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

par Xianjin Yang (King Abdullah University of Science and Technology)

vendredi 8 mars 2019, 11:00 → 12:00 Europe/Paris

XLIM Salle X.203

Effective Hamiltonians arise in multiple 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 existence and 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 previous methods in problems that are close to singular.