The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 1883-1915.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1051/m2an/2020036
Classification : 65M22, 35F21, 35B27
Mots-clés : Mean-field game, effective Hamiltonians, Mather measure 
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     title = {The {Hessian} {Riemannian} flow and {Newton{\textquoteright}s} method for effective {Hamiltonians} and {Mather} measures},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1883--1915},
     publisher = {EDP-Sciences},
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     zbl = {1472.65110},
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Gomes, Diogo A.; Yang, Xianjin. The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 1883-1915. doi : 10.1051/m2an/2020036. http://archive.numdam.org/articles/10.1051/m2an/2020036/

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