An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions
An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions
复制标题
DOI：
10.1016/j.jcp.2021.110294
复制
发表时间：
2020-05
期刊：
影响因子：
--
通讯作者：
Wenting Guo;Juntao Huang;Zhanjing Tao;Yingda Cheng
中科院分区：
文献类型：
--
作者：
Wenting Guo;Juntao Huang;Zhanjing Tao;Yingda Cheng
Hamilton-Jacobi （HJ）方程出现在最优控制和许多其他应用中。通常，这样的方程是在高维中提出的，这就提出了很大的数值挑战。本文提出了一种求解高维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，我们可以处理中等高维的情况。结合了自适应性来捕获解决方案的扭结和其他局部结构。采用正交Alpert多小波和插值多小波两类多小波来实现多分辨率。提供了多达四个维度的数值试验来验证该方法的性能。
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.