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

Abstract

The Hamilton-Jacobi (HJ) equations arise in optimal control and many other applications. Oftentimes, such equations are posed in high dimensions, and this presents great numerical challenges. In this paper, we propose an adaptive sparse grid (also called adaptive multiresolution) local discontinuous Galerkin (DG) method 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 including the orthonormal Alpert's multiwavelets and the interpolatory multiwavelets are used to achieve multiresolution. Numerical tests in up to four dimensions are provided to validate the performance of the method.