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Numerical Analysis Seminar
Seminar information archive ～04/03｜Next seminar｜Future seminars 04/04～
| Date, time & place | Tuesday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Norikazu Saito, Takahito Kashiwabara |
Seminar information archive
2024/11/27
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Yumiharu Nakano (Institute of Science Tokyo)
Schrödinger problems and diffusion generative models (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Yumiharu Nakano (Institute of Science Tokyo)
Schrödinger problems and diffusion generative models (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2024/10/16
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Kengo Nakai (Okayama University)
Data-driven modeling from biased small training data (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Kengo Nakai (Okayama University)
Data-driven modeling from biased small training data (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2024/07/09
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Bernardo Cockburn (University of Minnesota)
The transformation of stabilizations into spaces for Galerkin methods for PDEs (English)
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Bernardo Cockburn (University of Minnesota)
The transformation of stabilizations into spaces for Galerkin methods for PDEs (English)
[ Abstract ]
We describe a novel technique which allows us to transform the terms which render Galerkin methods stable into spaces (JJIAM, 2023). We begin by applying this technique to show that the Continuous and Discontinuous Galerkin (DG) methods for ODEs produce the very same approximation of the time derivative, and use this to obtain superconvergence points of the DG method. We then apply this technique to mixed methods for second-order elliptic equations to show that they can always be recast as hybridizable DG (HDG) methods. We then show that this recating makes the implementation from 10% to 20% better for polynomial degrees ranging from 1 to 20.We end by sketching or ongoing and future work.
[ Reference URL ]We describe a novel technique which allows us to transform the terms which render Galerkin methods stable into spaces (JJIAM, 2023). We begin by applying this technique to show that the Continuous and Discontinuous Galerkin (DG) methods for ODEs produce the very same approximation of the time derivative, and use this to obtain superconvergence points of the DG method. We then apply this technique to mixed methods for second-order elliptic equations to show that they can always be recast as hybridizable DG (HDG) methods. We then show that this recating makes the implementation from 10% to 20% better for polynomial degrees ranging from 1 to 20.We end by sketching or ongoing and future work.
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2024/05/29
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Satoshi Hayakawa (Sony Group Corporation)
Random convex hulls and kernel quadrature (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Satoshi Hayakawa (Sony Group Corporation)
Random convex hulls and kernel quadrature (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2024/05/15
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Koya Sakakibara (Kanazawa University)
Regularization via Bregman divergence for the discrete optimal transport problem (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Koya Sakakibara (Kanazawa University)
Regularization via Bregman divergence for the discrete optimal transport problem (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2024/04/24
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Yuka Hashimoto (NTT Network Service Systems Laboratories)
Generalization analysis of neural networks based on Koopman operators
(Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Yuka Hashimoto (NTT Network Service Systems Laboratories)
Generalization analysis of neural networks based on Koopman operators
(Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2024/03/13
16:30-17:30 Online
David Sommer (Weierstrass Institute for Applied Analysis and Stochastics)
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/
2024/03/13
17:30-18:30 Online
Andreas Rathsfeld (Weierstrass Institute for Applied Analysis and Stochastics)
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/
2024/01/09
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Takashi Matsubara (Osaka University)
Deep learning that learns from, becomes part of, or replaces numerical methods for differential equations (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Takashi Matsubara (Osaka University)
Deep learning that learns from, becomes part of, or replaces numerical methods for differential equations (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2023/11/14
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Ken Furukawa (RIKEN)
On some dynamical systems and their prediction using data assimilation (Japanese)
[ Reference URL ]
ハイブリッド開催です。参加の詳細は参考URLをご覧ください。
Ken Furukawa (RIKEN)
On some dynamical systems and their prediction using data assimilation (Japanese)
[ Reference URL ]
ハイブリッド開催です。参加の詳細は参考URLをご覧ください。
2023/10/24
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Kazuaki Tanaka (Waseda University)
Neural Network-based Enclosure of Solutions to Differential Equations and Reconsideration of the Sub- and Super-solution Method (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Kazuaki Tanaka (Waseda University)
Neural Network-based Enclosure of Solutions to Differential Equations and Reconsideration of the Sub- and Super-solution Method (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2023/10/17
