Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 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
Mots-clés : Min-plus eigenvalue problems, numerical analysis, Frenkel-kontorova model, Hamilton-Jacobi equations 
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     title = {Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, {Frenkel-Kontorova} models and homogenization of {Hamilton-Jacobi} equations},
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Bacaër, Nicolas. Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 1185-1195. http://archive.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

[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 | Zbl

[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

[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

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

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

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

[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 | Zbl

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

[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