A least-squares approximation of partial differential equations with high-dimensional random inputs

Journal Article · Wed Jul 01 04:00:00 EDT 2009 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2009.03.006· OSTI ID:21308092

Doostan, Alireza ^[1]; Iaccarino, Gianluca ^[1]

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1. Mechanical Engineering Department, Stanford University, Stanford, CA 94305 (United States)

Uncertainty quantification schemes based on stochastic Galerkin projections, with global or local basis functions, and also stochastic collocation methods in their conventional form, suffer from the so called curse of dimensionality: the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem. In this paper, to overcome the curse of dimensionality, a low-rank separated approximation of the solution of a stochastic partial differential (SPDE) with high-dimensional random input data is obtained using an alternating least-squares (ALS) scheme. It will be shown that, in theory, the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system. For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. Finally, different aspects of the proposed methodology are explored through its application to some numerical experiments.

• OSTI ID: 21308092
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 12 Vol. 228; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ALGORITHMS
APPROXIMATIONS
FUNCTIONS
LEAST SQUARE FIT
MATHEMATICAL SPACE
PARTIAL DIFFERENTIAL EQUATIONS
PROBABILITY
RANDOMNESS
STOCHASTIC PROCESSES