A Christoffel function weighted least squares algorithm for collocation approximations

TL;DR

This paper introduces a novel Christoffel function weighted least squares algorithm for polynomial approximation in uncertainty quantification, demonstrating improved performance over standard Monte Carlo methods through theoretical analysis and numerical validation.

Contribution

The paper proposes a new sampling and weighting scheme using Christoffel functions for polynomial collocation, enhancing Monte Carlo approximation accuracy.

Findings

The algorithm outperforms standard Monte Carlo methods in various scenarios.

Theoretical analysis supports the effectiveness of the Christoffel-weighted approach.

Numerical experiments validate the improved convergence and stability.

Abstract

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights 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.