|
Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations
Author(s):
Guy
Barles;
Espen
R.
Jakobsen.
Journal:
Math. Comp.
76
(2007),
1861-1893.
MSC (2000):
Primary 65M15, 65M06, 35K60, 35K70, 49L25
Posted:
April 20, 2007
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.
References:
-
- 1.
- G. Barles and E. R. Jakobsen.
On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations.
M2AN Math. Model. Numer. Anal. 36(1):33-54, 2002. MR 1916291 (2003h:65142) - 2.
- G. Barles and E. R. Jakobsen.
Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations.
SIAM J. Numer. Anal. 43(2):540-558, 2005. MR 2177879 - 3.
- G. Barles and P. E. Souganidis.
Convergence of approximation schemes for fully nonlinear second order equations.
Asymptotic Anal. 4(3):271-283, 1991. MR 1115933 (92d:35137) - 4.
- F. Bonnans, E. Ottenwaelter, and H. Zidani.
A fast algorithm for the two dimensional HJB equation of stochastic control.
M2AN Math. Model. Numer. Anal. 38(4):723-735, 2004. MR 2087732 (2005e:93165) - 5.
- F. Bonnans and H. Zidani.
Consistency of generalized finite difference schemes for the stochastic HJB equation.
SIAM J. Numer. Anal. 41(3):1008-1021, 2003. MR 2005192 (2004i:49061) - 6.
- F. Camilli and M. Falcone.
An approximation scheme for the optimal control of diffusion processes.
RAIRO Modél. Math. Anal. Numér. 29(1): 97-122, 1995. MR 1326802 (96a:49033) - 7.
- I. Capuzzo-Dolcetta and L. C. Evans.
Optimal switching for ordinary differential equations.
SIAM J. Control Optim. 22(1):143-161, 1984. MR 728678 (85b:49043) - 8.
- M. G. Crandall, H. Ishii, and P.-L. Lions.
User's guide to viscosity solutions of second order partial differential equations.
Bull. Amer. Math. Soc. (N.S.), 27(1):1-67, 1992. MR 1118699 (92j:35050) - 9.
- M. G. Crandall and P.-L. Lions.
Two approximations of solutions of Hamilton-Jacobi equations.
Math. Comp. 43(167):1-19, 1984. MR 744921 (86j:65121) - 10.
- H. Dong and N. V. Krylov.
On the rate of convergence of finte-difference approximations for Bellman equations with constant coefficients.
St. Petersburg Math. J. 17(2): 295-313, 2006. MR 2159586 (2006f:49050) - 11.
- H. Dong and N. V. Krylov.
On the Rate of Convergence of Finite-difference Approximations for Parabolic Equations with  and  Coeffcients.
Electron. J. Differential Equations 2005(102): 1-25, 2005. MR 2162263 (2006i:35008) - 12.
- H. J. Kushner and P. Dupuis.
Numerical methods for stochastic control problems in continuous time.
Springer-Verlag, New York, 2001. MR 1800098 (2001g:93002) - 13.
- L. C. Evans and A. Friedman.
Optimal stochastic switching and the Dirichlet problem for the Bellman equation.
Trans. Amer. Math. Soc. 253:365-389, 1979. MR 536953 (80f:93091) - 14.
- H. Ishii.
On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs.
Comm. Pure Appl. Math., 42(1):15-45, 1989. MR 973743 (89m:35070) - 15.
- H. Ishii and S. Koike.
Viscosity solutions for monotone systems of second-order elliptic PDEs.
Comm. Partial Differential Equations 16(6-7):1095-1128, 1991. MR 1116855 (92h:35066) - 16.
- H. Ishii and S. Koike.
Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games.
Funkcial. Ekvac., 34:143-155, 1991. MR 1116886 (92h:35067) - 17.
- E. R. Jakobsen.
On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems.
Math. Models Methods Appl. Sci. (M3AS) 13(5):613-644, 2003. MR 1978929 (2004g:49054) - 18.
- E. R. Jakobsen and K. H. Karlsen.
Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations.
J. Differential Equations 183:497-525, 2002. MR 1919788 (2003i:35135) - 19.
- N. V. Krylov.
On the rate of convergence of finite-difference approximations for Bellman's equations.
St. Petersburg Math. J., 9(3):639-650, 1997. MR 1466804 (98h:49033) - 20.
- N. V. Krylov.
On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients.
Probab. Theory Relat. Fields, 117:1-16, 2000. MR 1759507 (2001j:65134) - 21.
- N. V. Krylov.
Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies.
Electron. J. Probab. 4(2), 1999. MR 1668597 (2000b:49056) - 22.
- N. V. Krylov.
On the rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients.
Appl. Math. Optim. 52(2): 365-399, 2005. MR 2174020 (2006k:65219) - 23.
- P.-L. Lions.
Personal communication. - 24.
- P.-L. Lions and B. Mercier.
Approximation numérique des équations de Hamilton-Jacobi-Bellman.
RAIRO Anal. Numér. 14(4):369-393, 1980. MR 596541 (82b:65055) - 25.
- C.-W. Shu.
Total-variation-diminishing time discretizations.
SIAM J. Sci. Statist. Comput. 9(6):1073-1084, 1988. MR 963855 (90a:65196) - 26.
- P. E. Souganidis.
Approximation schemes for viscosity solutions of Hamilton-Jacobi equations.
J. Differential Equations 59(1):1-43, 1985. MR 803085 (86k:35028) - 27.
- A. Tourin.
Splitting methods for Hamilton-Jacobi equations.
Numer. Methods Partial Differential Equations, 22(2): 381-396, 2006. MR 2201439 (2006j:65254) - 28.
- N. Yamada.
Viscosity solutions for a system of elliptic inequalities with bilateral obstacles.
Funkcial. Ekvac. 30(2-3):417-425, 1987. MR 927191 (88m:35061)
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(2000):
65M15, 65M06, 35K60, 35K70, 49L25
Retrieve articles in all Journals with MSC
(2000):
65M15, 65M06, 35K60, 35K70, 49L25
Additional Information:
Guy
Barles
Affiliation:
Laboratoire de Mathématiques et Physique Théorique, University of Tours, 37200 Tours, France
Email:
barles@lmpt.univ-tours.fr
Espen
R.
Jakobsen
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
Email:
erj@math.ntnu.no
DOI:
10.1090/S0025-5718-07-02000-5
PII:
S 0025-5718(07)02000-5
Keywords:
Hamilton-Jacobi-Bellman equations,
switching system,
viscosity solution,
approximation schemes,
finite difference methods,
splitting methods,
convergence rate,
error bound
Received by editor(s):
June 24, 2005
Received by editor(s) in revised form:
June 29, 2006
Posted:
April 20, 2007
Additional Notes:
Jakobsen was supported by the Research Council of Norway, grant no. 151608/432
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org