Title: Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features
Speaker: Xianjin Yang, BIMSA
Time: 14:00-15:00, Thursday, April 28, 2022.
Venue: Offline: BIMSA 1110. Online: Tencent Meeting: 408-2490-3761.
Organizer: Hui Yu.
Abstract:
In this talk, I will present two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori probability estimators of GPs conditioned on the partial differential equation (PDE) system of the MFG at a finite number of sample points. To improve the performance, we introduce the FF method, whose insight comes from the recent trend of approximating positive definite kernels with random Fourier features. We give the existence and the convergence proofs for both algorithms. We show the efficacy of our algorithms through experiments on a stationary MFG with a non-local coupling and on a time-dependent planning problem.
Past Talks
Title: Thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds.
Speaker: Woojoo Shim, Research fellow at Korea Institute for Advanced Study.
Time: 10:00-11:00AM, Friday, April 22nd, 2022.
Venue: Tencent Meeting: 408-2490-3761.
Organizer: Hui Yu.
Abstract: We study emergent collective behaviors of a thermodynamic Cucker-Smale (TCS) ensemble on complete smooth Riemannian manifolds. For this, we extend the TCS model on the Euclidean space to a complete smooth Riemannian manifold by adopting the work for a CS ensemble and providing a sufficient framework to achieve velocity alignment and thermal equilibrium. Compared to the work for a CS ensemble, our model has an extra thermodynamic observable denoted by temperature, which is assumed to be nonidentical for each particle. However, for the isothermal case, our model reduces to the previous CS model on a manifold in a small velocity regime. As a concrete example, we study emergent dynamics of the TCS model on the unit d-sphere. We show that the asymptotic emergent dynamics of the proposed TCS model on the unit d-sphere exhibit a dichotomy, either convergence to zero velocity or an asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy in the asymptotic dynamics of the TCS particles on the 2-sphere.
题目: Introduction to mean-field models of particle systems（3）
报告人: 周百舸（清华大学）
时间：8:00-9:00 AM, March 23, 2022.
地点：清华大学六教6B109
腾讯会议：408-2490-3761
组织者：杜洁，蔚辉
摘要: Particle systems exhibit various dynamics such as collision, interaction, and collective motions that can lead to lots of applications in biology, physics, and engineering. One major modeling strategy is to start with individual-based models that consist of ordinary differential equations describing the dynamics of each particle due to physical laws, and then derive the mean-field and fluid models in the mesoscopic and macroscopic levels. The models in different scales are suitable for certain applications of great interest. We will present the classic theory on the connection between particle models and McKean-Vlasov equations, and discuss the open questions.
题目: Introduction to mean-field models of particle systems（2）
报告人: 周百舸（清华大学）
时间：8:00-9:00 AM, March 16, 2022.
地点：清华大学六教6B109
腾讯会议：408-2490-3761
组织者：杜洁，蔚辉
摘要: Particle systems exhibit various dynamics such as collision, interaction, and collective motions that can lead to lots of applications in biology, physics, and engineering. One major modeling strategy is to start with individual-based models that consist of ordinary differential equations describing the dynamics of each particle due to physical laws, and then derive the mean-field and fluid models in the mesoscopic and macroscopic levels. The models in different scales are suitable for certain applications of great interest. We will present the classic theory on the connection between particle models and McKean-Vlasov equations, and discuss the open questions.
题目: Introduction to mean-field models of particle systems
报告人: 周百舸（清华大学）
时间：8:00-9:00 AM, March 9, 2022.
地点：清华大学近春园西楼第一会议室
腾讯会议：408-2490-3761
组织者：杜洁，蔚辉
摘要: Particle systems exhibit various dynamics such as collision, interaction and collective motions that can lead to lots of applications in biology, physics and engineering. One major modelling strategy is to start with individual-based models that consists of ordinary differential equations describing the dynamics of each particle due to physical laws, and then derive the mean-field and fluid models in the mesoscopic and macroscopic levels. The models in different scales are suitable for certain applications of great interests. We will present the classic theory on the connection between particle models and McKean-Vlasov equations, and discuss the open questions.
题目 Title: A mixed precision Jacobi SVD algorithm
时间 Time: Friday (Jan 14) 9:30--10:30am
报告人 Speaker: Meiyue Shao, School of Data Science, Fudan University
组织者 Organizer: Xin Liang, YMSC and BIMSA
地点Venue: Tencent Meeting ID：970 908 883 口令 Password: 202201
摘要 Abstract: We propose a mixed precision Jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning,the proposed algorithm computes the SVD in a lower precision as an initial guess, and then performs one-sided Jacobi rotations in the working precision as iterative refinement.By carefully transforming a lower precision solution to a higher precision one, our algorithm achieves about 2x speedup on the x86 architecture compared to the usual one-sided Jacobi SVD algorithm in LAPACK, without sacrificing the accuracy.
