁnite (HPD) solution. In [M.M. Lin, C.-Y. Chiang, An accelerated technique
for solving one type of discrete-time algebraic Riccati equations, J. Comp. Appl.
Math., 338, 91-110, 2018], conditions for the existence of the HPD solution are
elegantly derived, a fast iterative algorithm for the solution is proposed, and it isshown that the speed of convergence can be of any desired order.
The work presented in this talk addresses the problem of veriﬁed computation for
the HPD solution, speciﬁcally, computing an interval matrix which is guaranteed
to contain the solution. To the author’s best knowledge, the veriﬁcation algorithm
designed speciﬁcally for the CDARE has not been written down in literatures.
The purpose of this talk is to propose an iterative veriﬁcation algorithm for the
HPD solution. This algorithm requires only cubic complexity per iteration, andmoreover veriﬁes that the solution contained in the interval is unique. Numericalresults show properties of the algorithm.
Keywords: conjugate discrete-time algebraic Riccati equation, Hermitian positive deﬁnite solution, veriﬁed numerical computation

Title: Inheritance properties of the Krylov subspace method for solving continuoustime algebraic Riccati equations
Speaker: Hung-Yuan Fan (National Taiwan Normal University)
Author(s): Liping Zhang and Eric King-wah Chu
Abstract: In this talk we consider the numerical solution of large-scale continuoustime algebraic Riccati equations arisen from the LQG optimal control problems.
We propose a projection method from a rational Krylov subspace interpretationof the doubling algorithm. More importantly, we prove that the solvability of the
projected algebraic Riccati equation does not have to be assumed but is inheritted,
under mild and reasonable assumptions. Some numerical examples are presented
for illustrating the inheritance properties of the proposed method.
Keywords: algebraic Riccati equation, Arnoldi process, Cayley transform, largescale problem, LQG optimal control, projection method, rational Krylov subspace

39

Room 63 (Building No.6)
Parallel Session 1 (23, June)
Title: Pseudospectral matrix element model for sloshing simulation in rectangulartank
Speaker: Ming-Jyh Chern (National Taiwan University of Science and Technology)
Author(s): Ming-Jyh Chern, Nima Vaziri
Abstract: Sloshing motion is a free surface ﬂow in a container due to external
excitation such as earthquake, vibration during rocket launch and wave impact
around a ship. Hydrodynamic loading of sloshing motion may damage the container. Especially, when resonance occurs, the container may be broken. The
proposed pseudospectral matrix element (PSME) model is to provide a high ordernumerical simulation of sloshing motion. A 3-D rectangular tank is considered inthis model. Prescribed tank motion is utilized to excite liquid sloshing in the tank.
The sigma transformation is used to capture the transient free surface proﬁle in the
PSME model. Eﬀects of liquid depth and the aspect ratio of the tank on sloshingmotion are discussed in the study. Free surface variations of sloshing are obtainedin the results. Hydrodynamic loadings on the tank are calculated for design of thetank. The ﬁrst order and the second order resonance phenomena are found in thesimulations and their eﬀects are discussed.
Keywords: sloshing, pseudospectral matrix element method, resonance, sigmatransformation

Title: Numerical simulation of tidal bore Bono at Kampar river
Speaker: Sri R. Pudjaprasetya (Institut Teknologi Bandung)
Author(s): A.C. Bayu, S.R. Pudjaprasetya, U.J. Wisha, S. Husrin
Abstract: Tidal bore is a natural phenomena sometimes encountered in rivers.
In Indonesia, this natural phenomenon is found in the Kampar River, which is
known as tidal bore Bono. Sometimes, these tidal bore phenomena disappear,
as happened to the Mascaret, tidal bore on the River Seine France. Through anunderstanding of the formation of tidal bore mechanism, there is hope that the
tidal bore Bono in Kampar River can be preserved. In this paper, the occurrenceof tidal bore Bono is simulated using the non-hydrostatic Saint-Venant equation
under a staggered grid formulation. To test the accuracy of the implementation,
several scenarios of hydraulic jumps were simulated ﬁrst. The numerical results
has shown to quantitatively conform the analytical formula of bore height and velocity, two parameters that are important to characterize a bore wave. Further, byadopting a model that incorporates the non-hydrostatic pressure, our simulation40

show the appearance of an undular bore accompanying the shock front. Finally,using tidal current data measured along Kampar River estuary, our simulation
using the actual river topography can show the appearance of tidal bore Bono.
Our simulations were shown to be in fair agreement with the measurement.
Keywords: hydraulic jump, undular bore, Saint-Venant equations, non-hydrostatic
numerical scheme

Title: Repulsive 2-body correlation function in two-dimensional point-vortex system evidenced by numerical simulatin on PEZY-SC2
Speaker: Yuichi Yatsuyanagi (Shizuoka University)
Author(s): Yuichi Yatsuyanagi(Shizuoka University), Hiroshi Ohtsuka(Kanazawa
University), Tadatsugu Hatori(NIFS), Toshikazu Ebisuzaki(RIKEN) and Aklio
Sanpei (Kyoto Institute of Technology)
Abstract: In a (neutral) plasma Debye shield is a dielectric phenomenon associated with counter-charged particles. Although there is no counter-charged particles
in non-neutral pure electron plasmas, Debye shield is accomplished by a density
depression, i.e. by the formation of a hole. To reveal its mechanism, we study a 2body correlation function in two-dimensional point vortex system as the dynamics
of the electrons in the perpendicular plane to the strong magnetic ﬁeld is identicalwith the two-dimensional Euler equation, which can be solved by the point vortex
model. The obtained solution shows a repulsive correlation in certain range of
distance, which corresponds a hole formation. To evidence the analytical result,numerical simulations of two-dimensional point vortex simulations with 100-1000
vortices are performed on a brand-new supercomputer, PEZY-SC2 at RIKEN. I
will mainly present numerical result.
Keywords: point vortex simulation, numerical simulation, PEZY-SC2, non-neutral
plasma experiments

