Commun. Comput. Phys., 12 (2012), pp. 1051-1069.


A Numerical Comparison Between Quasi-Monte Carlo and Sparse Grid Stochastic Collocation Methods

Juarez dos Santos Azevedo 1*, Saulo Pomponet Oliveira 2

1 CETEC-UFRB, Centro, 44380-000, Cruz das Almas-BA, Brazil.
2 DMAT-UFPR, Centro Politecnico, 81531-980, Curitiba-PR, Brazil.

Received 26 January 2011; Accepted (in revised version) 23 September 2011
Available online 28 March 2012
doi:10.4208/cicp.260111.230911a

Abstract

Quasi-Monte Carlo methods and stochastic collocation methods based on sparse grids have become popular with solving stochastic partial differential equations. These methods use deterministic points for multi-dimensional integration or interpolation without suffering from the curse of dimensionality. It is not evident which method is best, specially on random models of physical phenomena. We numerically study the error of quasi-Monte Carlo and sparse grid methods in the context of groundwater flow in heterogeneous media. In particular, we consider the dependence of the variance error on the stochastic dimension and the number of samples/collocation points for steady flow problems in which the hydraulic conductivity is a lognormal process. The suitability of each technique is identified in terms of computational cost and error tolerance.

AMS subject classifications: 60H15, 65C05, 65C20, 65C20

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Key words: Karhunen-Loeve expansion, Monte Carlo, quasi-Monte Carlo, sparse grid.

*Corresponding author.
Email: juarezsa@gmail.com (J. S. Azevedo), saulopo@ufpr.br (S. P. Oliveira)
 

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