Stochastic collocation methods via minimization of Transformed $L_1$ penalty

TL;DR

This paper introduces an improved sparse reconstruction method combining transformed  minimization with stochastic collocation, demonstrating enhanced recoverability and efficiency for uncertainty quantification in polynomial expansions.

Contribution

It presents new theoretical results on TL1 minimization recoverability and combines it with stochastic collocation for better polynomial coefficient identification.

Findings

Enhanced recoverability of sparse and non-sparse functions

Demonstrated efficiency in polynomial and analytical functions

Potential for practical uncertainty quantification applications

Abstract

We study the properties of sparse reconstruction of transformed $\ell_1$ (TL1) minimization and present improved theoretical results about the recoverability and the accuracy of this reconstruction from undersampled measurements. We then combine this method with the stochastic collocation approach to identify the coefficients of sparse orthogonal polynomial expansions for uncertainty quantification. In order to implement the TL1 minimization, we use the DCA-TL1 algorithm which was introduced by Zhang and Xin. In particular, when recover non-sparse functions, we adopt an adaptive DCA-TL1 method to guarantee the sparest solutions. Various numerical examples, including sparse polynomial functions recovery and non-sparse analytical functions recovery are presented to demonstrate the recoverability and efficiency of this novel method and its potential for problems of practical interests.