Title: On the precise structure of the impulse response for solutions of twodimensional Liouville type equations
Speaker: Hiroshi Ohtsuka (Kanazawa University)
Author(s): Tadatsugu Hatori (NIFS), Hiroshi Ohtsuka (Kanazawa Univ.), Yuichi
Yatsuyanagi (Shizuoka Univ.)
Abstract: Motivated by the experimental facts, the formation of a hole, observedin conﬁned non-neutral plasma, we are interested in the impulse response for solutions of two-dimensional Liouville type equations, that is, the asymptotic behavior
for solutions of Liouville type equations with one singular source as the singularity
vanishes. The solutions of Liouville type equations are considered to represent themean ﬁeld approximation of the density of particles in non-neutral plasma. In
41this talk, we will present several facts derived from the exact solution for singular
Liouville type equation established by Prajapat and Tarantello (2001) and will
discuss some phenomenological meaning of them.
Keywords: Liouville type equation, singular Liouville type equation, non-neutral
plasma experiments
Parallel Session 2 (23, June)
Title: Transmission dynamics of two strains of porcine reproductive and respiratory syndrome virus: cross-immunity, virulence, and vaccination
Speaker: Phithakdet Phoongurn (Mahidol University)
Author(s): Phithakdet Phoo-Ngurn and Farida Chamchod
Abstract: Porcine reproductive and respiratory syndrome is an important swine
disease that causes reproductive failure and respiratory problems in pigs. The
disease leads to signiﬁcant economic losses to the swine industry. In this present
study, a mathematical model for describing transmission dynamics of two different strains of porcine reproductive and respiratory syndrome virus (PRRSV)
with cross-immunity was developed to investigate the eﬀects of cross-immunity
and invasion of a highly pathogenic strain. We found that cross-immunity andvaccination strategies play an import role in determining severity of an outbreak,prevalence of PRRSV, and invasion of a new strain.
Keywords: PRRSV, mathematical modeling, cross-immunity, transmission dynamics
Title: Modeling the spread of rabies among vagrant dogs: short- and long-term
incubation, seasonality, and vaccination
Speaker: Pirommas Techitnutsarut (Mahidol University)
Author(s): Pirommas Techitnutsarut, Farida Chamchod
Abstract: Rabies is a fatal neurological disease that is endemic in many parts of
the world. The disease generally aﬀects mammals but domestic dogs are the main
reservoir. In this study, we developed a mathematical model that incorporates two
types of an incubation period, short- and long-term, to investigate the spread ofrabies among vagrant dogs. By taking seasonality in transmission and recruitment
into account, we predict that outbreaks of rabies among vagrant dogs peak at every
dry season. Moreover, we found that an incubation period may delay outbreaks
and vaccination may help reduce the prevalence of rabies.
Keywords: rabies, incubation period, vagrant dogs, transmission dynamics, seasonality

42Title: Traveling wave solutions of an epidemic model with free boundary
Speaker: Takeo Ushijima (Tokyo University of Science)
Author(s): Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima
Abstract: Free boundary problems are recently used to model phenomena of biological invasion for species such as migration into a new habitat (see, e.g., Du& Lin (2010) and references therein). These ideas are also applied to epidemic
models in order to investigate the front motion of infected individuals spreading
into a region where disease is not prevalent (see, e.g., Kim et al. (2013)). In thistalk, we consider a simple diﬀusive epidemic model. Hosono & Ilyas (1995) and
Kaellen (1984) proved the existence of traveling wave solutions of the model. We
extend the results in Kaellen (1984) to the model with free boundary, namely weprove the existence of a traveling wave solution. For free boundary problems, solutions propagating with the same proﬁle and the same speed are also called semiwave solutions in some literatures, e.g., Du & Lou (2015) and Kawai & Yamada(2016). We numerically investigate the existence and non-existence region of thesemi wave with respect to given parameter values in the model. We also observe
the front motion of the model with free boundary.
Keywords: free boundary, epidemic model

Parallel Session 3 (24, June)
Title: Global bounded variation solutions describing Fanno-Rayleigh ﬂuid ﬂows
in nozzles
Speaker: Shih-Wei Chou (National Central University)
Author(s): Shih-Wei Chou, John M. Hong, Bo-Chih Huang and Reyna Quita
Abstract: In this talk, we investigate the initial-boundary value problem of compressible Euler equations including friction and heating that model the transonicFanno-Rayleigh ﬂows through symmetric variable area nozzles. In particular, thecase of contracting nozzles is considered. A new version of a generalized Glimm
scheme (GGS) is presented for establishing the global existence of entropy solutions with bounded variation. The extended Glimm-Goodman’s type of wave
interaction estimates are investigated to determine the stability of the scheme andthe positivity of gas velocity that results in the existence of the weak solution.
Moreover, a quantitative relation between the shape of the nozzle, friction, andheat is proposed for the global existence result in the contracting nozzle. Numerical simulations of the contraction-expansion and expansion-contraction nozzles arepresented to validate the scheme.
Keywords: Fanno-Rayleigh ﬂows, transonic ﬂow, compressible Euler equations,entropy solutions, initial-boundary value problem, Riemann problem, boundary
Riemann problem, generalized Glimm scheme
43

