A Data-Driven Stochastic Multiscale Method

Abstract

We introduce a data-driven stochastic multiscale method to solve stochastic PDEs with high dimensionality in probability space.

This method consists of two steps: Offline and Online. In the Offline step, we construct a multiscale stochastic basis by approximating the covariance of the solution of the SPDE using Monte Carlo methods. In the Online step, we represent the stochastic solution as a truncated expansion using the multiscale stochastic basis. By solving a set of coupled PDEs of deterministic coefficients, we obtain the numerical solutions to SPDEs. One important property of this method is that the stochastic basis obtained in the offline computation can be used repeatedly in online computation for a large class of stochastic problems with different deterministic forcing coefficients or boundary conditions. This method effectively reduces the dimension of the stochastic PDEs. As a consequence, much smaller number of basis is required in the online computation to achieve the same level of error tolerance compared to the (generalized) polynomial chaos method or Wiener-Chaos Expansion method. Some numercal results will be presented to demonstrate the effectiveness of the method.
 

About the Speaker

Prof. Thomas Y. 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 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.