The Carleman convexification method for Hamilton-Jacobi equations on the whole space

TL;DR

This paper introduces the Carleman convexification method, a globally convergent numerical approach for solving Hamilton-Jacobi equations in any dimension by convexifying the mismatch functional with Carleman weights, ensuring unique solutions.

Contribution

The paper develops a new convexification theorem using Carleman weights, guaranteeing strict convexity and unique minimizers for Hamilton-Jacobi equations in multiple dimensions.

Findings

The mismatch functional becomes strictly convex with Carleman weights.

The minimizer approximates the viscosity solution of the Hamilton-Jacobi equation.

Numerical results demonstrate effectiveness in 1D and 2D cases.

Abstract

We propose a new globally convergent numerical method to solve Hamilton-Jacobi equations in $\mathbb{R}^d$, $d \geq 1$. This method is named as the Carleman convexification method. By Carleman convexification, we mean that we use a Carleman weight function to convexify the conventional least squares mismatch functional. We will prove a new version of the convexification theorem guaranteeing that the mismatch functional involving the Carleman weight function is strictly convex and, therefore, has a unique minimizer. Moreover, a consequence of our convexification theorem guarantees that the minimizer of the Carleman weighted mismatch functional is an approximation of the viscosity solution we want to compute. Some numerical results in 1D and 2D will be presented.