207 research outputs found
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
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 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 the gpc expansion are
contained in certain weighted -spaces for . Based on this we
show that reconstructions of the parametric solutions computed from the sampled
problems converge, with high probability, at the , resp.
convergence rates afforded by best -term approximations of the parametric
solution up to logarithmic factors.Comment: revised version, 27 page
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
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