On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations

Guy Barles; Espen Robstad Jakobsen

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

  • Volume: 36, Issue: 1, page 33-54
  • ISSN: 0764-583X

Abstract

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Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.

How to cite

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Barles, Guy, and Jakobsen, Espen Robstad. "On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.1 (2002): 33-54. <http://eudml.org/doc/244934>.

@article{Barles2002,
abstract = {Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.},
author = {Barles, Guy, Jakobsen, Espen Robstad},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Hamilton-Jacobi-Bellman equation; viscosity solution; approximation schemes; finite difference methods; convergence rate; approximation; finite difference method; convergence; monotone approximation schemes; dynamic programming},
language = {eng},
number = {1},
pages = {33-54},
publisher = {EDP-Sciences},
title = {On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations},
url = {http://eudml.org/doc/244934},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Barles, Guy
AU - Jakobsen, Espen Robstad
TI - On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 1
SP - 33
EP - 54
AB - Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.
LA - eng
KW - Hamilton-Jacobi-Bellman equation; viscosity solution; approximation schemes; finite difference methods; convergence rate; approximation; finite difference method; convergence; monotone approximation schemes; dynamic programming
UR - http://eudml.org/doc/244934
ER -

References

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