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

Technical Report · Sat Dec 01 04:00:00 EST 2007

DOI:https://doi.org/10.2172/934852· OSTI ID:934852

Webster, Clayton; Tempone, Raul ^[1]; Nobile, Fabio ^[2]

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1. Florida State University, Tallahassee, FL
2. Politecnico di Milano, Italy

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.

• Research Organization: Sandia National Laboratories
• Sponsoring Organization: USDOE
• DOE Contract Number: AC04-94AL85000
• OSTI ID: 934852
• Report Number(s): SAND2007-8093
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
99 GENERAL AND MISCELLANEOUS
APPROXIMATIONS
COMPARATIVE EVALUATIONS
CONVERGENCE
MONTE CARLO METHOD
Numerical grid generation (Numerical analysis)
PARTIAL DIFFERENTIAL EQUATIONS
Stochastic analysis.
Stochastic partial differential equations.