Volume 54, Number 6, November-December 2020
|Page(s)||1883 - 1915|
|Published online||16 September 2020|
The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures
King Abdullah University of Science and Technology (KAUST), AMCS Division, Thuwal 23955-6900, Saudi Arabia
* Corresponding author: email@example.com
Accepted: 8 May 2020
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.
Mathematics Subject Classification: 65M22 / 35F21 / 35B27
Key words: Mean-field game / effective Hamiltonians / Mather measure
© EDP Sciences, SMAI 2020
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