论文标题
适用于汉密尔顿 - 雅各比方程的自适应稀疏网格局部不连续的galerkin方法
An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions
论文作者
论文摘要
我们对在最佳控制和许多其他应用程序中产生的汉密尔顿 - 雅各布（HJ）方程的数值求解感兴趣。通常，这种方程式在高维度中提出，这构成了巨大的数值挑战。这项工作提出了一类自适应稀疏网格（也称为自适应多种解决方案）局部不连续的Galerkin（DG）方法，用于在高维度中求解汉密尔顿 - 雅各比方程。通过使用稀疏的网格技术，我们可以治疗中等高维的情况。适应适合捕获解决方案的扭结和其他局部结构。两类的多波网用于实现多解析，它们是正统的Alpert的多波管和插值多波武器。提供多达四个维度的数值测试以验证该方法的性能。
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.
