Stochastic collocation methods via $L_1$ minimization using randomized quadratures

TL;DR

This paper introduces a non-intrusive stochastic collocation method using $L_1$ minimization and randomized quadratures to efficiently approximate multivariate functions for uncertainty quantification, with theoretical validation and numerical demonstrations.

Contribution

It proposes a novel $L_1$ minimization approach with randomized quadratures for stochastic collocation, extending to various measures and providing theoretical analysis.

Findings

The method accurately approximates functions under different measures.

Theoretical analysis confirms the validity of the approach.

Numerical examples demonstrate the effectiveness of the method.

Abstract

In this work, we discuss the problem of approximating a multivariate function via $\ell_1$ minimization method, using a random chosen sub-grid of the corresponding tensor grid of Gaussian points. The independent variables of the function are assumed to be random variables, and thus, the framework provides a non-intrusive way to construct the generalized polynomial chaos expansions, stemming from the motivating application of Uncertainty Quantification (UQ). We provide theoretical analysis on the validity of the approach. The framework includes both the bounded measures such as the uniform and the Chebyshev measure, and the unbounded measures which include the Gaussian measure. Several numerical examples are given to confirm the theoretical results.