Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

Numerische Mathematik
Ivan G GrahamIan H Sloan

Abstract

In a previous paper (Graham et al. in J Comput Phys 230:3668-3694, 2011), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but no convergence theory was provided. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a...Continue Reading

Software Mentioned

interpolateSolution
Matlab
Vert
QMC4PDE
Matlab PDE
scriptstyle

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