Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method

TL;DR

This paper introduces a globally convergent numerical method called convexification, employing Carleman estimates to compute viscosity solutions of Hamilton-Jacobi equations through a convex cost functional.

Contribution

The paper develops a new convexification approach using Carleman estimates to ensure strict convexity and convergence in solving Hamilton-Jacobi equations numerically.

Findings

The convexification functional is strictly convex, ensuring unique minimizers.

Gradient descent can effectively find the minimizer approximating the viscosity solution.

Numerical examples demonstrate the method's effectiveness in approximating solutions.

Abstract

We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a fixed small number. By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation under consideration. The strict convexity of this functional is rigorously proved using a new Carleman estimate. We also prove that the unique minimizer of the this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the viscosity solution of the Hamilton-Jacobi equation as the noise contained in the boundary data tends to zero. Some interesting numerical illustrations are presented.