报告人简介Profile:
邵美悦，2014年毕业于瑞士洛桑联邦理工学院，获得计算数学博士学位。之后在美国劳伦斯伯克利国家实验室从事研究工作，先后担任博士后研究员（2014–2017）和项目科学家（2017–2019）。研究方向为数值线性代数和高性能计算。
Title: Strong convergence rate of a full discretization for nonlinear SPDE driven by multiplicative noise
Speaker: 黄灿
Time: Friday (Dec. 24th) 13:30—15:00
Tencent Meeting: 372-957-007
Abstract:
We consider a fully discrete scheme for nonlinear SPDE with non-globally Lipschitz coefficients driven by multiplicative noise in a multi-dimensional setting. Our method uses a polynomial based spectral method in space, so it does not require the elliptic operator and the covariance operator of noise in the equation commute, and thus successfully alleviates a restriction of Fourier spectral method for SPDEs pointed out by Jentzen, Kloeden and Winkel in their Annals of Applied Probability paper.
The discretization in time is a tamed semi-implicit scheme which treats the nonlinear term explicitly while being unconditionally stable. Under regular assumptions which are usually made for SPDEs with additive noise, we establish optimal strong convergence rates in both space and time for our fully discrete scheme.
个人介绍： 黄灿，厦门大学数学科学学院副教授，硕士生导师。研究方向为积分方程、分数阶方程数值解和随机偏微分方程数值解。2011年于美国Wayne State University获得博士学位，2011-2013年于美国Michigan State University从事博士后工作，博士后出站后受聘于厦门大学至今。主持完成国家自然科学基金一项，参与国家重点项目一项，在SIAM J. Numer. Anal、J.Comput. Phys.， IMA J. Numer. Anal.等期刊发表论文20余篇。
Title:Adaptive finite element methods for Poisson equation
Speaker:Nianyu Yi, Xiangtan University, China. (yinianyu@xtu.edu.cn)
Time:Thursday 10:00-11:00AM, December 9th.
Tencent Meeting:829 760 836
Abstract:
We will introduce the general framework of adaptive finite element methods(FEMs) for elliptic equations, and discuss a posteriori error estimation and the algorithms. In particular, we present the adaptive FEMs for the Poisson equation with Dirac delta sources, talk about the difficulties of the problems, and demonstrate the results with a series of numerical simulations.
报告人简介 Profile
易年余，2005毕业于天水师范学院数学与应用数学专业，获理学学士学位；2011年毕业于湘潭大学计算数学专业，获理学博士学位。现为湘潭大学数学与计算科学学院教授。主要从事有限元高精度理论与自适应方法研究，研究兴趣包括有限元超收敛分析与重构方法、后验误差估计与自适应算法、保持偏微分方程特征的高精度方法、相场问题的自适应有限元方法等。
Title: A spectrally accurate numerical method for computing the Bogoliubov-De Gennes excitations of dipolar Bose-Einstein Condensates
Speaker:Yong Zhang, Tianjin University
Time:Friday 10:00-11:00AM, December 10th.
Tencent Meeting: 879-873-442
Abstract:
In this talk, we propose an efficient and robust numerical method to study the elementary excitation of dipolar Bose-Einstein condensates (BEC), which is governed by the Bogoliubov-de Gennes equations (BdGEs) with nonlocal dipole-dipole interaction, around the mean field ground state. Analytical properties of the BdGEs are investigated, which could serve as benchmarks for the numerical methods. To evaluate the nonlocal interactions accurately and efficiently, we propose a new Simple Fourier Spectral Convolution method (SFSC). Then, integrating SFSC with the standard Fourier spectral method for spatial discretization and Implicitly Restarted Arnoldi Methods (IRAM) for the eigenvalue problem, we derive an efficient and spectrally accurate method, named as SFSC-IRAM method, for the BdGEs. Ample numerical tests are provided to illustrate the accuracy and efficiency. Finally, we apply the new method to study systematically the excitation spectrum and Bogoliubov amplitudes around the ground state with different parameters in different spatial dimensions.
About the speaker:
张勇教授2007年本科毕业于天津大学，2012年在清华大学获得博士学位。他先后在奥地利维也纳大学的Wolfgang Pauli 研究所，法国雷恩一大和美国纽约大学克朗所从事博士后研究工作。2015年7月获得奥地利自然科学基金委支持的薛定谔基金，2018年入选国家“青年千人”计划。研究兴趣主要是偏微分方程的数值计算和分析工作，尤其是快速算法的设计和应用。迄今发表论文20余篇，主要发表在包括SIAM Journal on Scientific Computing, SIAM Journal on Applied Mathematics, SIAM Multiscale Modeling and Simulation, Journal of Computational Physics, Mathematics of Computation, Computer Physics Communication 等计算数学顶尖杂志。
Title:Code optimization in Fortran and building higher level language interfaces using them
Time:Friday (Dec. 10th) 9:00--10:00pm
Speaker:Manas Rachh, Flatiron Institute, Simons Foundation
Organizer:王珺, YMSC and BIMSA
Zoom Meeting ID：849 963 1368
Passcode：YMSC
Abstract:In this talk, we will focus on using simd optimization for computing low level functions in Fortran and C, and building higher level language interfaces to MATLAB, Python and Julia using them. We will also talk about certain aspects of writing multithreaded codes in Fortran. Most of the examples will focus on the methods used in the development of our fast multipole library FMM3D (https://github.com/flatironinstitute/FMM3D).
