Enhancing $\ell_1$-minimization estimates of polynomial chaos expansions using basis selection

TL;DR

This paper introduces an adaptive basis selection method for polynomial chaos expansions that improves the accuracy of $ ext{l}_1$-minimization estimates by focusing on important dimensions and reducing unimportant terms.

Contribution

The proposed basis selection technique adaptively constructs anisotropic basis sets, enhancing polynomial chaos expansion accuracy and efficiency over fixed basis approaches.

Findings

Basis selection yields more accurate PCEs within the same computational budget.

The method effectively handles non-uniform random variables.

Gradient information can be incorporated to improve basis selection.

Abstract

In this paper we present a basis selection method that can be used with $\ell_1$-minimization to adaptively determine the large coefficients of polynomial chaos expansions (PCE). The adaptive construction produces anisotropic basis sets that have more terms in important dimensions and limits the number of unimportant terms that increase mutual coherence and thus degrade the performance of $\ell_1$-minimization. The important features and the accuracy of basis selection are demonstrated with a number of numerical examples. Specifically, we show that for a given computational budget, basis selection produces a more accurate PCE than would be obtained if the basis is fixed a priori. We also demonstrate that basis selection can be applied with non-uniform random variables and can leverage gradient information.