Stochastic Galerkin spectral method: Algorithm and application to SDE with discrete measure
报告专家:王汉权教授(云南师范大学)
报告时间:2022年3月31日（周四）15：00-16：00
腾讯会议 586-947-359
邀请人：唐庆粦教授
报告摘要：
There are many previous research on designing efficient and high-order numerical methods for stochastic differential equations (SDE) driven by discrete or continuous random variables. They mostly focus on proposing numerical methods for SDE with independent random variables, and scarcely solving SDE driven by dependent and discrete random variables. In this talk, we propose a Galerkin spectral method for solving SDE with dependent and discrete random variables. Our main technique are (1) starting from a set of linearly independent polynomials, we construct the orthogonal basis by adopting the Gauss-Schmidt orthogonalization; (2) based on the newly constructed orthogonal basis, we firstly assume the unknown function in the SDE has the generalized polynomial chaos expansion, secondly implement the Galerkin method for the SDE in the discrete measure space, and thirdly we obtain a deterministic differential equations for the coefficients of the expansion; (3) we employ a Fourier spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional stochastic Poisson equation, two-dimensional stochastic Poisson equation, one-dimensional stochastic heat equation, and two-dimensional stochastic heat equation, respectively. All the presented stochastic equation are driven by two discrete random variables, they are dependent and have Poisson distribution as their joint probability density.Discussion on the newly proposed method is made and some further enhancements related to the method will be discussed.
专家简介：
王汉权，云南师范大学数学学院教授。兼任四川大学数学学院博士生导师、云南财经大学特聘教授、博士生导师。中国数学会计算数学分会委员、云南省中青年学术带头人、云南省数学会常委理事，入选2013年教育部“新世纪优秀人才支持计划”。主要从事计算数学与科学工程计算，在偏微分方程数值解法等方面发表论著40余篇（SCI收录的高水平研究论文30余篇，科学出版社出版专著2部，部分论文发表在SIAM Journal of Numerical Analysis、SIAM Journal on Applied Mathematics、Journal of Computational Physics期刊上）。现主持国家自科基金面上项目1项、云南省基础研究重点项目1项。主持完成包括重大计划培育项目在内的国家自然科学基金3项，2014年获“云南省有突出贡献优秀专业技术人才”称号，2017年获云南省自然科学奖三等奖1项。
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