Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-29T21:12:48.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenizationof Hamilton-Jacobi equations

Published online by Cambridge University Press:  15 April 2002

Nicolas Bacaër
Nicolas Bacaër*
Affiliation:
Université Paris 6, Laboratoire d'Analyse Numérique, 175 rue du chevaleret, 75013 Paris, France. ([email protected])
Get access

Abstract

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.

Keywords

Min-plus eigenvalue problemsnumerical analysisFrenkel-Kontorova modelHamilton-Jacobi equations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 6 , November 2001 , pp. 1185 - 1195
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Aubry, S., The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Physica D 7 (1983) 240-258. CrossRef
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.
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.
F. Baccelli, G.J. Olsder, J.P. Quadrat and G. Cohen, Synchronization and Linearity. Wiley, Chichester (1992).
Chou, W. and Griffiths, R.B., Ground states of one-dimensional systems using effective potentials. Phys. Rev. B 34 (1986) 6219-6234. CrossRef
Chou, W. and Duffin, R.J., An additive eigenvalue problem of physics related to linear programming. Adv. in Appl. Math. 8 (1987) 486-498. CrossRef
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
Concordel, M., Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula. Indiana Univ. Math. J. 45 (1996) 1095-1117. CrossRef
Dudnikov, P.I. and Samborskii, S.N., Endomorphisms of semimodules over semirings with idempotent operation. Math. USSR-Izv. 38 (1992) 91-105. CrossRef
Evans, L.C. and Gomes, D., Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. CrossRef
J.S. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Longman Scientific & Technical, Harlow (1992).
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.
V.N. Kolokoltsov and V.P. Maslov, Idempotent Analysis and its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997).
Namah, G. and Roquejoffre, J.M., The "hump" effect in solid propellant combustion. Interfaces Free Bound 2 (2000) 449-467. CrossRef
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.