论文标题
使用张量类型离散和优化过程求解高维偏微分方程
Solving High Dimensional Partial Differential Equations Using Tensor Type Discretization and Optimization Process
论文作者
论文摘要
在本文中，我们提出了一种张量的离散化和优化过程，以解决高维部分微分方程。首先，我们为高维偏微分方程的试验功能设计张量的试验函数。基于试验函数的张量结构，我们可以在无蒙特卡洛方法的帮助下进行近似解决方案的直接数值集成。然后与Ritz或Galerkin方法结合使用，求解高维偏微分方程可以转换以解决相关的优化问题。提供了一些数值测试来验证所提出的数值方法。
In this paper, we propose a tensor type of discretization and optimization process for solving high dimensional partial differential equations. First, we design the tensor type of trial function for the high dimensional partial differential equations. Based on the tensor structure of the trial functions, we can do the direct numerical integration of the approximate solution without the help of Monte-Carlo method. Then combined with the Ritz or Galerkin method, solving the high dimensional partial differential equation can be transformed to solve a concerned optimization problem. Some numerical tests are provided to validate the proposed numerical methods.
