## Future seminars

Seminar information archive ～02/28｜Today's seminar 02/29 | Future seminars 03/01～

### 2024/03/11

#### FJ-LMI Seminar

13:30-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Fractional Nonlinear Diffusion Equation: Numerical Analysis and Large-time Behavior. (英語)

https://fj-lmi.cnrs.fr/seminars/

**Florian SALIN**(Université de Lyon - 東北大学)Fractional Nonlinear Diffusion Equation: Numerical Analysis and Large-time Behavior. (英語)

[ Abstract ]

This talk will discuss a fractional nonlinear diffusion equation on bounded domains. This equation arises by combining fractional (in space) diffusion, with a nonlinearity of porous medium or fast diffusion type. It is known that, in the porous medium case, the energy of the solutions to this equation decays algebraically, and in the fast diffusion case, solutions extinct in finite time. Based on these estimates, we will study the fine large-time asymptotic behavior of the solutions. In particular, we will show that the solutions approach separate variable solutions as the time converges to infinity in the porous medium case, or as it converges to the extinction time in the fast diffusion case. However, the extinction time is not known analytically, and to compute it, we will introduce a numerical scheme that satisfies the same decay estimates as the continuous equation.

[ Reference URL ]This talk will discuss a fractional nonlinear diffusion equation on bounded domains. This equation arises by combining fractional (in space) diffusion, with a nonlinearity of porous medium or fast diffusion type. It is known that, in the porous medium case, the energy of the solutions to this equation decays algebraically, and in the fast diffusion case, solutions extinct in finite time. Based on these estimates, we will study the fine large-time asymptotic behavior of the solutions. In particular, we will show that the solutions approach separate variable solutions as the time converges to infinity in the porous medium case, or as it converges to the extinction time in the fast diffusion case. However, the extinction time is not known analytically, and to compute it, we will introduce a numerical scheme that satisfies the same decay estimates as the continuous equation.

https://fj-lmi.cnrs.fr/seminars/

### 2024/03/12

#### Tuesday Seminar of Analysis

16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

On the generic regularity of min-max CMC hypersurfaces (English)

https://forms.gle/7mqzgLqhtBuAovKB8

**Kobe Marshall-Stevens**(University College London)On the generic regularity of min-max CMC hypersurfaces (English)

[ Abstract ]

Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.

[ Reference URL ]Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.

https://forms.gle/7mqzgLqhtBuAovKB8

### 2024/03/13

#### Numerical Analysis Seminar

16:30-17:30 Online

Approximating Langevin Monte Carlo with ResNet-like neural network architectures (English)

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

**David Sommer**(Weierstrass Institute for Applied Analysis and Stochastics)Approximating Langevin Monte Carlo with ResNet-like neural network architectures (English)

[ Abstract ]

We analyse a method to sample from a given target distribution by constructing a neural network which maps samples from a simple reference distribution, e.g. the standard normal, to samples from the target distribution. For this, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-2 distance are shown. The analysis heavily relies on the notion of sub-Gaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks (ResNets) and derive expressivity results for approximating the sample to target distribution map.

[ Reference URL ]We analyse a method to sample from a given target distribution by constructing a neural network which maps samples from a simple reference distribution, e.g. the standard normal, to samples from the target distribution. For this, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-2 distance are shown. The analysis heavily relies on the notion of sub-Gaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks (ResNets) and derive expressivity results for approximating the sample to target distribution map.

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

#### Numerical Analysis Seminar

17:30-18:30 Online

Analysis of the Scattering Matrix Algorithm (RCWA) for Diffraction by Periodic Surface Structures (English)

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

**Andreas Rathsfeld**(Weierstrass Institute for Applied Analysis and Stochastics)Analysis of the Scattering Matrix Algorithm (RCWA) for Diffraction by Periodic Surface Structures (English)

[ Abstract ]

The scattering matrix algorithm is a popular numerical method for the diffraction of optical waves by periodic surfaces. The computational domain is divided into horizontal slices and, by a clever recursion, an approximated operator, mapping incoming into outgoing waves, is obtained. Combining this with numerical schemes inside the slices, methods like RCWA and FMM have been designed.

