Volume 12, Issue 4
A Compressed Sensing Approach for Partial Differential Equations with Random Input Data

L. Mathelin & K. A. Gallivan

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

Published online: 2012-12

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  • 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 representationinthe retainedapproximationbasis, the proposedmethod makes the bestuse of the availableinformation and retrievesthe 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.


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@Article{CiCP-12-919, author = {L. Mathelin and K. A. Gallivan}, title = {A Compressed Sensing Approach for Partial Differential Equations with Random Input Data}, journal = {Communications in Computational Physics}, year = {2012}, volume = {12}, number = {4}, pages = {919--954}, 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 representationinthe retainedapproximationbasis, the proposedmethod makes the bestuse of the availableinformation and retrievesthe 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.


}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.151110.090911a}, url = {http://global-sci.org/intro/article_detail/cicp/7319.html} }
TY - JOUR T1 - A Compressed Sensing Approach for Partial Differential Equations with Random Input Data AU - L. Mathelin & K. A. Gallivan JO - Communications in Computational Physics VL - 4 SP - 919 EP - 954 PY - 2012 DA - 2012/12 SN - 12 DO - http://dor.org/10.4208/cicp.151110.090911a UR - https://global-sci.org/intro/cicp/7319.html KW - AB -

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 representationinthe retainedapproximationbasis, the proposedmethod makes the bestuse of the availableinformation and retrievesthe 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.


L. Mathelin & K. A. Gallivan. (1970). A Compressed Sensing Approach for Partial Differential Equations with Random Input Data. Communications in Computational Physics. 12 (4). 919-954. doi:10.4208/cicp.151110.090911a
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