## Numerical Analysis Seminar

Seminar information archive ～10/22｜Next seminar｜Future seminars 10/23～

Date, time & place | Tuesday 16:30 - 18:00 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2020/07/21

16:30-18:00 Online

Structure-preserving numerical schemes for constrained gradient flows of planar curves (Japanese)

[ Reference URL ]

https://forms.gle/3JiNEjWnrWLW8cFA9

**Tomoya Kemmochi**(Nagoya University)Structure-preserving numerical schemes for constrained gradient flows of planar curves (Japanese)

[ Reference URL ]

https://forms.gle/3JiNEjWnrWLW8cFA9

### 2020/06/30

16:30-18:00 Online

Structure-preserving numerical methods for interface problems (Japanese)

[ Reference URL ]

https://forms.gle/ztK741vNdBT7hfGSA

**Koya Sakakibara**(Okayama University of Science)Structure-preserving numerical methods for interface problems (Japanese)

[ Reference URL ]

https://forms.gle/ztK741vNdBT7hfGSA

### 2020/06/23

16:30-18:00 Online

Linearly implicit and high-order conservative schemes for ordinary differential equations with a quadratic invariant (Japanese)

[ Reference URL ]

https://forms.gle/hvvvFLAhH1314UQK8

**Shun Sato**(The University of Tokyo)Linearly implicit and high-order conservative schemes for ordinary differential equations with a quadratic invariant (Japanese)

[ Reference URL ]

https://forms.gle/hvvvFLAhH1314UQK8

### 2020/01/20

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Isogeometric Hierarchical Model Reduction: from analysis to patient-specific simulations (English)

**Yves A. B. C. Barbosa**(Politecnico di Milano)Isogeometric Hierarchical Model Reduction: from analysis to patient-specific simulations (English)

[ Abstract ]

In the field of hemodynamics, numerical models have evolved to account for the demands in speed and accuracy of modern diagnostic medicine. In this context, we studied in detail Hierarchical Model Reduction technique combined with Isogeometric Analysis (HigaMOD), a technique recently developed in [Perotto, Reali, Rusconi and Veneziani (2017)]. HigaMod is a reduction procedure used to downscale models when the phenomenon at hand presents a preferential direction of flow, e.g., when modelling the blood flow in arteries or the water flow in a channel network. The method showed a significant improvement in reducing the computational power and simulation time, while giving enough information to analyze the problem at hand.

Recently, we focused our work in solving the ADR problem and the Stokes problem in a patient-specific framework. Specifically, we evaluate the computational efficiency of HigaMod in simulating the blood flow in coronary arteries and cerebral arteries. The main goal is to assess the

mprovement that 1D enriched models can provide, with respect to traditional full models, when dealing with demanding 3D CFD simulations. The results obtained, even though preliminary, are promising [Brandes, Barbosa and Perotto (2019); Brandes, Barbosa, Perotto and Suito (2020)].

In the field of hemodynamics, numerical models have evolved to account for the demands in speed and accuracy of modern diagnostic medicine. In this context, we studied in detail Hierarchical Model Reduction technique combined with Isogeometric Analysis (HigaMOD), a technique recently developed in [Perotto, Reali, Rusconi and Veneziani (2017)]. HigaMod is a reduction procedure used to downscale models when the phenomenon at hand presents a preferential direction of flow, e.g., when modelling the blood flow in arteries or the water flow in a channel network. The method showed a significant improvement in reducing the computational power and simulation time, while giving enough information to analyze the problem at hand.

Recently, we focused our work in solving the ADR problem and the Stokes problem in a patient-specific framework. Specifically, we evaluate the computational efficiency of HigaMod in simulating the blood flow in coronary arteries and cerebral arteries. The main goal is to assess the

mprovement that 1D enriched models can provide, with respect to traditional full models, when dealing with demanding 3D CFD simulations. The results obtained, even though preliminary, are promising [Brandes, Barbosa and Perotto (2019); Brandes, Barbosa, Perotto and Suito (2020)].

### 2019/12/16

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

A second-order stabilization method for linearizing and decoupling nonlinear parabolic systems (Japanese)

**Yuki Ueda**(The Hong Kong Polytechnic University)A second-order stabilization method for linearizing and decoupling nonlinear parabolic systems (Japanese)

[ Abstract ]

We present a new time discretization method for strongly nonlinear parabolic systems. Our method is based on backward finite difference for the first derivative with second-order accuracy and the first-order linear discrete-time scheme for nonlinear systems which has been introduced by H. Murakawa. We propose a second-order stabilization method by combining these schemes.

