Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 6, p. 1185-1195

Using the min-plus version of the spectral radius formula, one proves: 1) that the unique eigenvalue of a min-plus eigenvalue problem depends continuously on parameters involved in the kernel defining the problem; 2) that the numerical method introduced by Chou and Griffiths to compute this eigenvalue converges. A toolbox recently developed at I.n.r.i.a. helps to illustrate these results. Frenkel-Kontorova models serve as example. The analogy with homogenization of Hamilton-Jacobi equations is emphasized.

Classification:  65J99,  65Z05
Keywords: Min-plus eigenvalue problems, numerical analysis, Frenkel-kontorova model, Hamilton-Jacobi equations
@article{M2AN_2001__35_6_1185_0,
author = {Baca\"er, Nicolas},
title = {Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {6},
year = {2001},
pages = {1185-1195},
zbl = {1037.65054},
mrnumber = {1873522},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_6_1185_0}
}

Bacaër, Nicolas. Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 6, pp. 1185-1195. http://www.numdam.org/item/M2AN_2001__35_6_1185_0/

[1] S. Aubry, The new concept of transitions by breaking of analyticity in a crystallographic model, in Solitons and Condensed Matter Physics, A.R. Bishop and T. Schneider, Eds., Springer-Verlag, Berlin (1978) 264-277.

[2] S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Physica D 7 (1983) 240-258. | Zbl 0559.58013

[3] N. Bacaër, Min-plus spectral theory and travelling fronts in combustion, in Proceedings of the Workshop on Max-Plus Algebras, Prague, August (2001). Submitted to S. Gaubert, Ed., Elsevier Science, Amsterdam.

[4] N. Bacaër, Can one use Scilab's max-plus toolbox to solve eikonal equations?, in Proceedings of the Workshop on Max-Plus Algebras, Prague, August (2001). Submitted to S. Gaubert, Ed., Elsevier Science, Amsterdam.

[5] F. Baccelli, G.J. Olsder, J.P. Quadrat and G. Cohen, Synchronization and Linearity. Wiley, Chichester (1992). | MR 1204266 | Zbl 0824.93003

[6] W. Chou and R.B. Griffiths, Ground states of one-dimensional systems using effective potentials. Phys. Rev. B 34 (1986) 6219-6234.

[7] W. Chou and R.J. Duffin, An additive eigenvalue problem of physics related to linear programming. Adv. in Appl. Math. 8 (1987) 486-498. | Zbl 0639.65033

[8] J. Cochet-Terrasson, G. Cohen, S. Gaubert, M. Mc Gettrick and J.P. Quadrat, Numerical computation of spectral elements in max-plus algebra. http://amadeus.inria.fr/gaubert/HOWARD.html

[9] M. Concordel, Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula. Indiana Univ. Math. J. 45 (1996) 1095-1117. | Zbl 0871.49025

[10] P.I. Dudnikov and S.N. Samborskii, Endomorphisms of semimodules over semirings with idempotent operation. Math. USSR-Izv. 38 (1992) 91-105. | Zbl 0746.16034

[11] L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. | Zbl 0986.37056

[12] J.S. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Longman Scientific & Technical, Harlow (1992). | MR 1163371 | Zbl 0780.16036

[13] R.B. Griffiths, Frenkel-Kontorova models of commensurate-incommensurate phase transitions, in Fundamental Problems in Statistical Mechanics. VII, H. van Beijeren, Ed., North-Holland, Amsterdam (1990) 69-110.

[14] V.N. Kolokoltsov and V.P. Maslov, Idempotent Analysis and its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997). | MR 1447629 | Zbl 0941.93001

[15] G. Namah and J.M. Roquejoffre, The “hump” effect in solid propellant combustion. Interfaces Free Bound 2 (2000) 449-467. | Zbl 0967.35156

[16] S.J. Sheu and A.D. Wentzell, On the solutions of the equation arising from the singular limit of some eigen problems, in Stochastic Analysis, Control, Optimization and Applications, W.M. McEneaney et al., Eds., Birkhäuser, Boston (1999) 135-150. | Zbl 0920.49015