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Medientyp:
E-Artikel
Titel:
COMPRESSIVE SENSING PETROV-GALERKIN APPROXIMATION OF HIGH-DIMENSIONAL PARAMETRIC OPERATOR EQUATIONS
Beteiligte:
RAUHUT, HOLGER;
SCHWAB, CHRISTOPH
Erschienen:
American Mathematical Society, 2017
Erschienen in:Mathematics of Computation
Sprache:
Englisch
ISSN:
0025-5718;
1088-6842
Entstehung:
Anmerkungen:
Beschreibung:
<label>Abstract</label>
<p>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 𝓁<sub>𝑝</sub>-spaces for 𝟶 < 𝑝 ≤ 𝟷. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the 𝐿₂, resp. 𝐿<sub>∞</sub>, convergence rates afforded by best 𝘴-term approximations of the parametric solution up to logarithmic factors.</p>