Our error estimate requires testing the error equation by two test functions and showing $W^{1,\infty}$-boundedness which is proved by ($H^2$ or) $H^3$ energy estimate. We overcome the difficulty for establishing energy estimate by using the generating function technique which is popular in studying ordinary differential equations. Several numerical examples are provided to support the theoretical result.

We present a new time discretization method for strongly nonlinear parabolic systems. Our method is based on backward finite difference for the first derivative with second-order accuracy and the first-order linear discrete-time scheme for nonlinear systems which has been introduced by H. Murakawa. We propose a second-order stabilization method by combining these schemes.

Our error estimate requires testing the error equation by two test functions and showing $W^{1,\infty}$-boundedness which is proved by ($H^2$ or) $H^3$ energy estimate. We overcome the difficulty for establishing energy estimate by using the generating function technique which is popular in studying ordinary differential equations. Several numerical examples are provided to support the theoretical result.

### 2019/12/09

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

Point-wise error estimation for the finite element solution to Poisson's equation --- new approach based on Kato-Fujita's method (Japanese)

**Xuefeng Liu**(Niigata University)Point-wise error estimation for the finite element solution to Poisson's equation --- new approach based on Kato-Fujita's method (Japanese)

[ Abstract ]

In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.

Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.

Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

### 2019/09/25

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Finite volume method for the Keller-Segel system (Japanese)

**Guanyu Zhou**(Tokyo University of Science)Finite volume method for the Keller-Segel system (Japanese)

### 2019/08/19

13:00-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Staggered hybridisation for discontinuous Galerkin methods (英語)

DG and HDG methods for the variational inequality problems (英語)

A new HDG method using a hybridized flux (英語)

Numerical approximation of the Stokes–Darcy problem using discontinuous linear elements (英語)

**Eric Chung**(The Chinese University of Hong Kong) 13:00-14:00Staggered hybridisation for discontinuous Galerkin methods (英語)

[ Abstract ]

In this talk, we present a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations and nonlinear Stokes equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion.

In this talk, we present a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations and nonlinear Stokes equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion.

**Feifei Jing**(Northwestern Polytechnical University) 14:30-15:30DG and HDG methods for the variational inequality problems (英語)

[ Abstract ]

There exist many numerical methods for solving the fluid dynamics equations, the main difference between them lies in the partitions of geometric domain and the discrete forms of governing equations. Due to the discontinuous piecewise polynomial subspaces, DG and HDG methods can be easily implemented on highly unstructured meshes, e.g. general polygonal mesh, and volume integrals could be calculated on physical elements, without reference elements and mappings between physical and reference elements. In this talk, DG and HDG methods employed to a class of variational inequality problems arising in hydrodynamics are studied. Some theoretical results will be shown, as well as the implementations of these methods are also put into practice.

There exist many numerical methods for solving the fluid dynamics equations, the main difference between them lies in the partitions of geometric domain and the discrete forms of governing equations. Due to the discontinuous piecewise polynomial subspaces, DG and HDG methods can be easily implemented on highly unstructured meshes, e.g. general polygonal mesh, and volume integrals could be calculated on physical elements, without reference elements and mappings between physical and reference elements. In this talk, DG and HDG methods employed to a class of variational inequality problems arising in hydrodynamics are studied. Some theoretical results will be shown, as well as the implementations of these methods are also put into practice.

**Issei Oikawa**(Hitotsubashi University) 16:00-16:30A new HDG method using a hybridized flux (英語)

[ Abstract ]

We propose a new hybridizable discontinuous Galerkin (HDG) method for steady-state diffusion problems. In our method, both the trace and flux of the exact solution are hybridized. The Lehrenfeld-Schöberl stabilization is implicitly included in the method, so that the orders of convergence in all variables are optimal without postprocessing and computation of any projection. Numerical results are present to show the validation of our method.

We propose a new hybridizable discontinuous Galerkin (HDG) method for steady-state diffusion problems. In our method, both the trace and flux of the exact solution are hybridized. The Lehrenfeld-Schöberl stabilization is implicitly included in the method, so that the orders of convergence in all variables are optimal without postprocessing and computation of any projection. Numerical results are present to show the validation of our method.

**Takahito Kashiwabara**(The University of Tokyo) 16:30-17:00Numerical approximation of the Stokes–Darcy problem using discontinuous linear elements (英語)

[ Abstract ]

We consider the Stokes–Darcy interface problem supplemented with the Beavers– Joseph–Saffman condition on the interface separating two domains. This condition allows for discontinuity in the tangential velocities and in the pressures along the interface. To effectively express it, we propose to use discontinuous linear finite elements to approximate all of the velocities/pressures in the Stokes/Darcy regions. The continuity of velocity in the normal direction is weakly enforced by adopting either the penalty method or Nitsche’s method. We present stability and error estimates for the proposed scheme, taking into account the situation where a curved interface is approximated by a polygonal curve or polyhedral surface.

