Year: 2018
Communications in Computational Physics, Vol. 24 (2018), Iss. 1 : pp. 286–308
Abstract
In this paper, we discuss a gradient-enhanced $ℓ_1$ approach for the recovery of sparse Fourier expansions. By $gradient$-$enhanced$ approaches we mean that the directional derivatives along given vectors are utilized to improve the sparse approximations. We first consider the case where both the function values and the directional derivatives at sampling points are known. We show that, under some mild conditions, the inclusion of the derivatives information can indeed decrease the coherence of measurement matrix, and thus leads to the improved the sparse recovery conditions of the $ℓ_1$ minimization. We also consider the case where either the function values or the directional derivatives are known at the sampling points, in which we present a sufficient condition under which the measurement matrix satisfies RIP, provided that the samples are distributed according to the uniform measure. This result shows that the derivatives information plays a similar role as that of the function values. Several numerical examples are presented to support the theoretical statements. Potential applications to function (Hermite-type) interpolations and uncertainty quantification are also discussed.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2018-0006
Communications in Computational Physics, Vol. 24 (2018), Iss. 1 : pp. 286–308
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Gradient-enhanced $ℓ_1$ minimization compressed sensing sparse Fourier expansions restricted isometry property mutual incoherence.
-
A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions
Guo, Ling | Narayan, Akil | Zhou, TaoJournal of Computational Physics, Vol. 367 (2018), Iss. P.49
https://doi.org/10.1016/j.jcp.2018.04.026 [Citations: 34] -
The Sufficient Conditions for Orthogonal Matching Pursuit to Exactly Reconstruct Sparse Polynomials
Huang, Aitong | Feng, Renzhong | Wang, AndongMathematics, Vol. 10 (2022), Iss. 19 P.3703
https://doi.org/10.3390/math10193703 [Citations: 1] -
Constructing Least-Squares Polynomial Approximations
Guo, Ling | Narayan, Akil | Zhou, TaoSIAM Review, Vol. 62 (2020), Iss. 2 P.483
https://doi.org/10.1137/18M1234151 [Citations: 30]