Commun. Comput. Phys., 6 (2009), pp. 509-535.


A Comparative Study of Stochastic Collocation Methods for Flow in Spatially Correlated Random Fields

Haibin Chang 1*, Dongxiao Zhang 2

1 Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China.
2 Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China; and Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA.

Received 27 February 2008; Accepted (in revised version) 24 December 2008
Available online 6 February 2009

Abstract

Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years. Similar to Monte Carlo methods, the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it. The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method, Smolyak method, Stroud 2 or 3 cubature method, and adaptive Stroud method. Another type of collocation method, the probabilistic collocation method (PCM), has also been proposed and applied to flow in porous media. In this paper, we discuss these methods in terms of their accuracy, efficiency, and applicable range for flow in spatially correlated random fields. These methods are compared in details under different conditions of spatial variability and correlation length. This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency. The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods. Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.

AMS subject classifications: 60H15, 65M70, 76S05, 76M22

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Key words: Stochastic collocation method, probabilistic collocation method, stochastic partial differential equations, Smolyak sparse grid method.

*Corresponding author.
Email: haibinch@usc.edu (H. Chang), donzhang@usc.edu (D. Zhang)
 

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