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

Guy Barles; Espen Robstad Jakobsen

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 36.1 (2010): 33-54. <http://eudml.org/doc/194095>.

@article{Barles2010,
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},
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},
month = {3},
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/194095},
volume = {36},
year = {2010},
}

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
DA - 2010/3//
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/194095
ER -

References

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  12. N.V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients. Probab. Theory Relat. Fields117 (2000) 1-16.  
  13. H.J. Kushner, Numerical Methods for Approximations in Stochastic Control Problems in Continuous Time. Springer-Verlag, New York (1992).  
  14. P.-L. Lions, Existence results for first-order Hamilton-Jacobi equations. Ricerche Mat.32 (1983) 3-23.  
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  16. P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part II: Viscosity solutions and uniqueness. Comm. Partial Differential Equations8 (1983) 1229-1276.  
  17. P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part III, in Nonlinear Partial Differential Equations and Appl., Séminaire du Collège de France, Vol. V, Pitman, Ed., Boston, London (1985).  
  18. P.-L. Lions and B. Mercier, Approximation numérique des équations de Hamilton-Jacobi-Bellman. RAIRO Anal. Numér.14 (1980) 369-393.  
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Citations in EuDML Documents

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  2. Dan Goreac, Oana-Silvia Serea, Linearization techniques for 𝕃 See PDF-control problems and dynamic programming principles in classical and 𝕃 See PDF-control problems
  3. Dan Goreac, Oana-Silvia Serea, Linearization techniques for See PDF -control problems and dynamic programming principles in classical and See PDF -control problems
  4. J. Frédéric Bonnans, Élisabeth Ottenwaelter, Housnaa Zidani, A fast algorithm for the two dimensional HJB equation of stochastic control
  5. G. Barles, A. Briani, E. Chasseigne, A Bellman approach for two-domains optimal control problems in ℝN
  6. J. Frédéric Bonnans, Élisabeth Ottenwaelter, Housnaa Zidani, A fast algorithm for the two dimensional HJB equation of stochastic control
  7. Dan Goreac, Viability, invariance and reachability for controlled piecewise deterministic Markov processes associated to gene networks
  8. Dan Goreac, Viability, invariance and reachability for controlled piecewise deterministic Markov processes associated to gene networks
  9. Dan Goreac, Viability, invariance and reachability for controlled piecewise deterministic Markov processes associated to gene networks

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