On the rate of convergence in homogenization of Hamilton-Jacobi equations

Let u ε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O ( ε ) of u ε → u as ε → 0 + for a large class of convex Hamiltonians H ( x , y , p ) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n = 1.

Download Full-text

Rate of Convergence in Periodic Homogenization of Hamilton–Jacobi Equations: The Convex Setting

Archive for Rational Mechanics and Analysis ◽

10.1007/s00205-019-01371-y ◽

2019 ◽

Vol 233 (2) ◽

pp. 901-934 ◽

Cited By ~ 2

Author(s):

Hiroyoshi Mitake ◽

Hung V. Tran ◽

Yifeng Yu

Keyword(s):

Rate Of Convergence ◽

Periodic Homogenization ◽

Jacobi Equations ◽

Convex Setting

Download Full-text

HOMOGENIZATION OF FIRST-ORDER EQUATIONS WITH u/∊-PERIODIC HAMILTONIAN: RATE OF CONVERGENCE AS ∊ → 0 AND NUMERICAL METHODS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511005349 ◽

2011 ◽

Vol 21 (06) ◽

pp. 1317-1353 ◽

Cited By ~ 1

Author(s):

YVES ACHDOU ◽

STEFANIA PATRIZI

Keyword(s):

Numerical Methods ◽

Rate Of Convergence ◽

Effective Hamiltonian ◽

First Order ◽

Jacobi Equations ◽

Homogenization Problems

We consider homogenization problems for first-order Hamilton–Jacobi equations with u∊/∊ periodic dependence, recently introduced by Imbert and Monneau, and also studied by Barles: this unusual dependence leads to nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter ∊ tends to 0. We obtain the same rates as those obtained by Capuzzo Dolcetta and Ishii for the more usual homogenization problems without the dependence in u∊/∊. In the second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.

Download Full-text

Rate of convergence for multiscale homogenization of Hamilton—Jacobi equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210507000704 ◽

2009 ◽

Vol 139 (3) ◽

pp. 519-539 ◽

Cited By ~ 4

Author(s):

Claudio Marchi

Keyword(s):

Optimal Control ◽

Finite Number ◽

Rate Of Convergence ◽

Differential Games ◽

Homogenization Problem ◽

Fully Nonlinear ◽

First Order ◽

Jacobi Equations ◽

Multiscale Homogenization

This paper concerns the homogenization problem for fully nonlinear first-order equations of Hamilton—Jacobi type with a finite number of scales. Under some coercivity and periodicity assumptions we provide an estimate of the rate of convergence. Finally, some examples arising from optimal control and deterministic differential games are discussed.

Download Full-text