Computing Multiscale Stochastic PDEs using Data-Driven Stochastic Basis
Abstract
One of the major challenges in uncertainty quantification is to solve stochastic PDEs with both multiscale features in the physical space and high dimensional random input variables. To solve these problems, we not only need to use a very fine mesh to resolve the small scales of the solution in the physical space, but also need to approximate the solution in the stochastic space with high input dimension. To overcome this difficulty, the speaker and his research group have developed several data-driven numerical methods for solving stochastic PDEs. These include the Dynamically Bi-Orthogonal Method (DyBO) for time dependent problems, a multiscale data-driven stochastic method, a multiscale multilevel Monte Carlo method, and a heterogeneous stochastic FEM framework for elliptic PDEs in the multi-query setting. More recently, the speaker and his research group develop localized stochastic methods that are targeted at stochastic problems with small correlation length. Such problems are known to be extremely challenging. Even if we use the Karhunen-Loeve expansion to extract data-driven basis, the number of unknowns is still prohibitively expensive. Inspired by the recent developments in information theory, the speaker and his research group introduce the low rank plus sparsity criteria in constructing local stochastic basis. The speaker will give several examples and present careful analysis of numerical complexities to demonstrate the advantage of the new criteria over other existing methods.
About the speaker
Prof. Thomas Hou is the Charles Lee Powell Professor of Applied and Computational Mathematics at Caltech, and is one of the leading experts in vortex dynamics and multiscale problems. His research interests are centered around developing analytical tools and effective numerical methods for vortex dynamics, interfacial flows, and multiscale problems. He received his PhD from UCLA in 1987. Upon graduating from UCLA, he joined the Courant Institute as a postdoc and then became a faculty member in 1989. He moved to the Applied Math Department at Caltech in 1993, and is currently the Executive Director of Applied and Computational Mathematics.
Prof. Hou has received a number of honors and awards, including Fellow of the American Academy of Arts and Sciences in 2011, the SIAM Fellow in 2009, the Computational and Applied Sciences Award from USACM in 2005, the Morningside Gold Medal in Applied Mathematics in 2004, the SIAM Wilkinson Prize in Numerical Analysis and Scientific Computing in 2001, the Francois N. Frenkiel Award from the Division of Fluid Mechanics of APS in 1998, the Feng Kang Prize in Scientific Computing in 1997, a Sloan fellow from 1990 to 1992. He was an invited plenary speaker at the International Congress of Industrial and Applied Mathematics in 2003, and an invited speaker of the International Congress of Mathematicians in 1998. He was also the founding Editor-in-Chief of the SIAM Journal on Multiscale Modeling and Simulation from 2002 to 2007.