Title: Multiscale reconstruction algorithm in time harmonic viscoelastic equation
for MR elastography
Speaker: Suguru Maekawa (Kyoto University)
Author(s): Suguru Maekawa
Abstract: In this talk, we will discuss on coeﬃcient recostruction of time harmonic viscoelastic equation for MR Elastography, which is one of modalities to
identify stiﬀness of human tissue. We will consider a minimizing problem of acertain cost function as a formularization for reconstruction. For this problem wecan apply gradient method. However the results by some conventional gradientmethods are unreliable. In this presentation, we will propose a multiscale reconstruction algorithm for fast and reliable reconstruction.
Keywords: MR elastography, time harmonic viscoelastic equation, coeﬃcient reconstruction

Title: An ε-Stokes problem with several pressure boundary conditions
Speaker: Kazunori Matsui (Kanazawa University)
Author(s): Masato Kimura, Kazunori Matsui, Adrian Muntean, Hirofumi Notsu
Abstract: We propose an ε-Stokes problem connecting the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter ε > 0.
We show that the solution to the ε-Stokes problem, convergences as ε tends to
0 or ∞ to the Stokes and pressure-Poisson problem, respectively. We also establish error estimates in suitable norms between the solutions to ε-Stokes problem,
pressure-Poisson problem and Stokes problem, respectively. Numerical results illustrating our analysis work are presented.
Keywords: Stokes problem, pressure-Poisson equation, asymptotic analysis, ﬁnite element method

Parallel Session 4 (24, June)
Title: Two-layer non-hydrostatic scheme for simulations of wave runup
Speaker: Maria Artanta Ginting (Institut Teknologi Bandung)
Author(s): M. A. Ginting, S. R. Pudjaprasetya, D. Adytia
Abstract: A non-hydrostatic model is implemented to simulate a wave runup.
This model is approximated by resolving the ﬂuid depth into two-layer with equal
depths. The equations are then solved by applying the staggered grid scheme. For
wave generation mechanism the embedded inﬂuxing as proposed by Liam et.al.
(2014) is adopted. The ﬁrst benchmark test was simulating a periodic wave runupof Carrier and Greenspan. Good agreement between numerical result and the44

analytical formula are obtained for both; surface proﬁle as well as the moving
shoreline. Further, runup simulation were conducted for solitary waves climbing
up a sloping beach, following the experimental set up by Synolakis. Two simulations with solitary waves of small and large amplitudes were conducted. Again
good agreement were obtained, especially for the prediction of run up height.
Keywords: non-hydrostatic model, wave runup, embedded inﬂuxing
Title: The collective motion of camphor disks in a annular water channel
Speaker: Mamoru Okamoto (Hokkaido University)
Author(s): Mamoru Okamoto, Masaharu Nagayama, Satoshi Nakata
Abstract: Billiard and jamming like motions of camphor papers placed over water
have recently been observed in annular water channels. We investigate the mechanisms of these motions by constructing a mathematical model for the camphor
system. In particular, we study the motion of two camphor papers by means of
numerical simulation and mathematical analysis. As a result of our investigations,
we have uncovered various morphologies of the camphor disk motions. Moreover,
we were able to obtain the existence and stability of rotational and cluster motions numerically, by means of computer aided analysis. We have found that thebilliard-like phenomenon is caused by the coexistence of uniform rotational andsymmetrical oscillatory motions, and that the jamming phenomenon arises from a
Hopf bifurcation of the uniform rotating motion.
Keywords: mathematical modeling, self-propelled system, collective motion
Title: The eﬀect of cubic basic concentration gradient on the onset of doublediﬀusive convection in micropolar ﬂuids
Speaker: Ruwaidiah Idris (Universiti Malaysia Terengganu)
Author(s): Ruwaidiah Idris
Abstract: In this paper, the linear stability analysis is perform to study the eﬀectof cubic basic concentration gradient to the onset of double-diﬀusive convection in
micropolar ﬂuid. The inﬂuence of various parameters on the onset of convection
has been analyzed. It is observed that, the basic concentration gradient has minimize inﬂuence to the suspended particle then destabilize the system.
Keywords: cubic concentration, double-diﬀusive, micropolar ﬂuid
Title: Energy-preserving model reduction for Poisson systems
Speaker: Yuto Miyatake (Osaka University)
Author(s): Yuto Miyatake
Abstract: Model reduction methods have been widely used to solve very large dynamical systems eﬃciently. The basic idea of the methods is to approximate such45

large systems with much lower and simpler ones such that the reduced systems
capture the given solution data as precise as possible. However, most conventional
approaches could destroy other aspects of the original system. This motivates us
to consider new types of morel reduction methods that can keep some underlying
structures of the problem. In the last few years, the main focus has been put on thesymplecticity or energy-preservation of Hamiltonian systems. In this talk, keeping
in mind that discretizing Hamiltonian PDEs often leads to Poisson systems, we
propose an energy-preserving model reduction technique for Poisson systems. We
also show numerical results to conﬁrm the eﬃciency taking the KdV equation as
our working example.
Keywords: structure-preserving numerical methods, model reduction

