A weighted L1-minimization approach for sparse polynomial chaos expansions

TL;DR

This paper introduces a weighted l1-minimization method for sparse polynomial chaos expansions that leverages prior coefficient decay information to improve high-dimensional stochastic function approximation.

Contribution

It develops a weighted l1-minimization algorithm with theoretical recovery guarantees tailored for high-dimensional polynomial chaos approximation.

Findings

Weighted l1-minimization outperforms standard methods in numerical tests.

The method effectively recovers solutions to high-dimensional stochastic differential equations.

The approach provides theoretical conditions for guaranteed recovery.

Abstract

This work proposes a method for sparse polynomial chaos (PC) approximation of high-dimensional stochastic functions based on non-adapted random sampling. We modify the standard l1 -minimization algorithm, originally proposed in the context of compressive sampling, using a priori information about the decay of the PC coefficients and refer to the resulting algorithm as weighted l1 -minimization. We provide conditions under which we may guarantee recovery using this weighted scheme. Numerical tests are used to compare the weighted and non-weighted methods for the recovery of solutions to two differential equations with high-dimensional random inputs: a boundary value problem with a random elliptic operator and a 2-D thermally driven cavity flow with random boundary condition.