Solving High Dimensional Partial Differential Equations Using Tensor Type Discretization and Optimization Process

TL;DR

This paper introduces a tensor-based discretization and optimization approach for efficiently solving high-dimensional partial differential equations, avoiding Monte Carlo methods and enabling direct numerical integration.

Contribution

It presents a novel tensor trial function design and an optimization framework that simplifies high-dimensional PDE solutions without stochastic sampling.

Findings

Numerical tests validate the effectiveness of the proposed methods.

The approach enables direct integration without Monte Carlo methods.

Transforming PDE solving into an optimization problem improves computational efficiency.

Abstract

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.