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Makoto Okumura (Konan University)
Structure-preserving schemes for the Cahn-Hilliard equation with dynamic boundary conditions in two spatial dimensions (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Makoto Okumura (Konan University)
Structure-preserving schemes for the Cahn-Hilliard equation with dynamic boundary conditions in two spatial dimensions (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2023/06/27
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Toshihiro Yamada (Hitotsubashi University)
Solving high-dimensional partial differential equations via deep learning and probabilistic methods (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Toshihiro Yamada (Hitotsubashi University)
Solving high-dimensional partial differential equations via deep learning and probabilistic methods (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2023/06/06
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Hideyuki Azegami (Nagoya Industrial Science Research Institute)
Relation between regularity and numerical solutions of shape optimization problems
(Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Hideyuki Azegami (Nagoya Industrial Science Research Institute)
Relation between regularity and numerical solutions of shape optimization problems
(Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2023/05/23
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Masaaki Imaizumi (The University of Tokyo)
Theory of Deep Learning and Over-Parameterization (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Masaaki Imaizumi (The University of Tokyo)
Theory of Deep Learning and Over-Parameterization (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2023/05/16
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Yuuki Shimizu (The University of Tokyo)
Numerical analysis of the Plateau problem by the method of fundamental solutions (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Yuuki Shimizu (The University of Tokyo)
Numerical analysis of the Plateau problem by the method of fundamental solutions (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2023/04/25
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Taihei Oki (The University of Tokyo)
Combinatorial preprocessing methods for differential-algebraic equations (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Taihei Oki (The University of Tokyo)
Combinatorial preprocessing methods for differential-algebraic equations (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2021/07/06
16:30-18:00 Online
Ken Hayami (National Institute of Informatics (Professor Emeritus))
Iterative solution methods for least squares problems and their applications
(Japanese)
[ Reference URL ]
https://forms.gle/B5Hwxa7o8F36hZKr7
Ken Hayami (National Institute of Informatics (Professor Emeritus))
Iterative solution methods for least squares problems and their applications
(Japanese)
[ Reference URL ]
https://forms.gle/B5Hwxa7o8F36hZKr7
2021/06/22
17:00-18:30 Online
Taiji Suzuki (The University of Tokyo)
On approximation ability and adaptivity of deep neural network (Japanese)
[ Reference URL ]
https://forms.gle/HwetNGXCzbCyMC7B7
Taiji Suzuki (The University of Tokyo)
On approximation ability and adaptivity of deep neural network (Japanese)
[ Reference URL ]
https://forms.gle/HwetNGXCzbCyMC7B7
2021/06/08
16:30-18:00 Online
Kohei Soga (Keio University)
Action minimizing random walks and numerical analysis of Hamilton-Jacobi equations (Japanese)
[ Reference URL ]
https://forms.gle/kjhqne4nV6fqEFWB8
Kohei Soga (Keio University)
Action minimizing random walks and numerical analysis of Hamilton-Jacobi equations (Japanese)
[ Reference URL ]
https://forms.gle/kjhqne4nV6fqEFWB8
2021/05/11
16:30-18:00 Online
Takuya Tsuchiya (Ehime University )
Topics on finite element error analysis on anisotropic meshes (Japanese)
[ Reference URL ]
https://forms.gle/CoaM4vSE1GvDRuDR6
Takuya Tsuchiya (Ehime University )
Topics on finite element error analysis on anisotropic meshes (Japanese)
[ Reference URL ]
https://forms.gle/CoaM4vSE1GvDRuDR6
2021/04/27
16:30-18:00 Online
Akitoshi Takayasu (University of Tsukuba)
Rigorous numerics for nonlinear heat equations in the complex plane of time (Japanese)
[ Reference URL ]
https://forms.gle/qW5ktphBB6dsh8Np7
Akitoshi Takayasu (University of Tsukuba)
Rigorous numerics for nonlinear heat equations in the complex plane of time (Japanese)
[ Reference URL ]
https://forms.gle/qW5ktphBB6dsh8Np7
2021/01/12
16:30-18:00 Online
Takaharu Yaguchi (Kobe University)
DGNet: Deep Energy-Based Modeling of Discrete-Time Physics and Related Topics (Japanese)
[ Reference URL ]
https://forms.gle/DpuhGupZ7NYbot5d7
Takaharu Yaguchi (Kobe University)
DGNet: Deep Energy-Based Modeling of Discrete-Time Physics and Related Topics (Japanese)
[ Reference URL ]
https://forms.gle/DpuhGupZ7NYbot5d7
2020/12/15
16:30-18:00 Online
Ming-Cheng Shiue (National Chiao Tung University)
Iterated pressure-correction projection methods for the 2d Navier-Stokes equations based on the scalar auxiliary variable approach (English)
https://forms.gle/y7w2nmaYtHNeoDSn8
Ming-Cheng Shiue (National Chiao Tung University)
Iterated pressure-correction projection methods for the 2d Navier-Stokes equations based on the scalar auxiliary variable approach (English)
[ Abstract ]
In this talk, the first-order iterated pressure-correction projection methods based on the scalar auxiliary variable approach is proposed and studied for the 2d Navier-Stokes equations and Boussinesq equations.
