On the Rate of Convergence of Difference Approximations for Uniformly Nondegenerate Elliptic Bellman’s Equations

Applied Mathematics & Optimization - Tập 69 - Trang 431-458 - 2013
N. V. Krylov1
1University of Minnesota, Minneapolis, USA

Tóm tắt

We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman’s equations is of order at least h 2/3, where h is the mesh size. The equations are considered in smooth bounded domains.

Tài liệu tham khảo

Barles, G., Jakobsen, E.R.: On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. Modél. Math. Anal. Numér. 36(1), 33–54 (2002) Barles, G., Jakobsen, E.R.: Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal. 43(2), 540–558 (2005) Barles, G., Jakobsen, E.R.: Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comput. 76(260), 1861–1893 (2007) Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271–283 (1991) Crandall, M.G., Lions, P.-L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1–19 (1984) Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982) Dong, Hongjie, Krylov, N.V.: On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients. Algebra Anal. 17(2), 108–132 (2005). St. Petersburg Math. J. 17(2) 295–313 (2006) Dong, Hongjie, Krylov, N.V.: The rate of convergence of finite-difference approximations for parabolic Bellman equations with Lipschitz coefficients in cylindrical domains. Appl. Math. Optim. 56(1), 37–66 (2007) Fleming, W.L., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Stochastic Modelling and Applied Probability, vol. 25. Springer, New York (2006) Froese, B.D., Oberman, A.M.: Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49(4), 1692–1714 (2011) Krylov, N.V.: Controlled Diffusion Processes. Nauka, Moscow (1977). in Russian; English translation, Springer, 1980 Krylov, N.V.: On control of the solution of a stochastic integral equation with degeneration. Izv. Akad. Nauk SSSR, Ser. Mat. 36(1), 248–261 (1972). in Russian; English translation: Math. USSR Izv. 6(1), 249–262 (1972) Krylov, N.V.: On the rate of convergence of finite–difference approximations for Bellman’s equations. Algebra Anal. 9(3), 245–256 (1997). in Russian; English translation: St. Petersburg Math. J. 9(3), 639–650 (1998) Krylov, N.V.: Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies. Electron. J. Probab. 4, 1–19 (1999). http://www.math.washington.edu/~ejpecp/EjpVol4/paper2.abs.html Krylov, N.V.: A priori estimates of smoothness of solutions to difference Bellman equations with linear and quasi-linear operators. Math. Comput. 76, 669–698 (2007) Krylov, N.V.: On factorizations of smooth nonnegative matrix-values functions and on smooth functions with values in polyhedra. Appl. Math. Optim. 58(3), 373–392 (2008) Krylov, N.V.: On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications. Probl. Mat. Anal., 59, 3–24 (2011). in Russian; English translation: Journal of Math. Sci. New York 177(1), 1–26 (2011) Krylov, N.V.: Interior estimates for the first-order differences for finite-difference approximations for elliptic Bellman’s equations. Appl. Math. Optim. 65(3), 349–370 (2012) Krylov, N.V.: Interior estimates for second differences of solutions of finite-difference elliptic Bellman’s equations. Math. Comp. 82(283), 1463–1487 (2013) Kushner, H.J.: Probability Methods for Approximations in Stochastic Control and for Elliptic Equations. Mathematics in Science and Engineering, vol. 129. Academic Press, New York (1977) Kushner, H.J., Dupuis, P.G.: Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, Berlin (1992) Kuo, H.-J., Trudinger, N.S.: Linear elliptic difference inequalities with random coefficients. Math. Comput. 55(191), 37–53 (1990) Kuo, H.-J., Trudinger, N.S.: Discrete methods for fully nonlinear elliptic equations. SIAM J. Numer. Anal. 29(1), 123–135 (1992) Kuo, H.-J., Trudinger, N.S.: Local estimates for parabolic difference operators. J. Differ. Equ. 122, 398–413 (1995) Kuo, H.-J., Trudinger, N.S.: Positive difference operators on general meshes. Duke Math. J. 83(2), 415–433 (1996) Motzkin, T., Wasow, W.: On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Phys. 37, 253–259 (1953) Oberman, A.M.: 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(1), 221–238 (2008) Pragarauskas, G.: Approximation of controlled solutions of Itô equations by controlled Markov chains. Lit. Mat. Sbornik 23(1), 175–188 (1983). in Russian; English translation: Lith. Math. J. 23(1), 98–108 (1983) Souganidis, P.E.: Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differ. Equ. 59(1), 1–43 (1985) Safonov, M.V.: Classical solution of second-order nonlinear elliptic equations. Izv. Akad. Nauk SSSR, Ser. Mat. 52(6), 1272–1287 (1988). in Russian; English translation: Math. USSR-Izv. 33(3), 597–612 (1989)