Parallel Session 5 (24, June)
Title: Riemannian optimization methods for IEPs
Speaker: Xiao-Qing Jin (University of Macau)
Author(s): Xiao-Qing Jin
Abstract: In this talk, we use the Riemannian inexact Newton method to solve
some inverse eigenvalue problems from matrix computations. Both theoretical andnumerical results will be given.
Keywords:

Title: Continuation Methods for Computing Eigenpairs of Nonnegative Tensors
Speaker: Yueh-Cheng Kuo (National university of Kaohsiung)
Author(s): Yueh-Cheng Kuo, Wen-Wei Lin, Ching-Sung Liu
Abstract: In this talk, a homotopy continuation method for the computationof nonnegative Z-/H-eigenpairs of a nonnegative tensor is presented. We showthat the homotopy continuation method is guaranteed to compute a nonnegative
eigenpair. Additionally, using degree analysis, we show that the number of positive
Z-eigenpairs of an irreducible nonnegative tensor is odd. A novel homotopy continuation method is proposed to compute an odd number of positive Z-eigenpairs,
and some numerical results are presented.
Keywords: continuation method, nonnegative tensor, Z-eigenpair

Title: Structure-preserving ﬂows of symplectic matrix pairs
Speaker: Shih-Feng Shieh (National Taiwan Normal University)
Author(s): Shih-Feng Shieh
Abstract: We construct a nonlinear diﬀerential equation of matrix pairs that is46

invariant (the in a class of symplectic matrix pairs. Its solution also preserves invariant subspaces on the whole orbit (the Eigenvector-Preserving Property). Sucha ﬂow is called a structure-preserving ﬂow and is governed by a Riccati diﬀerential equation (RDE) for some suitable Hamiltonian matrix. In addition, Radon’slemma leads to an explicit form of this equation. Therefore, blow-ups for the
structure-preserving ﬂows may happen at a ﬁnite t. To continue, we then utilize
the Grassmann manifolds to extend the domain of the structure-preserving ﬂow to
the whole R subtracting some isolated points. On the other hand, the StructurePreserving Doubling Algorithm (SDA) is an eﬃcient numerical method for solving
algebraic Riccati equations and nonlinear matrix equations. In conjunction withthe structure-preserving ﬂow, we consider the two special classes of symplectic pairs
corresponding algorithms SDA-1 and SDA-2. It is shown that at t = 2k−1 , k ∈ Z
this ﬂow passes through the iterates generated by SDA-1 and SDA-2, respectively.
Therefore, the SDA and its corresponding structure-preserving ﬂow have identical
asymptotic behaviors, including the stability, instability, periodicity, and quasiperiodicity of the dynamics.
Keywords:

47

Room 64 (Building No.6)
Parallel Session 1 (23, June)
Title: A self-propelled particle model for representing collective cell migrations
Speaker: Takamichi Sushida (Hokkaido University)
Author(s): Takamichi Sushida, Hitomi Mori, Sumire Ishida, Kazuya Furusawa,
Hisashi Haga, and Masakazu Akiyama
Abstract: Recently, in the fruiting body formation of Dictyostelium discoideum(D. discoideum) and the somite formation of the zebraﬁsh covered with the basement membrane, since collective rotational migrations are observed when cellulartissues elongate, the relationship between collective rotational migrations and tissue elongations has been attracted. In this study, in order to describe cellularmechanisms causing collective rotational migrations, we construct a mathematicalmodel assuming typical cellular mechanisms. As a result of numerical simulations, when individual cells release diﬀusible substances such as cyclic AMP of D.
discoideum, it is shown that chemotaxis induces collective rotational migrations
robustly. In addition, such as cells in a somite of the zebraﬁsh, when cell clusters
are covered with the basement membrane, by representing the membrane using
Phase-ﬁeld method, it is shown that the repulsive force from the membrane occurs
collective rotational migrations. In particular, by a nonlinearity of the magnitude of driving force and the speed of cell migration introducing into our model,
we show that switch rotational migrations which repeat inversion of rotational
direction occur. Moreover, by changing the cellular culture environment taking
numerical results of our model as a hint, we discover a new experimental resultthat cells exert switch rotational migrations.
Keywords: collective cell migration, morphogenesis, self-propelled particle model,
phase-ﬁeld method, numerical simulation