In the literature, enormous amounts of work have contributed to the study of numerical schemes for computing the Navier-Stokes equations. In general, two of the main numerical difficulties for solving Navier-Stokes equations are the incompressible condition and the nonlinear term. One of the approaches to deal with the incompressible condition is the so-called projection. The typical projection method only needs to solve the Poisson type of equations depending on the nonlinear term's treatment, which is efficient. However, the pressure-correction projection methods suffer from the splitting error, leading to spurious numerical boundary layers and the limitation of accuracy in time. In the literature, an iterated pressure-correction projection method has been proposed to overcome the difficulty.
As for the nonlinear term treatment, it is better to treat the nonlinear term explicitly so that one only requires to solve the corresponding linear system with constant coefficients at each time step. However, such treatment often results in a restricted time step due to the stable issue. Recently, the scalar auxiliary variable approach has been constructed to have an unconditional energy stable numerical scheme.
In this work, a new iterated pressure-correction projection method based on the scalar auxiliary variable's simple choice is proposed. We find that this new scheme can enjoy two properties, including reducing the splitting errors and having unconditional energy stability. The proofs of the energy stability and error convergence are provided and analyzed. Finally, numerical examples are provided to illustrate the theoretical work. This is joint work with Tony Chang.
[ Reference URL ]In this talk, the first-order iterated pressure-correction projection methods based on the scalar auxiliary variable approach is proposed and studied for the 2d Navier-Stokes equations and Boussinesq equations.
In the literature, enormous amounts of work have contributed to the study of numerical schemes for computing the Navier-Stokes equations. In general, two of the main numerical difficulties for solving Navier-Stokes equations are the incompressible condition and the nonlinear term. One of the approaches to deal with the incompressible condition is the so-called projection. The typical projection method only needs to solve the Poisson type of equations depending on the nonlinear term's treatment, which is efficient. However, the pressure-correction projection methods suffer from the splitting error, leading to spurious numerical boundary layers and the limitation of accuracy in time. In the literature, an iterated pressure-correction projection method has been proposed to overcome the difficulty.
As for the nonlinear term treatment, it is better to treat the nonlinear term explicitly so that one only requires to solve the corresponding linear system with constant coefficients at each time step. However, such treatment often results in a restricted time step due to the stable issue. Recently, the scalar auxiliary variable approach has been constructed to have an unconditional energy stable numerical scheme.
In this work, a new iterated pressure-correction projection method based on the scalar auxiliary variable's simple choice is proposed. We find that this new scheme can enjoy two properties, including reducing the splitting errors and having unconditional energy stability. The proofs of the energy stability and error convergence are provided and analyzed. Finally, numerical examples are provided to illustrate the theoretical work. This is joint work with Tony Chang.
https://forms.gle/y7w2nmaYtHNeoDSn8
2020/12/01
16:30-18:00 Online
Hiroyuki Sato (Kyoto University)
Conjugate gradient methods for optimization problems on manifolds (Japanese)
[ Reference URL ]
https://forms.gle/Ubeccm8neLkacjbk8
Hiroyuki Sato (Kyoto University)
Conjugate gradient methods for optimization problems on manifolds (Japanese)
[ Reference URL ]
https://forms.gle/Ubeccm8neLkacjbk8
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