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Sparse Recovery via ℓq-Minimization for Polynomial Chaos Expansions

Part of: Approximations and expansions Numerical approximation and computational geometry Nontrigonometric harmonic analysis

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)
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Abstract

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 ℓ2 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 ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.

Keywords

Uncertainty quantificationstochastic collocationℓq-minimizationpolynomial chaos expansions

MSC classification

Secondary: 65D05: Interpolation 42C05: Orthogonal functions and polynomials, general theory 41A10: Approximation by polynomials
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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