We consider the Stokes–Darcy interface problem supplemented with the Beavers– Joseph–Saffman condition on the interface separating two domains. This condition allows for discontinuity in the tangential velocities and in the pressures along the interface. To effectively express it, we propose to use discontinuous linear finite elements to approximate all of the velocities/pressures in the Stokes/Darcy regions. The continuity of velocity in the normal direction is weakly enforced by adopting either the penalty method or Nitsche’s method. We present stability and error estimates for the proposed scheme, taking into account the situation where a curved interface is approximated by a polygonal curve or polyhedral surface.

### 2019/07/08

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Parameter estimation and discretization errors for ordinary differential models (Japanese)

**Takeru Matsuda**(University of Tokyo)Parameter estimation and discretization errors for ordinary differential models (Japanese)

### 2019/07/01

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

The effect of preventing scale formation by ceramic balls and its effect on the human body (Japanese)

**Hideo Kawarada**(AMSOK)The effect of preventing scale formation by ceramic balls and its effect on the human body (Japanese)

### 2019/05/13

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Iterative refinement for symmetric eigenvalue problems (Japanese)

**Kensuke Aishima**(Hosei University)Iterative refinement for symmetric eigenvalue problems (Japanese)

### 2019/04/22

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Superconvergence of the HDG method (Japanese)

**Issei Oikawa**(Hitotsubashi University )Superconvergence of the HDG method (Japanese)

### 2019/04/08

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Crack growth model of viscoelastic material with the phase field approach (Japanese)

**Takeshi Takaishi**(Musashino University)Crack growth model of viscoelastic material with the phase field approach (Japanese)

### 2018/11/05

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Tosio Kato as an applied mathematician (Japanese)

**Hisashi Okamoto**(Gakushuin University)Tosio Kato as an applied mathematician (Japanese)

[ Abstract ]

Tosio Kato (1917-1999) is nowadays considered to be a rigorous analyst or theorist. Many people consider his contributions in quantum mechanics to be epoch-making, his work on nonlinear partial differential equations elegant and inspiring. However, around the time when he visited USA for the first time in 1954, he was studying problems of applied mathematics, too, notably numerical computation of eigenvalues. I wish to shed light on the historical background of his study of applied mathematics. This is a joint work with Prof. Hiroshi Fujita.

Tosio Kato (1917-1999) is nowadays considered to be a rigorous analyst or theorist. Many people consider his contributions in quantum mechanics to be epoch-making, his work on nonlinear partial differential equations elegant and inspiring. However, around the time when he visited USA for the first time in 1954, he was studying problems of applied mathematics, too, notably numerical computation of eigenvalues. I wish to shed light on the historical background of his study of applied mathematics. This is a joint work with Prof. Hiroshi Fujita.

### 2018/10/22

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Residual smoothing technique for short-recurrence Krylov subspace methods (Japanese)

**Kensuke Aihara**(Tokyo City University)Residual smoothing technique for short-recurrence Krylov subspace methods (Japanese)

### 2018/10/15

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Möbius invariant discretizations and decomposition of the Möbius energy (Japanese)

**Takeyuki Nagasawa**(Saitama University)Möbius invariant discretizations and decomposition of the Möbius energy (Japanese)

### 2018/07/31

14:00-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Recent advances on numerical analysis and simulation of invisibility cloaks with metamaterials (English)

**Jichun Li**(University of Nevada Las Vegas)Recent advances on numerical analysis and simulation of invisibility cloaks with metamaterials (English)

[ Abstract ]

In the June 23, 2006's issue of Science magazine, Pendry et al. and Leonhardt independently published their seminar papers on electromagnetic cloaking. Since then, there is a growing interest in using metamaterials to design invisibility cloaks. In this talk, I will first give a brief introduction to invisibility cloaks with metamaterials, then I will focus on some time-domain cloaking models we studied in the last few years. Well-posedness study and time-domain finite element method for these models will be presented. I will conclude the talk with some open issues.

In the June 23, 2006's issue of Science magazine, Pendry et al. and Leonhardt independently published their seminar papers on electromagnetic cloaking. Since then, there is a growing interest in using metamaterials to design invisibility cloaks. In this talk, I will first give a brief introduction to invisibility cloaks with metamaterials, then I will focus on some time-domain cloaking models we studied in the last few years. Well-posedness study and time-domain finite element method for these models will be presented. I will conclude the talk with some open issues.

