An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions
arXiv:2006.05250 · doi:10.1016/j.jcp.2021.110294
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.
arXiv commentarXiv admin note: text overlap with arXiv:2004.08525
References in corpus (8)
- Solving high-dimensional partial differential equations using deep learning
- On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations
- Sparse Grid Discontinuous Galerkin Methods for High-Dimensional Elliptic Equations
- A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations
- Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures
- An adaptive high-order piecewise polynomial based sparse grid collocation method with applications
- An adaptive multiresolution discontinuous Galerkin method with artificial viscosity for scalar hyperbolic conservation laws in multidimensions
- Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations