A Generalized Newton Method for Homogenization of Hamilton-Jacobi Equations

We propose a new approach to the numerical solution of cell problems arising in the homogenization of Hamilton--Jacobi equations. It is based on a Newton-like method for solving inconsistent systems of nonlinear equations, coming from the discretization of the corresponding cell problems. We show that our method is able to solve efficiently cell problems in very general contexts, e.g., for first and second order scalar convex and nonconvex Hamiltonians, weakly coupled systems, dislocation dynamics, and mean field games, also in the case of more competing populations. A large collection of numerical tests in dimensions one and two shows the performance of the proposed method, both in terms of accuracy and computational time.

[1]  Simone Cacace,et al.  A numerical method for Mean Field Games on networks , 2015 .

[2]  Boualem Khouider,et al.  Computing the Effective Hamiltonian in the Majda-Souganidis Model of Turbulent Premixed Flames , 2002, SIAM J. Numer. Anal..

[3]  Régis Monneau,et al.  A numerical study for the homogenisation of one-dimensional models describing the motion of dislocations , 2008, Int. J. Comput. Sci. Math..

[4]  Régis Monneau,et al.  Homogenization of some particle systems with two-body interactions and of the dislocation dynamics , 2008 .

[5]  M. Rorro,et al.  An approximation scheme for the effective Hamiltonian and applications , 2006 .

[6]  Adam M. Oberman,et al.  Computing the Effective Hamiltonian using a Variational Approach , 2005, CDC/ECC.

[7]  Maurizio Falcone,et al.  On a Variational Approximation of the Effective Hamiltonian , 2008 .

[8]  S. Osher,et al.  One-sided difference approximations for nonlinear conservation laws , 1981 .

[9]  Adi Ben-Israel.  A Newton-Raphson method for the solution of systems of equations , 1966 .

[10]  Lawrence C. Evans,et al.  Some new PDE methods for weak KAM theory , 2003 .

[11]  Antonin Chambolle,et al.  A posteriori error estimates for the effective Hamiltonian of dislocation dynamics , 2012, Numerische Mathematik.

[12]  Nicolas Bacaër.  Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations , 2001 .

[13]  Hongkai Zhao,et al.  A New Approximation for Effective Hamiltonians for Homogenization of a class of Hamilton-Jacobi Equations , 2011, Multiscale Model. Simul..

[14]  P. Lions,et al.  Mean field games , 2007 .

[15]  F. Camilli,et al.  HOMOGENIZATION OF HAMILTON–JACOBI EQUATIONS: NUMERICAL METHODS , 2008 .

[16]  Adam M. Oberman,et al.  Homogenization of Metric Hamilton-Jacobi Equations , 2009, Multiscale Model. Simul..

[17]  Andrea Davini,et al.  Aubry Sets for Weakly Coupled Systems of Hamilton-Jacobi Equations , 2012, SIAM J. Math. Anal..

[18]  Diogo Aguiar Gomes,et al.  A stochastic analogue of Aubry-Mather theory , 2001 .

[19]  Fabio Camilli,et al.  Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs , 2009 .

[20]  Marco Cirant,et al.  Multi-population Mean Field Games systems with Neumann boundary conditions , 2015 .

[21]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[22]  Yves Achdou,et al.  Mean Field Games: Numerical Methods , 2010, SIAM J. Numer. Anal..