Reweighted minimization method for stochastic elliptic differential equations

We consider elliptic stochastic partial differential equations (SPDEs) with random coefficients and solve them by expanding the solution using generalized polynomial chaos (gPC). Under some mild conditions on the coefficients, the solution is “sparse” in the random space, i.e., only a small number of gPC basis makes considerable contribution to the solution. To exploit this sparsity, we employ reweighted l 1 minimization to recover the coefficients of the gPC expansion. We also combine this method with random sampling points based on the Chebyshev probability measure to further increase the accuracy of the recovery of the gPC coefficients. We first present a one-dimensional test to demonstrate the main idea, and then we consider 14 and 40 dimensional elliptic SPDEs to demonstrate the significant improvement of this method over the standard l 1 minimization method. For moderately high dimensional (∼10) problems, the combination of Chebyshev measure with reweighted l 1 minimization performs well while for higher dimensional problems, reweighted l 1 only is sufficient. The proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources.