报告题目：Efficient Sparse Grid Spectral Methods for High-dimensional PDEs
报告摘要：Many scientific and engineering applications require solving high-dimensional Partial Differential Equations (PDEs). Traditional methods solving high-dimensional PDEs using tensor-product discretizations need N^d total points if N points are used in each dimension, which soon become infeasible for problems with d > 3. This is so-called curse of dimensionality. However, for functions with special regularity, one can use sparse grid or hyperbolic-cross to approximate them. This approach is first introduced by Smolyak for quadrature and interpolation problems in 1960s, and then is extended to solve PDEs by Bungartz, Griebel, et al., where lower order finite element bases is used for problems with boundaries, Fourier bases and wavelets are used for problems without boundaries.
In this talk, I will introduce some efficient spectral sparse grid solvers for elliptic PDEs with non-periodic boundary conditions and PDEs in unbounded domains. The efficient implementation is achieved by an efficient sparse grid transform algorithm on Chebyshev-Gauss-Lobatto points. The treatments of high-dimensional PDEs with singularities in equation coefficients will covered as well.
This talk is based on joint works with Prof. Jie Shen and Dr. Yingwei Wang.
Haijun received his Ph.D. in 2007 at Peking University, and then worked as postdoc and visiting assistant professor
at Princeton and Purdue University for about 3 year, before moving back to Institute of Computational Mathematics, CAS
in 2010. His research includes the numerical methods for complex fluids and numerical methods for high-dimensional
PDEs, especially spectral method. He is promoted to associate professor in 2012 and received Chen Jingrun Future
Star Award in 2013.