Title: A two-item inventory-dependent demand model with return and all-units
discount
Speaker: Dharma Lesmono (Universitas Katolik Parahyangan)
Author(s): Dharma Lesmono, Tauﬁk Limansyah, Neilshan Loedy
Abstract: In this paper, we propose a two-item inventory model with deterministic demands, return and all-units discount. We consider a retailer who sells
two products, where demands of these two products are deterministic and haveinventory-dependent demand feature in the sense that demands are related to thenumber of inventory available at certain time. In developing our model, we assume
that supplier oﬀers quantity discount to the retailer, these two products have different constant deterioration rates and retailer can return some of unsold products48

at some costs near the end of the period. The objective of the model is to ﬁnd the
optimal return time and the optimal ordering quantity that minimizes the total
cost for the retailer. In dealing with this model, we consider two replenishment
policies, namely the individual and joint replenishment policy. Some numericalexamples are given to illustrate our model and to compare the individual and joint
replenishment policies by considering quantity discount or not in terms of their
total costs. We found that joint replenishment policy by considering quantity discount mostly gives the minimum total cost.
Keywords: inventory-dependent demand model, two-item, deterioration, allunits discountTitle: Mathematical model of transmission dynamics and optimal control strategies for 2009 A/H1N1 inﬂuenza in the Republic of Korea
Speaker: Soyoung Kim (Konkuk University)
Author(s): Soyoung Kim, Jonggul Lee, and Eunok Jung
Abstract: A mathematical model for the transmission dynamics of the 2009
A/H1N1 inﬂuenza epidemic in the Republic of Korea is developed. The simulation period is separated into three consecutive periods based on the government’s
intervention strategies: the nonpharmaceutical strategy is used during Period 1.
The nonpharmaceutical and antiviral strategies are executed during Period 2 andthe vaccine strategy is added during Period 3. During Period 1, we estimate thereduction in the transmission rate due to the government’s intervention policies
as a diﬀerence between the data-ﬁtted and uncontrolled transmission rate that isderived from the basic reproductive number, R0, of the model without intervention. This quantiﬁed reduced transmission rate is used as an upperbound of thenonpharmaceutical control for studying optimal control strategies, which is a new
approach for determining the realistic upperbound of control. In this study, we
also explore the real-time prediction of incidence using the mathematical model
during the early stage of the epidemic. We investigate the impact of vaccination
coverage and timing with respect to the cumulative incidence. The result implies
that early vaccination plays a signiﬁcant role for preventing the epidemic.
Keywords: pandemic inﬂuenza, mathematical model, vaccine strategy, optimal
control, predicition

Title: Transmission dynamics of methicillin-resistant Staphylococcus aureus (MRSA)
among patients in an intensive care unit and nursing home residents: optimal control eﬀorts and inﬂuences of patient and resident numbers
Speaker: Farida Chamchod (Mahidol University)
Author(s): Farida Chamchod
49

Abstract: Owing to advancing age, chronic illness, and scarce resources, nursing
homes have been viewed as an important reservoir of methicillin-resistant Staphylococcus aureus (MRSA). As residents are frequently transferred from nursing
homes to hospitals, MRSA can possibly be repeatedly introduced into those twointerconnected institutions. In this study, a mathematical model was developed toinvestigate the eﬀects of patient sharing, patient turnover, and control eﬀorts on
the prevalence levels of MRSA in nursing homes and hospitals. The results help
underline how characteristics of nursing homes and residents, and the presence of
hospitalized residents can lead to the increased prevalence of MRSA. Moreover,control eﬀorts towards nursing home residents in both settings may help reduce
the overall prevalence of MRSA.
Keywords: MRSA, transmission dynamics, mathematical modeling, optimal control

Parallel Session 2 (23, June)
Title: A mathematical modeling of sensory receptors in skin toward understanding a tactile illusion
Speaker: Masaaki Uesaka (Hokkaido University)
Author(s): Masaaki Uesaka, Masafumi Nakatani, Zixia Zhao, Hiroyuki Kitahata,
Masaharu Nagayama
Abstract: Tactile sensation provides reliable information on the geometric properties of objects and texture of a surface. However, it is not necessarily faithfulto actual geometric properties and this discrepancy is recognized as the tactile
illusion. We approach ”ﬁshbone tactile illusion”, which is one of such illusion byusing the mathematical model. Our model simulates a set of process of active
potential ﬁring of nerves when human ﬁnger touches the texture and consists of
the Maxwell model of viscoelastic material and the model of mechanoreceptors inhuman skin and Hodgkin-Huxley model. We discuss the validity of the model andshow the simulation results.
Together with this model, we performed a psycological experiments corresponding
to our mathematical model. Comparing this experiment results and simulation
results, we propose the interpretation between the intensity of illusion and simulation results.
Keywords: ﬁshbone tactile illusion, mathematical modeling, tactile sensation

Title: Eﬀects of drug regimen and drug-induced death rate on a avascular tumor:a mathematical modeling study
Speaker: Chayasin Saetia (Mahidol University)50

Author(s): Chayasin Sae-Tia, Farida Chamchod
Abstract: An aim of this present study is to investigate factors that may inﬂuencedrug and cell kinetics during cancer chemotherapy in a patient. A mathematicalmodel is based on a system of ordinary and partial diﬀerential equations for describing cell velocity, drug concentration, and tumor radius. We found that strategies ofa drug regimen and drug-induced death rate may aﬀect the tumor radius outcome
after the chemotherapy. Our work also suggests possible treatment regimens that
may be useful to help design an eﬃcient strategy for the cancer therapy.
Keywords: drug delivery, drug kinetics, cancer modeling, cancer therapy

