A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions

TL;DR

This paper introduces a new sampling and preconditioning algorithm for sparse polynomial chaos expansions that outperforms traditional Monte Carlo methods, especially for high-degree polynomials and various weight functions.

Contribution

The authors develop a generalized algorithm using weighted equilibrium measures and Christoffel function preconditioning for improved sparse polynomial recovery.

Findings

The proposed method outperforms Monte Carlo sampling in high-degree polynomial recovery.

Numerical results show improved accuracy over Legendre and Hermite specific algorithms.

The algorithm is applicable to a wide range of weight functions on bounded and unbounded domains.

Abstract

In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality. However MC methods result in poor function recovery when the polynomial degree is high. Here we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided that demonstrate that our proposed Christoffel Sparse Approximation algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermite specific algorithms.