A Compressed Sensing Approach for Partial Differential Equations with Random Input Data (Communications in Computational Physics, 2012)
In this paper, a novel approach for quantifying the parametric uncertaintyassociated with a stochastic problem output is presented. As with Monte-Carlo andstochastic collocation methods, only point-wise evaluations of the stochastic outputresponse surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncertain problem ofinterest. The new approach differs from these standard methods in that it is basedon ideas directly linked to the recently developed compressed sensing theory. Thetechnique allows the retrieval of the modes that contribute most significantly to theapproximation of the solution using a minimal amount of information. The generation of this information, via many solver calls, is almost always the bottle-neck of anuncertainty quantification procedure. If the stochastic model output has a reasonablycompressible representation in the retained approximation basis, the proposed methodmakes the best use of the available information and retrieves the dominant modes. Uncertainty quantification of the solution of both a 2-D and 8-D stochastic Shallow Water problem is used to demonstrate the significant performance improvement of thenew method, requiring up to several orders of magnitude fewer solver calls than theusual sparse grid-based Polynomial Chaos (Smolyak scheme) to achieve comparableapproximation accuracy.