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 (2010)

  • 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 35.6 (2010): 1185-1195. <http://eudml.org/doc/197454>.

@article{Bacaër2010,
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},
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; Hamilton-Jacobi equations},
language = {eng},
month = {3},
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/197454},
volume = {35},
year = {2010},
}

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
DA - 2010/3//
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; Hamilton-Jacobi equations
UR - http://eudml.org/doc/197454
ER -

References

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  12. J.S. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Longman Scientific & Technical, Harlow (1992).  Zbl0780.16036
  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).  Zbl0941.93001
  15. G. Namah and J.M. Roquejoffre, The "hump" effect in solid propellant combustion. Interfaces Free Bound2 (2000) 449-467.  Zbl0967.35156
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