Applied and Computational Math Seminar: Spectral stochastic finite elements for two nonlinear problems

Speaker: Bedrich Sousedik, UMBC
Title: Spectral stochastic finite elements for two nonlinear problems

Abstract: We study two applications of spectral stochastic finite element methods (SSFEM): Navier-Stokes equations and eigenvalue problems. In the first part we focus the steady-state Navier-Stokes equations assuming that the viscosity is a random field given as a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration. In the second part we focus on random eigenvalue problems. Given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration and Newton iteration, which are based on the stochastic Galerkin finite element method. Our approach allows the computation of interior eigenvalues, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. For both applications we compare the accuracy with that of Monte Carlo and stochastic collocation methods, and we demonstrate effectiveness of the algorithms using a set of benchmark problems. This is a joint work with Howard C. Elman and Kookjin Lee, and the research was supported by the U.S. National Science Foundation under grant DMS1521563. 

Time: Friday, November 22, 2019, 1:30-2:30pm

Place: Exploratory Hall, Room 4106