Dr Handy Zhang
Division of Applied Mathematics - Brown University
Numerical methods for stochastic partial differential equations with white noise: a spectral approach
Thursday, Apr 24, 2014, 4:00pm to 5:00pm | Room 5-134
Deterministic integration methods in random space, such as polynomial chaos methods and stochastic collocation methods, have been extensively use for numerical methods of stochastic partial differential equation with color noise for their high accuracy. However, these deterministic integration methods are only efficient for problems with low random dimensionality. Thus, these deterministic integration methods are only efficient for short time integration of stochastic partial differential equation with temporal white noise as the number of increments of Brownian motion increases with time steps. For linear stochastic partial differential equations, we apply deterministic integration methods using the linear property of these equations and recursive strategy in time for a longer-time numerical integration.
We show that polynomial chaos methods and stochastic collocation methods are comparable in computational performance and can be more efficient than Monte Carlo methods, when the recursive strategy in time is used. In these deterministic integration methods, we truncate the Brownian motion with its orthogonal expansions other than piecewise linear approximation. The orthogonal expansions can lead to higher order schemes with proper time discretization when polynomial chaos expansion methods are used. However, we usually keep only a small number of truncation terms in the orthogonal expansion of Brownian motion to efficiently use deterministic integration methods for temporal noise. Further, we consider the spectral approximation using truncated orthogonal expansions of Brownian motion for semilinear elliptic equations with spatial additive noise. We show that when the solution is smooth enough, the spectral approximation is superior to the piecewise linear approximation while both approximations are comparable when the solution is not smooth.
Zhongqiang (Handy) Zhang is now a postdoctoral researcher at Division of Applied Mathematics of Brown University. During his graduate study at Brown University, he worked with Professors George Em Karniadakis and Boris Rozovksy on numerical methods for stochastic partial differential equations using stochastic spectral methods. Before he joined Brown University, he received a B.S. degree in mathematics at Qufu Normal University in 2004 and the Sc. M. and PhD degrees in mathematics at Shanghai University in 2006 and 2011, respectively. His research interest is numerical methods for stochastic differential equations and their applications.