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Abstract
We devise and implement quasiMonte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients. Our motivation comes from fluid flow in random porous media, where relevant functionals include the fluid pressure/velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field. Our emphasis is on situations where a very large number of random variables is needed to model the coefficient field. As an alternative to classical Monte Carlo, we here employ quasiMonte Carlo methods, which use deterministically chosen sample points in an appropriate (usually highdimensional) parameter space. Each realization of the PDE solution requires a finite element (FE) approximation in space, and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid (not necessarily the same as the FE grid). In the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using FFT. In this way we avoid the use of a truncated KarhunenLoeve expansion, but introduce high nominal dimension in parameter space. Numerical experiments with 2dimensional rough random fields, high variance and small length scale are reported, showing that the quasiMonte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O(N1/2) convergence rate, where N is the number of samples. Moreover, the rate of convergence of the quasiMonte Carlo method does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 106 are reported.
Original language  English 

Pages (fromto)  36683694 
Number of pages  27 
Journal  Journal of Computational Physics 
Volume  230 
Issue number  10 
Early online date  9 Feb 2011 
DOIs  
Publication status  Published  10 May 2011 
Keywords
 random porous media
 quasiMonte Carlo
 fluid flow
 circulant embedding
 fast Fourier transform
 highdimensional quadrature
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Projects
 1 Finished

Multilevel Monte Carlo Methods for Elliptic Problems
Engineering and Physical Sciences Research Council
1/07/11 → 30/06/14
Project: Research council