@Article{CiCP-12-4,
author = {},
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 = {<p style="text-align: justify;">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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.151110.090911a},
url = {https://global-sci.com/article/80805/a-compressed-sensing-approach-for-partial-differential-equations-with-random-input-data}
}