### 2018/07/26

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

$L^\infty$ error estimates of the finite element method for elliptic and parabolic equations with the Neumann boundary condition in smooth domains (日本語)

**Takahito Kashiwabara**(University of Tokyo)$L^\infty$ error estimates of the finite element method for elliptic and parabolic equations with the Neumann boundary condition in smooth domains (日本語)

### 2018/07/10

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Free surface flow using orthogonal basis bubble function finite element method (Japanese)

**Junichi Matsumoto**(AIST)Free surface flow using orthogonal basis bubble function finite element method (Japanese)

### 2018/06/19

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Small data global existence for the semi-discrete scheme of a model system of hyperbolic balance laws (Japanese)

**Shuji Yoshikawa**(Oita University)Small data global existence for the semi-discrete scheme of a model system of hyperbolic balance laws (Japanese)

### 2018/05/31

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Parallel Computing Methods for Quantitative Finance: the Parareal Algorithm for American Options (English)

**Olivier Pironneau**(Sorbonne University and Academy of Sciences)Parallel Computing Methods for Quantitative Finance: the Parareal Algorithm for American Options (English)

[ Abstract ]

With parallelism in mind we investigate the parareal method for American contracts both theoretically and numerically. Least-Square Monte-Carlo (LSMC) and parareal time decomposition with two or more levels are used, leading to an efficient parallel implementation which scales linearly with the number of processors and is appropriate to any multiprocessor-memory architecture in its multilevel version. We prove $L^2$ superlinear convergence for an LSMC backward in time computation of American contracts, when the conditional expectations are known, i.e. before Monte-Carlo discretization. In all cases the computing time is increased only by a constant factor, compared to the sequential algorithm on the finest grid, and speed-up is guaranteed when the number of processors is larger than that constant. A numerical implementation will be shown to confirm the theoretical error estimates.

With parallelism in mind we investigate the parareal method for American contracts both theoretically and numerically. Least-Square Monte-Carlo (LSMC) and parareal time decomposition with two or more levels are used, leading to an efficient parallel implementation which scales linearly with the number of processors and is appropriate to any multiprocessor-memory architecture in its multilevel version. We prove $L^2$ superlinear convergence for an LSMC backward in time computation of American contracts, when the conditional expectations are known, i.e. before Monte-Carlo discretization. In all cases the computing time is increased only by a constant factor, compared to the sequential algorithm on the finest grid, and speed-up is guaranteed when the number of processors is larger than that constant. A numerical implementation will be shown to confirm the theoretical error estimates.

### 2018/05/08

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Various aspects of numerical analysis (Japanese)

**Norikazu Saito**(University of Tokyo)Various aspects of numerical analysis (Japanese)

### 2018/04/17

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Introduction to Machine learning and its application to Medical diagnosis (Japanese)

**Yoshiki Sugitani**(Tohoku University)Introduction to Machine learning and its application to Medical diagnosis (Japanese)

### 2018/02/19

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Existence, multiplicity, and orbital stability for travelling waves in a nonlinearly supported beam (English)

**Michael Plum**(Karlsruhe Insitute of Technology)Existence, multiplicity, and orbital stability for travelling waves in a nonlinearly supported beam (English)

[ Abstract ]

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions "close to" the approximate ones. Furthermore we investigate the orbital stability of these solutions by making use of both analytical and computer-assisted techniques.

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions "close to" the approximate ones. Furthermore we investigate the orbital stability of these solutions by making use of both analytical and computer-assisted techniques.

### 2018/02/19

16:15-17:15 Room #056 (Graduate School of Math. Sci. Bldg.)

An approach to computer-assisted existence proofs for nonlinear space-time fractional parabolic problems (English)

**Kaori Nagatou**(Karlsruhe Insitute of Technology)An approach to computer-assisted existence proofs for nonlinear space-time fractional parabolic problems (English)

[ Abstract ]

We consider an initial boundary value problem for a space-time fractional parabolic equation, which includes the fractional Laplacian, i.e. a nonlocal operator. We treat a corresponding local problem which is obtained by the Caffarelli-Silvestre extension technique, and show how to enclose a solution of the extended problem by computer-assisted means.

We consider an initial boundary value problem for a space-time fractional parabolic equation, which includes the fractional Laplacian, i.e. a nonlocal operator. We treat a corresponding local problem which is obtained by the Caffarelli-Silvestre extension technique, and show how to enclose a solution of the extended problem by computer-assisted means.