Homogenization of Metric Hamilton–Jacobi Equations

Abstract

Abstract. In this work we provide a novel approach to homogenization for a class of static Hamilton-Jacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach allows us to conclude that the homogenized equation also induces a metric. The advantage of the method is that we can solve just one auxiliary equation to recover the homogenized HamiltonianH(p). This is significant improvement over existing methods which require the solution of the cell problem (or a variational problem) for each value of p. Computational results are presented and compared with analytic results when available for piece-wise constant periodic and random speed functions.

Variational Formula for the Time Constant of First‐Passage Percolation

“…and the algorithm continues to produce a minimizer. Periodic homogenization has been well studied, and there are many algorithms to produce the effective Hamiltonian; see, for example, [18] or [36]. Our contribution here is that the algorithm works even if .0; ; !/ takes an uncountable number of values; i.e., the period is infinite.…”

“…Periodicity only forces the additional constraint P(A i ) = 1/n, and except for this, the problem is nearly unchanged. Periodic homogenization has been well-studied and there are many algorithms to produce the effective Hamiltonian; see for example, Gomes and Oberman [17] or Oberman et al [31].…”

A Multiscale Domain Decomposition Algorithm for Boundary Value Problems for Eikonal Equations

“…There are several approaches to reducing the computational cost of numerically solving Eikonal equations. For certain periodic functions r ǫ , one approach is homogenization [16,18]. The goal of homogenization is to derive an effective function, r, that accurately describes the effective properties of r ǫ in the solution.…”

Singular Perturbation of Optimal Control Problems on MultiDomains

“…However, this essential Hamiltonian H E is not necessarily Lipschitz continuous, which is a significant difficulty. There are some works [5,19] dealing with the homogenization of metric Hamilton-Jacobi equations where the Hamiltonians are continuous and coercive. But when the Hamiltonians become discontinuous, this problem remains a difficult issue.…”

“…But when the Hamiltonians become discontinuous, this problem remains a difficult issue. In [19], an algorithm has been introduced to solve the piecewise-periodic problems numerically where the Hamiltonians are not continuous, but there is no general theoretical result for this method.…”