The Hessian Riemannian flow and Newton's method for Effective Hamiltonians and Mather measures

TL;DR

This paper introduces a novel approach combining Hessian Riemannian flow and Newton's method to efficiently compute effective Hamiltonians and Mather measures, with proven convergence and improved stability in numerical experiments.

Contribution

It develops a new algorithmic framework that unifies the computation of effective Hamiltonians and Mather measures, with theoretical convergence guarantees and enhanced numerical stability.

Findings

Proves convergence of Hessian Riemannian flow in continuous and discrete settings.

Demonstrates improved stability and non-negativity preservation in numerical experiments.

Provides a new method for approximating stationary mean-field games.

Abstract

Effective Hamiltonians arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton's method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than {related} methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.