Short bio:Manas Rachh is a research scientist at the Center for Computational Mathematics at the Flatiron institute. His research interests include developing fast and accurate solvers for applications in electromagnetics, microfluidics, biomedical imaging and data visualization.
Before working at the Flatiron institute, he obtained his Ph.D from the Courant institute of mathematical sciences followed by a Gibbs assistant professorship at Yale.
题目:Effects of Air Quality on Housing Location: A Predictive Dynamic Continuum User-Optimal Approach
报告人:杨良泽（BIMSA）
时间：14:00-15:00, Friday, December 3, 2021.
地点：雁栖湖应用数学研究院1129B
腾讯会议：829 760 836
组织者：杜洁
摘要:Recent decades have seen increasing concerns regarding air quality in housing locations. This study proposes a predictive continuum dynamic user-optimal model with combined choice of housing location, destination, route, and departure time. A traveler’s choice of housing location is modeled by a logit-type demand distribution function based on air quality, housing rent, and perceived travel costs. Air quality, or air pollutants, within the modeling region are governed by the vehicle-emission model and the advection-diffusion equation for dispersion. In this study, the housing-location problem is formulated as a fixed-point problem, and the predictive continuum dynamic user-optimal model with departure-time consideration is formulated as a variational inequality problem. The Lax-Friedrichs scheme, the fast-sweeping method, the Goldstein-Levitin-Polyak projection algorithm, and self-adaptive successive averages are adopted to discretize and solve these problems. A numerical example is given to demonstrate the characteristics of the proposed housing-location choice problem with consideration of air quality and to demonstrate the effectiveness of the solution algorithms.
题目:相对论磁流体力学方程组的高精度保界数值方法
报告人:吴开亮（南方科技大学）
时间：10:00-11:00AM, Friday, November 26, 2021.
腾讯会议：829 760 836
组织者：杜洁
摘要:本报告将介绍近年来相对论磁流体力学方程组的高精度保界（保持密度和压力为正、速度小于光速）数值方法的研究进展。由于相对论效应，流体的压力和速度等原始变量无法由方程组的守恒变量显式地表示，因而上述有关界的约束是强非线性的。我们提出了一种将非线性约束等价转化为线性约束（“用额外辅助变量换取线性性”）的几何拟线性化(GQL)方法。基于该方法的理论分析首次揭示了保界性与磁场无散度条件之间的内在联系。利用磁流体力学方程组的可对称化形式，建立了一种设计高精度保界数值方法的一般框架，给出了保界性的严格数学证明。在不加任何人工处理的情况下，所发展的高精度保界方法成功模拟了强磁场中的相对论磁流体爆炸问题（等离子体beta达到10的负10次方量级）和高马赫数的射流等极具挑战性问题。
报告人简介：吴开亮，南方科技大学数学系副教授、博士生导师。2011年获华中科技大学数学学士学位；2016年获北京大学计算数学博士学位；2016-2020年先后在美国犹他大学和俄亥俄州立大学从事博士后研究；2021年1月加入南科大、任副教授。研究方向包括计算流体力学与数值相对论、机器学习与数据驱动建模、微分方程数值解、高维逼近与不确定性量化等。研究成果发表在SINUM、M3AS、Numer. Math.、SISC、J. Comput. Phys.、JSC、ApJS、Phys. Rev. D等重要期刊上。曾获中国数学会计算数学分会 优秀青年论文奖一等奖（2015）和中国数学会 钟家庆数学奖（2019），入选国家高层次人才计划（青年），主持国家自然科学基金面上项目。
Title:An essentially oscillation-free discontinuous Galerkin method for hyperbolic systems
Time：10:00-11:00AM, Friday, November 19, 2021.
Tencent meeting：829 760 836
Speaker:刘勇（中国科学院数学与系统科学研究院）
Abstract:In this talk, we introduce an essentially oscillation-free discontinuous Galerkin (OFDG) method for systems of hyperbolic conservation laws. Based on the standard discontinuous Galerkin (DG) method, the numerical damping terms are introduced so as to control the spurious oscillations. We use both the classical Runge-Kutta method and the modified exponential Runge-Kutta method in time discretization. Particularly,the latter one could avoid additional restrictions of time step size due to the numerical damping. Extensive numerical experiments are shown to demonstrate our algorithm is robust and effective.
Short Bio：刘勇，中国科学院数学与系统科学研究院，华罗庚数学中心博士后。分别于2015年，2020年获中国科学技术大学学士和博士学位。2018年--2020年在美国布朗大学应用数学系联合培养。主要研究领域为高精度数值计算方法，包括间断有限元方法的算法设计及其数值分析、磁流体力学方程的数值模拟及应用等方面。在SIAM Journal on Numerical Analysis, Journal of Computational Physics, Journal of Scientific Computing 等杂志发表论文10余篇。