Title: The eﬀect of spatial heterogeneity on the viral dynamics during antiretroviral treatment
Speaker: Chang-Yuan Cheng (National Pingtung University)
Author(s): Chang-Yuan Cheng
Abstract: Classical virus models were proposed based on the concept of wellmixture within the host. However, during a drug treatment, the drug concentration may be low in some regions (drug sanctuary) and then the spatial heterogeneity can be an issue in the research. We explore an virus model in heterogeneousenvironments, which imitates the complexity of the human body. How the heterogeneous environments aﬀect the the drug eﬃcacy and then the viral dynamics are
signiﬁcant in administrating a drug therapy. Especially, it is interesting to study
the optimal phase diﬀerence between multi-drug therapy and establish the optimal
distribution of drugs in the multi-compartmental environment.
Keywords: viral dynamics, spatial heterogeneity

Parallel Session 3 (24, June)
Title: Backward bifurcation of a network-based SIS epidemic model with saturated treatment function
Speaker: Chun-Hsien Li (National Kaohsiung Normal University)
Author(s): Yi-Jie Huang, Chun-Hsien Li
Abstract: In this talk, we consider a network-based SIS epidemic model withsaturated treatment function to characterize the saturation phenomenon of thelimited medical resources. In this model, we ﬁrst obtain a threshold value R0 andthen we show that a backward bifurcation occurs under some condition. More precisely, the saturation treatment function leads to a backward bifurcation. In thiscase, R0¡1 is not suﬃcient to eradicate the disease from the population. Numericalexperiments are provided to support the theoretical results.
Keywords: complex network, epidemic model, saturated treatment function,51

backward bifurcation
Title: Simulation of inﬂuenza A/H5N1 in hostSpeaker: Hidenori Yasuda (Josai University)
Author(s): Hidenori Yasuda, Shohji Kawachi, Kazuo Suzuki
Abstract: To elucidate the pathogenesis associated with highly pathogenic avian
inﬂuenza A/H5N1 infection, simulations were conducted using a mathematicalmodel of mice. Simulated results suggest antibody therapy showed prominent improvement.
Keywords: highly pathogenic avian inﬂuenza A/H5N1, mathematical model, antibody therapy, leukopenia
Title: 3D image morphing by using large deformation diﬀeomorphic metric map
with discrete Green kernel
Speaker: Chin-Tien Wu (National Chiao-Tung University)
Author(s): Chin-Tien Wu
Abstract: Major challenges in 3D image processing includes (1) geometry processing, (2) texture mapping (3) image morphing and (4) managing huge amountof data in 3D, etc. In this talk, we shall brieﬂy address these problems and focus
on discussing issues in 3D image morphing. To manage 3D image data, Gu andYau proposed to compress the data using mean curvature H and conformal factor
lambda. We have followed Gu and Yau’s idea on 3D video compressing where key
frames in the video are selected and 3d images between key frames are obtained byinterpolation according to the deformation map built in the conformal parameters(H,lambda) space. In order to capture detail geometric features and avoid meshtangling, an energy-minimizing conformal parameterization is proposed. Moreover,a discrete large deformation diﬀeomorphic metric map (DLDDMM) is introduced
to generate the geometry and texture images on the init disk. The discrete Green
kernel in DLDDMM is computed by ﬁnite element method. We shall also brieﬂyreview the conformal parameterization and LDDMM methods. Finally, some of
our numerical results will be presented.
Keywords: conformal parameterization, LDDMM, 3d image
Parallel Session 4 (24, June)
Title: Hopf-Fold bifurcation in a delayed Hindmarsh-Rose model
Speaker: Shyanshiou Chen (National Taiwan Normal University)
Author(s): ShyanShiou Chen
Abstract: In this talk, we will show the existence of the travelling waves in a52

Hindmarsh-Rose-type (HRT) equation with delay and spatial terms by ﬁning theirheteroclinic orbits. A Hopf bifurcation had been studied for the model with delay
term and without spatial term. However, it is not enough to study a single neuron
since a population of neurons has been thought of a function to cooperate withanother population. Hence, to understand dynamics of a single population seems
to be an important way to realize some of possible cooperation. For the purpose,
one of schemes is to study their Hopf and Fold-Hopf (FH) bifurcation which isapplied to show the existence of periodic orbits and heteroclinic orbits for a single
population. By the bifurcation theory in functional diﬀerential equation, the HRT
will be reduced in a normal form with abstract ordinary diﬀerential equation andthe center manifold theory. Then some conditions of parameters for the existenceof Hopf and FH bifurcation are given.
Keywords: bifurcation, neuron

Title: Self-sustained shear driven dynamos
Speaker: Kengo Deguchi (Monash University)
Author(s): Kengo Deguchi
Abstract: A large Reynolds number matched asymptotic expansion is concerned
for nonlinear 3D magneto-hydrodynamic (MHD) dynamo states driven by a shear.
The theory has emerged out of a nice combination of the vortex-wave interaction
theory by Hall & Smith (1991) and the resonant absorption theories for Alfv ?enand cusp waves, developed in solar physics community. The dynamos are selfsustained, in the sense that they are maintained without any linear instability
mechanism of the basic ﬂow. There are two classes of dynamos can be sought.
The ﬁrst class of dynamos appears when we consider the similar scaling for bothhydrodynamic and magnetic ﬁelds. In this case the dynamos must be supported bya small external magnetic ﬁeld, but induce much bigger magnetic ﬁeld. The external magnetic ﬁeld can actually be switched oﬀ to yield the second class of dynamos,
which is now purely driven by shear. The theory is compared with the unstable
invariant solutions of the full MHD equations at ﬁnite Reynolds numbers. The
dynamo solutions can be found by continuing the hydrodynamic solution branchof plane Couette ﬂow. Using an external magnetic ﬁeld as a homotopy parameter, the dynamo solutions can indeed be found even at zero-external magnetic
ﬁeld condition. Their asymptotic development can be excellently explained by theasymptotic theory.
Keywords: magnetohydrodynamics, asymptotic analysis, invariant solutions

