A non-adapted sparse approximation of PDEs with stochastic inputs
We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a black box. The method converges in probability (with probabilistic error bounds) as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples. We show that the method is well suited for truly high-dimensional problems.
Additional Information© 2011 Elsevier Inc. Received 10 June 2010. Received in revised form 24 October 2010. Accepted 4 January 2011. Available online 9 January 2011. A preliminary version of this work has appeared in a 2009 CTR Annual Research Brief report. The first author acknowledges the support of the United States Department of Energy under Stanford's Predictive Science Academic Alliance Program (PSAAP) for the preliminary stages of his work. The second author acknowledges the support of the National Science Foundation via NSF Grant CMMI-092600 and of the United States Department of Energy under Caltech's Predictive Science Academic Alliance Program (PSAAP).
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