Sparse Pseudospectral Approximation Method

TL;DR

This paper introduces a new sparse pseudospectral approximation method that improves the accuracy of polynomial coefficient estimation in high-dimensional uncertainty quantification problems by leveraging Smolyak's algorithm.

Contribution

It reexamines Smolyak's algorithm and exploits interpolation-projection connections to develop a sparse pseudospectral method that accurately reproduces basis function coefficients.

Findings

Accurate reproduction of polynomial coefficients using the new method.

Numerical results demonstrate improved approximation accuracy.

Proper use of sparse grid rules enhances pseudospectral methods.

Abstract

Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials. By reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients of basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.