Sparse Recovery via ℓq-Minimization for Polynomial Chaos Expansions. Issue 4 (12th September 2017)

• Record Type: Journal Article
• Authors: Guo, Ling; Liu, Yongle; Yan, Liang
• Is Part Of: Numerical mathematics. Volume 10:Issue 4(2017)
• Page Start: 775
• Page End: 797
• Publication Date: 2017-09-12
• Subjects: 65D05, -- 42C05, -- 41A10; Uncertainty quantification, -- stochastic collocation, -- ℓq-minimization, -- polynomial chaos expansions; Numerical analysis -- Periodicals; Numerical analysis; Periodicals; 518.05
• DOI: 10.4208/nmtma.2017.0001
• Languages: English
• ISSNs: 1004-8979

Abstract

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 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.