An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

TL;DR

This paper introduces an adaptive sparse grid collocation method using high-order piecewise polynomials, allowing for discontinuities, with applications in high-dimensional problems and uncertainty quantification.

Contribution

It provides a systematic framework for collocation on high-order discontinuous piecewise polynomial spaces, including error analysis and practical numerical comparisons.

Findings

Effective in high-dimensional function interpolation

Improves accuracy in uncertainty quantification tasks

Compatible with various polynomial interpolation schemes

Abstract

This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space that is allowed to be discontinuous. We consider both Lagrange and Hermite interpolation methods on nested collocation points. Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method. Error estimates are provided, and the numerical results in function interpolation, integration and some benchmark problems in uncertainty quantification are used to compare different collocation schemes.