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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
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by Holger Rauhut and Christoph Schwab PDF
Math. Comp. 86 (2017), 661-700 Request permission

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
  • Email: rauhut@mathc.rwth-aachen.de
  • Christoph Schwab
  • Affiliation: Seminar for Applied Mathematics, ETH Zurich, Rämistr. 101, 8092 Zürich, Switzerland
  • MR Author ID: 305221
  • Email: christoph.schwab@sam.math.ethz.ch
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 661-700
  • MSC (2010): Primary 35B30; Secondary 65N30, 41A58, 94A20
  • DOI: https://doi.org/10.1090/mcom/3113
  • MathSciNet review: 3584544