Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
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2014-10
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Report
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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, the parametric solution is then computed via compressive sensing methods from samples of (a functional of) the solution. A key ingredient in our analysis of independent interest consists in 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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published
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2014-29
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Seminar for Applied Mathematics, ETH Zurich
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Subject
Affine-parametric operator equations; Tensorized Chebyshev polynomial chaos approximation; Compressive sensing; Parametric diffusion equation; s-term approximation; High-dimensional approximation
Organisational unit
03435 - Schwab, Christoph / Schwab, Christoph
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
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Funding
247277 - Automated Urban Parking and Driving (EC)
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Is previous version of: http://hdl.handle.net/20.500.11850/128371