An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions

TL;DR

This paper introduces an adaptive sparse grid local discontinuous Galerkin method tailored for high-dimensional Hamilton-Jacobi equations, effectively capturing local solution features and managing computational complexity.

Contribution

It develops a novel adaptive sparse grid DG framework using multiwavelets for high-dimensional HJ equations, addressing computational challenges and solution accuracy.

Findings

Effective in up to four dimensions

Captures local solution features accurately

Reduces computational cost in high dimensions

Abstract

We are interested in numerically solving the Hamilton-Jacobi (HJ) equations, which arise in optimal control and many other applications. Oftentimes, such equations are posed in high dimensions, and this poses great numerical challenges. This work proposes a class of adaptive sparse grid (also called adaptive multiresolution) local discontinuous Galerkin (DG) methods for solving Hamilton-Jacobi equations in high dimensions. By using the sparse grid techniques, we can treat moderately high dimensional cases. Adaptivity is incorporated to capture kinks and other local structures of the solutions. Two classes of multiwavelets are used to achieve multiresolution, which are the orthonormal Alpert's multiwavelets and the interpolatory multiwavelets. Numerical tests in up to four dimensions are provided to validate the performance of the method.