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

2019 ◽  
Vol 233 (2) ◽  
pp. 901-934 ◽  
Author(s):  
Hiroyoshi Mitake ◽  
Hung V. Tran ◽  
Yifeng Yu
Keyword(s):  
Rate Of Convergence ◽  
Periodic Homogenization ◽  
Jacobi Equations ◽  
Convex Setting

scholarly journals Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

2021 ◽  
Vol 121 (2) ◽  
pp. 171-194
Author(s):  
Son N.T. Tu
Keyword(s):  
Rate Of Convergence ◽  
Classical Mechanics ◽  
One Dimension ◽  
Separable Potential ◽  
Periodic Homogenization ◽  
Periodic Functions ◽  
Effective Equation ◽  
Optimal Rate Of Convergence ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

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.

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

2011 ◽  
Vol 21 (06) ◽  
pp. 1317-1353 ◽  
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.

Rate of convergence for multiscale homogenization of Hamilton—Jacobi equations

2009 ◽  
Vol 139 (3) ◽  
pp. 519-539 ◽  
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.

scholarly journals Cauchy problem and periodic homogenization for nonlocal Hamilton–Jacobi equations with coercive gradient terms

2019 ◽  
Vol 150 (6) ◽  
pp. 3028-3059
Author(s):  
Martino Bardi ◽  
Annalisa Cesaroni ◽  
Erwin Topp
Keyword(s):  
Cauchy Problem ◽  
Uniform Convergence ◽  
Unique Solution ◽  
Comparison Principle ◽  
Nonlocal Operator ◽  
Periodic Homogenization ◽  
Effective Equation ◽  
Superlinear Growth ◽  
The Cauchy Problem ◽  
Jacobi Equations

AbstractThis paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

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