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Sparse Recovery via ℓq -Minimization for Polynomial Chaos Expansions

Part of:Nontrigonometric harmonic analysisApproximations and expansionsNumerical approximation and computational geometry

Published online by Cambridge University Press:  12 September 2017

Ling Guo,
Yongle Liu and
Liang Yan
Ling Guo*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, China
Yongle Liu*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, China
Liang Yan*
Affiliation:
School of Mathematics, Southeast University, Nanjing, 210096, China
*
*Corresponding author. Email addresses:lguo@shnu.edu.cn (L. Guo), 1000378341@smail.shnu.edu.cn (Y. Liu), yanliang@seu.edu.cn (L. Yan)
*Corresponding author. Email addresses:lguo@shnu.edu.cn (L. Guo), 1000378341@smail.shnu.edu.cn (Y. Liu), yanliang@seu.edu.cn (L. Yan)
*Corresponding author. Email addresses:lguo@shnu.edu.cn (L. Guo), 1000378341@smail.shnu.edu.cn (Y. Liu), yanliang@seu.edu.cn (L. Yan)

Abstract

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In this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ 2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ 1 and reweighted ℓ 1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ 1 and reweighted ℓ 1 minimization.

Keywords

Uncertainty quantificationstochastic collocationℓq -minimizationpolynomial chaos expansions

MSC classification

Secondary:65D05: Interpolation42C05: Orthogonal functions and polynomials, general theory41A10: Approximation by polynomials

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017, pp. 775 - 797
Copyright
Copyright © Global-Science Press 2017 

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