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Mathematics of Computation

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Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations

Authors: Holger Rauhut and Christoph Schwab
Journal: Math. Comp. 86 (2017), 661-700
MSC (2010): Primary 35B30; Secondary 65N30, 41A58, 94A20
Published electronically: May 17, 2016
MathSciNet review: 3584544
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Abstract: We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called “non-intrusive” sampling of the high-dimensional parameter space, reminiscent of Monte Carlo sampling. In contrast to Monte Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists of a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of gpc expansion are contained in certain weighted $\ell _p$-spaces for $0<p\leq 1$. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the $L_2$, resp. $L_\infty$, convergence rates afforded by best $s$-term approximations of the parametric solution up to logarithmic factors.

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Additional Information

Holger Rauhut
Affiliation: Lehrstuhl C für Mathematik (Analysis), RWTH Aachen University, Pontdriesch 10, 52062 Aachen, Germany
MR Author ID: 720385

Christoph Schwab
Affiliation: Seminar for Applied Mathematics, ETH Zurich, Rämistr. 101, 8092 Zürich, Switzerland
MR Author ID: 305221

Keywords: Compressive sensing, affine-parametric operator equations, parametric diffusion equation, $s$-term approximation, high-dimensional approximation, tensorized Chebyshev polynomial chaos approximation
Received by editor(s): October 18, 2014
Received by editor(s) in revised form: August 7, 2015, and September 21, 2015
Published electronically: May 17, 2016
Article copyright: © Copyright 2016 American Mathematical Society