The key for the analysis is the scattering problem with special radiation conditions for inhomogeneous cover materials. If the numerical scheme inside the slices is the FEM, then the scattering matrix algorithm is nothing else than a clever version of a domain decomposition method.

[ Reference URL ]The scattering matrix algorithm is a popular numerical method for the diffraction of optical waves by periodic surfaces. The computational domain is divided into horizontal slices and, by a clever recursion, an approximated operator, mapping incoming into outgoing waves, is obtained. Combining this with numerical schemes inside the slices, methods like RCWA and FMM have been designed.

The key for the analysis is the scattering problem with special radiation conditions for inhomogeneous cover materials. If the numerical scheme inside the slices is the FEM, then the scattering matrix algorithm is nothing else than a clever version of a domain decomposition method.

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

### 2024/03/14

#### Colloquium

14:30-17:00 Room #大講義室(auditorium) (Graduate School of Math. Sci. Bldg.)

If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at [Reference URL].

Many years from now (JAPANESE)

https://forms.gle/m38f1KRi67ECuA7MA

Mathematics, which I eventually found that I like: from the viewpoint of some marginal areas (JAPANESE)

https://forms.gle/m38f1KRi67ECuA7MA

If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at [Reference URL].

**Toshiyasu Arai**(Graduate School of Mathematical Sciences, The University of Tokyo) 14:30-15:30Many years from now (JAPANESE)

[ Abstract ]

I have been studying proof theory since the 1980's. In this talk I will talk about what happened to me in these 40 years, and let me report the latest result on ordinal analysis.

[ Reference URL ]I have been studying proof theory since the 1980's. In this talk I will talk about what happened to me in these 40 years, and let me report the latest result on ordinal analysis.

https://forms.gle/m38f1KRi67ECuA7MA

**Masahiro Yamamoto**(Graduate School of Mathematical Sciences, The University of Tokyo) 16:00-17:00Mathematics, which I eventually found that I like: from the viewpoint of some marginal areas (JAPANESE)

[ Abstract ]

Looking back on my experiences over 40 years, I have been convinced that I have been loving my own mathematics among others.

After all, I can sum up as that all my mathematics are concerned with the three topics: control theories, inverse problems and time-fractional partial differential equations. Some of these research fields has already developed to major topics, while others keep still minor interests.

When I started studies on inverse problems in 1980's, there were very few population of mathematicians as specialists in Japan. In particular, inverse problems did not call great attention of mathematicians and were understood as marginal mathematical topics in spite of practical significance and demands On the other hand, possibly available methodologies and ideas have been exploited and integrated gradually. As consequence, main research partners have been outside Japan.

I have been enjoying not only the research contents, but also such wider collaboration.

Aiming at non-meaningless reference for the youngers, and trying not to be too retrospective, I will describe how I have done in mathematics as well as my research contents.

[ Reference URL ]Looking back on my experiences over 40 years, I have been convinced that I have been loving my own mathematics among others.

After all, I can sum up as that all my mathematics are concerned with the three topics: control theories, inverse problems and time-fractional partial differential equations. Some of these research fields has already developed to major topics, while others keep still minor interests.

When I started studies on inverse problems in 1980's, there were very few population of mathematicians as specialists in Japan. In particular, inverse problems did not call great attention of mathematicians and were understood as marginal mathematical topics in spite of practical significance and demands On the other hand, possibly available methodologies and ideas have been exploited and integrated gradually. As consequence, main research partners have been outside Japan.

I have been enjoying not only the research contents, but also such wider collaboration.

Aiming at non-meaningless reference for the youngers, and trying not to be too retrospective, I will describe how I have done in mathematics as well as my research contents.

https://forms.gle/m38f1KRi67ECuA7MA