Computing the Effective Hamiltonian Using a Variational Approach

A numerical method for homogenization of Hamilton--Jacobi equations is presented and implemented as an $L^\infty$ calculus of variations problem. Solutions are found by solving a nonlinear convex optimization problem. The numerical method is shown to be convergent, and error estimates are provided....

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Bibliographic Details
Published in:SIAM journal on control and optimization 2004-01, Vol.43 (3), p.792-812
Main Authors: Gomes, Diogo A., Oberman, Adam M.
Format: Article
Language:English
Subjects:
Applied mathematics
Calculus
Calculus of variations
Convex analysis
Homogenization
Numerical analysis
Optimization
Propagation
Viscosity
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Summary:A numerical method for homogenization of Hamilton--Jacobi equations is presented and implemented as an $L^\infty$ calculus of variations problem. Solutions are found by solving a nonlinear convex optimization problem. The numerical method is shown to be convergent, and error estimates are provided. One and two dimensional examples are worked in detail, comparing known results with the numerical ones and computing new examples. The cases of nonstrictly convex Hamiltonians and Hamiltonians for which the cell problem has no solution are treated.
ISSN:0363-0129
1095-7138
DOI:10.1137/S0363012902417620