53

Title: An invariant for the comparison of composition operators for reproducing
kernel Hilbert spaces
Speaker: Isao Ishikawa (RIKEN Center for Advanced Inteligent Project / Keio
University)
Author(s): Keisuke Fujii, Masahiro Ikeda, Yuka Hashimoto, Yoshinobu Kawahara
Abstract: Classifying time series datas (sequences in the Euclid space) is important problem for the real world. Our task is giving a ”good” mathematicalinvariant to classify them. Dynamical system is one of the eﬀective mathematicalmodel to analyze them, namely, we assume each time series data is generated bya dynamical system. In terms of the theory of reproducing kernel Hilbert spaces(RKHS), the time series datas and the dynamical systems are regarded as sequences of vectors and composition operators in RKHS, respectively. We deﬁne
an invariant to measure ”distance” between two dynamical systems (composition
operators) in the framework of RKHS. This invariant has an explicit formula which
can be easily implemented, on the other hand it includes mathematically diﬃcult
problems such as the bounded-ness of composition operators. This study is thejoint work with Ikeda (RIKEN AIP/ Keio Univ), Hashimoto (RIKEN AIP/ Keio
Univ), Fujii (RIKEN AIP) and Kawahara (RIKEN AIP/ Osaka Univ.).
Keywords: dynamical system, reproducing kernel Hilbert space

Title: Topological data analysis of across-sniﬀ habituation in olfactory cortex
Speaker: Keiji Miura (Kwansei Gakuin University)
Author(s): Yuki Usui, Ayaka Onishi, Keiji Miura
Abstract: Although it is well known that animals habituate to odors across sniﬀs,
its neural basis is still unclear. While many neurons we recorded from the olfactory cortex (anterior piriform cortex) decreased ﬁring rates for the continued odor
stimulation, some increased ﬁring rates. Therefore, it is necessary to characterize
the responses dynamically as a whole to understand the habituation in the olfactory cortex. The principal component analysis revealed that the neural responses
at the ﬁrst and second sniﬀs after the odor delivery looked diﬀerent. In fact, thetopological data analysis revealed that the orbit of the neural responses for thetwo sniﬀs did not necessarily repeat the same loop twice. These ﬁndings suggestthat the olfactory cortex could underlie the odor habituation across sniﬀs.
Keywords: topological data analysis, neuroscience54

Parallel Session 5 (24, June)
Title: Fractal structure of a nonlinear cellular automaton and the singular function
Speaker: Akane Kawaharada (Kyoto University of Education)
Author(s): Akane Kawaharada, Takao Namiki
Abstract: The authors study the relation between two-dimensional symmetricelementary cellular automata and the singular function. It is well known thateach linear cellular automaton with prime number of states has its “limit set” (alimit of a series of spatio-temporal patterns contracted by time), however, in caseof nonlinear automata, the existence of a limit set has not been reported even if
a case study. In this talk, the authors show the existence of a nonlinear cellular automaton which has the limit set and calculate the fractal dimension (thebox-counting dimension) of the boundary of its limit set. We also show that the
spatio-temporal patterns of the automaton are characterized by the singular function that is a fractal curve.
Keywords: cellular automaton, fractal geometry, singular function
Title: Mathematical modeling for stabilization of stratum granulosum and epidermal desquamation
Speaker: Takeshi Gotoda (Hokkaido university)
Author(s): Takeshi Gotoda, Masaaki Uesaka, Yusuke Yasugahira, Yasuaki Kobayashi,
Hiroyuki Kitahata, Mitsuhiro Denda, Masaharu Nagayama
Abstract: The stratum granulosum is a thin layer of cells migrating from thestratum spinosum and found just below the stratum corneum. Tight junctions
occur in the second layer of the stratum granulosum and plays a crucial role in
the functional barrier in the epidermal together with the stratum corneum. Usinga mathematical model of the epidermis, we propose a stabilization mechanism of
the stratum granulosum. We show that, in the mathematical model, the stratum
granulosum is formed and its layer structure is maintained stably by assuming thata stimulant, which promotes the diﬀerentiation process of cells, is released whencells undergo corniﬁcation. We also introduce our trial model for understanding
the mechanism of occurence of tight junctions in the stratum granulosum from theviewpoint of the mathematical modeling. Regarding the epidermal desquamation,
we propose a mathematical model of the stratum corneum with the desquamation
process, considering the proteases that play a central role in the degradation ofmacula adherens.
Keywords: mathematical modeling, numerical simulation, stratum granulosum,
desquamation of stratum corneum

55

Title: Wave transmission coeﬃcient of trapezoidal submerged breakwater
Speaker: Ikha Magdalena (Institut Teknologi Bandung)
Author(s): I. Magdalena
Abstract: In this research, a wave transmission coeﬃcient of the trapezoidal
breakwater is derived analytically. We use shallow water equations to study thedamping mechanism that is explained by the transmission coeﬃcient. Numerically, we solve the equations using ﬁnite volume method on a staggered grid.
Moreover, our numerical method is free from damping error that is very important in this case, because we want to get the actual amplitude reduction only fromthe trapezoidal breakwater. For validation, we compare our numerical result withthe analytical solution.
Keywords: wave transmission coeﬃcient, trapezoidal breakwater, shallow waterequations, staggered grid56

