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.

汉密尔顿-雅各比（HJ）方程出现在最优控制和许多其他应用中。通常，这样的方程式是高维提出的，这带来了巨大的数值挑战。在本文中，我们提出了一种用于解决高维Hamilton-Jacobi方程的自适应稀疏网格（也称为自适应多分辨率）局部不连续Galerkin（DG）方法。通过使用稀疏网格技术，我们可以处理中等大小的情况。引入了适应性以捕获解决方案的纽结和其他局部结构。两类多小波包括正交Alpert多小波和插值多小波，用于实现多分辨率。提供了多达四个维度的数值测试，以验证该方法的性能。