Commun. Comput. Phys., 12 (2012), pp. 919-954.


A Compressed Sensing Approach for Partial Differential Equations with Random Input Data

L. Mathelin 1*, K. A. Gallivan 2

1 LIMSI-CNRS, BP 133, 91403 Orsay, France.
2 Mathematics Dpt., 208 Love Building, 1017 Academic Way, Florida State University, Tallahassee FL 32306-4510, USA.

Received 15 November 2010; Accepted (in revised version) 9 September 2011
Available online 28 March 2012
doi:10.4208/cicp.151110.090911a

Abstract

In this paper, a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output is presented. As with Monte-Carlo and stochastic collocation methods, only point-wise evaluations of the stochastic output response surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncertain problem of interest. The new approach differs from these standard methods in that it is based on ideas directly linked to the recently developed compressed sensing theory. The technique allows the retrieval of the modes that contribute most significantly to the approximation 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 an uncertainty quantification procedure. If the stochastic model output has a reasonably compressible representation in the retained approximation basis, the proposed method makes 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 the new method, requiring up to several orders of magnitude fewer solver calls than the usual sparse grid-based Polynomial Chaos (Smolyak scheme) to achieve comparable approximation accuracy.

AMS subject classifications: 60-08, 35J35, 35Q30

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Key words: Uncertainty quantification, compressed sensing, collocation technique, stochastic spectral decomposition, Smolyak sparse approximation, stochastic collocation.

*Corresponding author.
Email: mathelin@limsi.fr (L. Mathelin), gallivan@math.fsu.edu (K. A. Gallivan)
 

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