The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data.

Webster, Clayton ; Tempone, Raul (Florida State University, Tallahassee, FL) ; Nobile, Fabio (Politecnico di Milano, Italy)

关键词: 97;    APPROXIMATIONS;    CONVERGENCE;    MONTE CARLO METHOD;    PARTIAL DIFFERENTIAL EQUATIONS;    COMPARATIVE EVALUATIONS Stochastic partial differential equations.;    Numerical grid generati;   

DOI  :  10.2172/934852
RP-ID  :  SAND2007-8093
PID  :  OSTI ID: 934852
Others  :  TRN: US200815%%178

学科分类：社会科学、人文和艺术（综合）

美国|英语

来源: SciTech Connect

【 摘 要 】

This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo.