Semi-Lagrangian Schemes For Linear And Fully Non-Linear Diffusion Equations

Mathematics of Computation

Volumn 82, Issue 283, 2013, Pages 1433-1462

  Debrabant, Kristian ^a   Jakobsen, Espen R ^b  

^a UNIVERSITY OF SOUTHERN DENMARK   (Denmark)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Convergence; Degenerate parabolic equations; Difference interpolation methods; Error bound; Hamilton Jacobi Bellman equations; Monotone approximation schemes; Stability; Viscosity solution

Indexed keywords

EID: 84878201774     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-2012-02632-9     Document Type: Article

References (27)

1. Guy Barles and Espen Robstad Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN Math. Model. Numer. Anal. 36 (2002), no. 1, 33-54. MR1916291 (2003h:65142)
2. Guy Barles and Espen Robstad Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations, Math. Comp. 76 (2007), no. 260, 1861-1893 (electronic). MR2336272 (2008i:65161)
3. Guy Barles and Panagiotis E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no. 3, 271-283. MR1115933 (92d:35137)
4. Olivier Bokanowski, Benjamin Bruder, Stefania Maroso, and Hasnaa Zidani, Numerical approximation for a superreplication problem under gamma constraints, SIAM J. Numer. Anal. 47 (2009), no. 3, 2289-2320. MR2519604 (2010j:91213)
5. Joseph Frédéric Bonnans, Elisabeth Ottenwaelter, and Hasnaa Zidani, A fast algorithm for the two dimensional HJB equation of stochastic control, M2AN Math. Model. Numer. Anal. 38 (2004), no. 4, 723-735. MR2087732 (2005e:93165)
6. Joseph Frédéric Bonnans and Hasnaa Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation, SIAM J. Numer. Anal. 41 (2003), no. 3, 1008-1021. MR2005192 (2004i:49061)
7. Fabio Camilli and Maurizio Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Model. Math. Anal. Numer. 29 (1995), no. 1, 97-122. MR1326802 (96a:49033)
8. Fabio Camilli and Espen Robstad Jakobsen, A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal. 47 (2009), no. 4, 2407-2431. MR2525605 (2010j:65227)
9. Italo Capuzzo Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim. 10 (1983), no. 4, 367-377. MR713483 (84j:49024)
10. Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67. MR1118699 (92j:35050)
11. Michael G. Crandall and Pierre-Louis Lions, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numer. Math. 75 (1996), no. 1, 17-41. MR1417861 (97j:65134)
12. John Crank and Phyllis Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Cambridge Philos. Soc. 43 (1947), 50-67. MR0019410 (8,409b)
13. Timothy A. Davis, Algorithm 832: UMFPACK V4.3-an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software 30 (2004), no. 2, 196-199. MR2075981
14. Hongjie Dong and Nicolai V. Krylov, On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients, Algebra i Analiz 17 (2005), no. 2, 108-132. MR2159586 (2006f:49050)
15. Stanley C. Eisenstat, Kenneth R. Jackson, and John W. Lewis, The order of monotone piecewise cubic interpolation, SIAM J. Numer. Anal. 22 (1985), no. 6, 1220-1237. MR811195 (87d:65014)
16. Maurizio Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim. 15 (1987), no. 1, 1-13. MR866164 (88c:49025)
17. Frederic N. Fritsch and Ralph E. Carlson, Monotone piecewise cubic interpolation, SIAM J. Numer. Anal. 17 (1980), no. 2, 238-246. MR567271 (81g:65012)
18. Nicolai V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probab. Theory Related Fields 117 (2000), no. 1, 1-16. MR1759507 (2001j:65134)
19. Harold J. Kushner and Paul Dupuis, Numerical methods for stochastic control problems in continuous time, second ed., Applications of Mathematics (New York), vol. 24, Springer-Verlag, New York, 2001, Stochastic Modelling and Applied Probability. MR1800098 (2001g:93002)
20. Jose-Luis Menaldi, Some estimates for finite difference approximations, SIAM J. Control Optim. 27 (1989), no. 3, 579-607. MR993288 (90m:65137)
21. Theodore Samuel Motzkin and Wolfgang Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Physics 31 (1953), 253-259. MR0052895 (14,693i)
22. Rémi Munos and Hasnaa Zidani, Consistency of a simple multidimensional scheme for Hamilton-Jacobi-Bellman equations, C.R. Math. Acad. Sci. Paris 340 (2005), no. 7, 499502. MR2135230 (2005k:49088)
23. Adam M. Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems, SIAM J. Numer. Anal. 44 (2006), no. 2, 879-895 (electronic). MR2218974 (2007a:65173)
24. Adam M. Oberman, Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian, Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 1, 221-238. MR2399429 (2009f:35101)
25. David M. Pooley, Peter A. Forsyth, and Ken R. Vetzal, Numerical convergence properties of option pricing PDEs with uncertain volatility, IMA J. Numer. Anal. 23 (2003), no. 2, 241-267. MR1974225 (2004b:65132)
26. Philip J. Rasch and David L. Williamson, On shape-preserving interpolation and semi-Lagrangian transport, SIAM J. Sci. Statist. Comput. 11 (1990), no. 4, 656-687. MR1054632(91f:65020)
27. Jiongmin Yong and Xun Yu Zhou, Stochastic controls, Applications of Mathematics (New York), vol. 43, Springer-Verlag, New York, 1999, Hamiltonian systems and HJB equations. MR1696772 (2001d:93004)