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

Nicolas Bacaër

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 6, page 1185-1195
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.6 (2001): 1185-1195. <http://eudml.org/doc/194091>.

@article{Bacaër2001,
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.},
author = {Bacaër, Nicolas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Min-plus eigenvalue problems; numerical analysis; Frenkel-kontorova model; Hamilton-Jacobi equations; Frenkel-Kontorova models; min-plus integral eigenvalue problem; convergence; solid-state physics; homogenization},
language = {eng},
number = {6},
pages = {1185-1195},
publisher = {EDP-Sciences},
title = {Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations},
url = {http://eudml.org/doc/194091},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Bacaër, Nicolas
TI - Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 6
SP - 1185
EP - 1195
AB - 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.
LA - eng
KW - Min-plus eigenvalue problems; numerical analysis; Frenkel-kontorova model; Hamilton-Jacobi equations; Frenkel-Kontorova models; min-plus integral eigenvalue problem; convergence; solid-state physics; homogenization
UR - http://eudml.org/doc/194091
ER -

References

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  9. [9] M. Concordel, Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula. Indiana Univ. Math. J. 45 (1996) 1095–1117. Zbl0871.49025
  10. [10] P.I. Dudnikov and S.N. Samborskii, Endomorphisms of semimodules over semirings with idempotent operation. Math. USSR-Izv. 38 (1992) 91–105. Zbl0746.16034
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  13. [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. [14] V.N. Kolokoltsov and V.P. Maslov, Idempotent Analysis and its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997). Zbl0941.93001MR1447629
  15. [15] G. Namah and J.M. Roquejoffre, The “hump” effect in solid propellant combustion. Interfaces Free Bound 2 (2000) 449–467. Zbl0967.35156
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