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A Gradient-Enhanced l(1) Approach for the Recovery of Sparse Trigonometric Polynomials
Xu, Zhiqiang; Zhou, Tao
AbstractIn 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.
KeywordGradient-enhanced l(1) minimization compressed sensing sparse Fourier expansions restricted isometry property mutual incoherence
Funding ProjectNSFC[91630203] ; NSFC[11422113] ; NSFC[11331012] ; NSFC[11688101] ; National Basic Research Program of China (973 Program)[2015CB856000] ; NSF of China[91630203] ; NSF of China[11688101] ; NSF of China[91630312] ; NSF of China[11571351] ; NSF of China[11731006] ; science challenge project[TZ2018001] ; NCMIS ; youth innovation promotion association (CAS)
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000455953800014
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Document Type期刊论文
Corresponding AuthorZhou, Tao
AffiliationChinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math, LSEC, Beijing, Peoples R China
Recommended Citation
GB/T 7714
Xu, Zhiqiang,Zhou, Tao. A Gradient-Enhanced l(1) Approach for the Recovery of Sparse Trigonometric Polynomials[J]. COMMUNICATIONS IN COMPUTATIONAL PHYSICS,2018,24(1):286-308.
APA Xu, Zhiqiang,&Zhou, Tao.(2018).A Gradient-Enhanced l(1) Approach for the Recovery of Sparse Trigonometric Polynomials.COMMUNICATIONS IN COMPUTATIONAL PHYSICS,24(1),286-308.
MLA Xu, Zhiqiang,et al."A Gradient-Enhanced l(1) Approach for the Recovery of Sparse Trigonometric Polynomials".COMMUNICATIONS IN COMPUTATIONAL PHYSICS 24.1(2018):286-308.
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