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International Journal for Uncertainty Quantification

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.v1.i1.20
pages 19-33

ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA

Howard C. Elman
Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, USA
Christopher W. Miller
Department of Applied Mathematics and Scientific Computation, University of Maryland, USA
Eric T. Phipps
Sandia National Laboratories, PO Box 5800, MS 1318, Albuquerque, NM 87185, USA
Raymond S. Tuminaro
Sandia National Laboratories, PO Box 969, MS 9159, Livermore, CA 94551, USA

ABSTRACT

We compare the performance of two methods, the stochastic Galerkin method and the stochastic collocation method, for solving partial differential equations (PDEs) with random data. The stochastic Galerkin method requires the solution of a single linear system that is several orders larger than linear systems associated with deterministic PDEs. The stochastic collocation method requires many solves of deterministic PDEs, which allows the use of existing software. However, the total number of degrees of freedom in the stochastic collocation method can be considerably larger than the number of degrees of freedom in the stochastic Galerkin system. We implement both methods using the Trilinos software package and we assess their cost and performance. The implementations in Trilinos are known to be efficient, which allows for a realistic assessment of the computational complexity of the methods. We also develop a cost model for both methods which allows us to examine asymptotic behavior.