Scalable Fixed-Point Framework for High-Dimensional Hamilton-Jacobi Equations

TL;DR

This paper introduces a mesh-free, gradient-free fixed-point method leveraging the Hopf-Lax formula to efficiently solve high-dimensional Hamilton-Jacobi equations, demonstrating scalability and accuracy in up to 100 dimensions.

Contribution

It presents a novel fixed-point framework that avoids grids and differentiation, enabling scalable solutions for high-dimensional HJ equations.

Findings

Achieves high accuracy in up to 100 dimensions

Computational time largely independent of dimensionality

Effective for control problems and non-smooth solutions

Abstract

We propose a novel, mesh-free, and gradient-free fixed-point approach for computing viscosity solutions of high-dimensional Hamilton-Jacobi (HJ) equations. By leveraging the Hopf-Lax formula, our approach iteratively solves the associated variational problem via a Picard iteration, enabling efficient evaluation of both the solution and its corresponding control without relying on grids, characteristics, or differentiation. We demonstrate the practical efficacy and scalability of the approach through numerical experiments in up to 100 dimensions, including control problems and non-smooth solutions. Our results show that the proposed scheme achieves high accuracy, is highly efficient, and exhibits computational times that are largely independent of dimensionality, highlighting its suitability for high-dimensional problems.