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

Abstract

We analyze the convergence of compressive sensing based sampling techniques for the ecient evaluation of functionals of solutions for a class of high-dimensional, ane-parametric, linear operator equations which depend on possibly innitely 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 sucient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coecients of gpc expansion are contained in certain weighted 'p-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. L1 convergence rates aorded by best s-term approximations of the parametric solution up to logarithmic factors.