Room B (Building No.6)
Parallel Session 7 (25, June)
Title: Delay-induced blow-up in some oscillation models
Speaker: Tetsuya Ishiwata (Shibaura Institute of Technology)
Author(s): Emiko Ishiwata, Tetsuya Ishiwata, Yukihiko Nakata
Abstract: It is well-known that time-lags or histories sometimes play a essential
role in the phenomena and thus many mathematical model with delay-eﬀects arestudied mathematically and numerically. It is also well-known that the delayeﬀects cause an instability or an oscillatory. In this talk we consider the eﬀects
of time-delay for such instabilities from the viewpoint of a ﬁnite time blow-up
of the solutions and show ”delay-induced blow-up” phenomena for a very simple
oscillation model with a constant delay. We also show the emergence of inﬁnity
many periodic solutions, while the non-delay system has only one stable limit cycle.
Finally we show some numerical examples and give our observations.
Keywords: delay diﬀerential equation, blow-up, periodic solution

Title: Dipole problems for the one-dimensional thin-ﬁlm equation
Speaker: Mark Bowen (Waseda University)
Author(s): Mark Bowen, Thomas P. Witelksi, J. R. King, J. Hulshof
Abstract: We investigate the dynamics of a thin liquid ﬁlm spreading in a semiinﬁnite domain x¿0, so that x=0 corresponds to an edge over which ﬂuid candrain. In particular, we investigate self-similar solutions of the one-dimensional”thin-ﬁlm” equation (which is a fourth order degenerate parabolic equation) on thehalf-line x > 0. Such solutions typically feature an anomalous exponent and aresaid to be similarity solutions of the second-kind (generally requiring the numericalsolution of a nonlinear eigenvalue problem in order to obtain the scaling exponent);
when there are no free exponents, the self-similar solution is of the ﬁrst kind.
In order to make progress, we employ a four-dimensional phase-space analysis,
matched asymptotic expansions and numerical simulations.
We will also discuss the extension of our results to (non-physical) self-similar solutions featuring sign-changes and describe how classes of ﬁrst and second-kindsimilarity solutions interact. We will also comment on the diﬃculties of obtaining
sign-change solutions to degenerate diﬀusion equations.
Keywords: thin-ﬁlm, degenerate parabolic equation, self-similar solutions

57

Room C (Building No.6)
Parallel Session 7 (25, June)
Title: Cavitation and corresponding temperature distribution in soft tissue during
HIFU thermal therapy
Speaker: Ekaterina Zilonova (National Taiwan University)
Author(s): E. Zilonova, M. Solovchuk, T.W.H. Sheu
Abstract: High intensity focused ultrasound is a rapidly developing medical technology for fully noninvasive surgery. Nowadays, it has a wide range of therapeutic applications (such as tumor ablation in diﬀerent parts of the body, acoustic
hemostasis, drug and gene delivery, treatment of neurological disorders). Highintensity waves propagating through the medium may lead to the formation ofcavities, which can be ﬁlled with gas or vapor. This eﬀect is known as acoustic
cavitation. The present study investigates the cavitation presence impact on the
HIFU thermal therapy process in soft tissue. For these purposes, a new coupledmathematical model is introduced that incorporates bubble dynamics model, viscoelastic model for soft tissue simulation, two temperature models for bubble’s
interior and exterior and diﬀusion model for the calculation of vapor mass concentration within the bubble. Dependency of the bubble dynamics on soft tissue
viscoelastic properties has been investigated. The continuous temperature distribution inside and outside the oscillating bubble is going to be presented. The
impact of the vapor mass ﬂux eﬀect on the temperature distribution has beenstudied. The observed signiﬁcant temperature elevation in tissue outside the bubble emphasizes that the presence of cavitation during thermal therapy can’t beignored.
Keywords: cavitation in soft tissue, temperature distribution, vapor mass diﬀusion, high intensity focused ultrasound, heat deposition

Title: Delay-embedding time-series analysis based on Sturm-Liouville frameworkfor dynamical reconstruction
Speaker: Naoto Nakano (Kyoto University)
Author(s): Naoto Nakano
Abstract: Delay embedding is well-known for non-linear time-series analysis, and
it is used in several research ﬁelds. Takens theorem ensures validity of the delay
embedding analysis: embedded data preserves the topological properties possessedby the original dynamics so long as one embeds it within some phase space ofsuﬃciently large dimension. This means that, for example, an attractor can be reconstructed by the delay coordinate system topologically. However, conﬁgurationof embedded data may easily vary with the delay width and the delay dimension,
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namely, “the way of embedding”. Here, we investigate the mathematical structureof the framework of delay-embedding to provide Ansatz to choose the appropriate
way of embedding, in order to utilise for time-series prediction. In short, mathematical theories of the Hilbert-Schmidt integral operator and the corresponding
Sturm-Liouville eigenvalue problem underlie the framework. We will show somenumerical results of the present method.
Keywords: delay-embedding, dynamical reconstruction, time-series analysis, SturmLiouville problem, data-driven dynamical modelling59