×

zbMATH — the first resource for mathematics

A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations. (English) Zbl 1204.35070
The paper shows that a broad class of fully nonlinear, second-order parabolic or elliptic distributed parameter systems (PDEs) can be realized as the Hamilton-Jacobi-Bellman equations of deterministic two-person games. Namely, given a PDE, one can identify a deterministic, discrete-time, two-person game whose value function converges in the continuous-time limit to the viscosity solution of the desired equation. This game is a deterministic analogue of the corresponding stochastic game. In the parabolic setting with no \(u\)-dependence, it amounts to a semidiscrete numerical scheme whose time step is a min-max. The results of the work are quite interesting, because the usual control-based interpretations of second-order PDEs involve stochastic rather than deterministic control.

MSC:
35F21 Hamilton-Jacobi equations
35Q93 PDEs in connection with control and optimization
91A05 2-person games
35K55 Nonlinear parabolic equations
35J60 Nonlinear elliptic equations
91A50 Discrete-time games
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Armstrong, A finite-difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc. · Zbl 1239.91011
[2] Armstrong, An infinity Laplace equation with gradient term and mixed boundary conditions, Preprint (2009)
[3] Bardi, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations (1997) · Zbl 0890.49011
[4] Barles, A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal. 162 (4) pp 287– (2002) · Zbl 1052.35084
[5] Barles, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations 26 (11-12) pp 2323– (2001) · Zbl 0997.35023
[6] Barles, A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its application, Comm. Partial Differential Equations 23 (11-12) pp 1995– (1998) · Zbl 0919.35009
[7] Barles, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (3) pp 271– (1991) · Zbl 0729.65077
[8] Barron, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc. 360 (1) pp 77– (2008) · Zbl 1125.35019
[9] Bellettini, The level set method for systems of PDEs, Comm. Partial Differential Equations 32 (7-9) pp 1043– (2007) · Zbl 1132.35051
[10] Bernhard, Robust control approach to option pricing: a representation theorem and fast algorithm, SIAM J. Control Optim. 46 (6) pp 2280– (2007) · Zbl 1152.49029
[11] Buckdahn, A representation formula for the mean curvature motion, SIAM J. Math. Anal. 33 (4) pp 827– (2001) · Zbl 1074.93037
[12] Caffarelli, A rate of convergence for monotone finite difference approximations of fully nonlinear, uniformly elliptic PDEs, Comm. Pure Appl. Math 61 (1) pp 1– (2007)
[13] Caffarelli, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math. 180 (2) pp 301– (2010) · Zbl 1192.35048
[14] Carlini, Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2007, the 7th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, September 2007 pp 671– (2008)
[15] Carlini, A semi-Lagrangian scheme for the curve shortening flow in codimension-2, J. Comput. Phys. 225 (2) pp 1388– (2007) · Zbl 1255.65157
[16] Catté, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal. 32 (6) pp 1895– (1995) · Zbl 0841.68124
[17] Cheridito, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Comm. Pure Appl. Math. 60 (7) pp 1081– (2007) · Zbl 1121.60062
[18] Cox, Option pricing: A simplified approach, J. Financial Economics 7 (3) pp 229– (1979) · Zbl 1131.91333
[19] Crandall, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1) pp 1– (1992) · Zbl 0755.35015
[20] Crandall, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numer. Math. 75 (1) pp 17– (1996) · Zbl 0874.65066
[21] Evans, Some min-max methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J. 33 (1) pp 31– (1984) · Zbl 0543.35012
[22] Evans, Partial differential equations (1997)
[23] Evans, Perspectives in nonlinear partial differential equations pp 245– (2007)
[24] Evans, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J. 33 (5) pp 773– (1984) · Zbl 1169.91317
[25] Fleming, Controlled Markov processes and viscosity solutions (2005)
[26] Fleming, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J. 38 (2) pp 293– (1989) · Zbl 0686.90049
[27] Friedman, Handbook of game theory with economic applications II pp 781– (1994)
[28] Giga, Recent advances in scientific computing and partial differential equations (Hong Kong, 2002) pp 73– (2003)
[29] Giga, Viscosity solutions with shocks, Comm. Pure Appl. Math. 55 (4) pp 431– (2002) · Zbl 1028.35044
[30] Giga, Nonlinear phenomena with energy dissipation pp 103– (2008)
[31] Giga, A billiard-based game interpretation of the Neumann problem for the curve shortening equation, Adv. Differential Equations 14 (3-4) pp 201– (2009) · Zbl 1170.35437
[32] Giga, A level set approach to semicontinuous viscosity solutions for Cauchy problems, Comm. Partial Differential Equations 26 (5-6) pp 813– (2001) · Zbl 1005.49025
[33] Imbert, Repeated games for eikonal equations, integral curvature flows, and nonlinear parabolic integro-differential equations, Preprint (2009)
[34] Juutinen, On the evolution governed by the infinity Laplacian, Math. Ann. 335 (4) pp 819– (2006) · Zbl 1110.35037
[35] Kohn, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math. 59 (3) pp 344– (2006) · Zbl 1206.53072
[36] Kohn, 6th International Congress on Industrial and Applied Mathematics: Zurich, Switzerland, 16-20 July 2007 pp 239– (2009)
[37] Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math. 99 (2) pp 365– (2004) · Zbl 1070.65082
[38] Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp. 74 (251) pp 1217– (2005) · Zbl 1094.65110
[39] Peres, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (1) pp 167– (2009) · Zbl 1206.91002
[40] Peres, Tug of war with noise: a game theoretic view of the p-Laplacian, Duke Math. J. 145 (1) pp 91– (2008) · Zbl 1206.35112
[41] Soner, Dynamic programming for stochastic target problems and geometric flows, J. Eur. Math. Soc. (JEMS) 4 (3) pp 201– (2002) · Zbl 1003.49003
[42] Soner, A stochastic representation for the level set equations, Comm. Partial Differential Equations 27 (9-10) pp 2031– (2002) · Zbl 1036.49010
[43] Soner, A stochastic representation for mean curvature type geometric flows, Ann. Probab. 31 (3) pp 1145– (2003) · Zbl 1080.60076
[44] Taras’ev, Approximation schemes for the construction of minimax solutions of Hamilton-Jacobi equations, Prikl. Mat. Mekh. 58 (2) pp 22– (1994)
[45] Tsai, A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations, Math. Comp. 72 (241) pp 159– (2003) · Zbl 1013.65088
[46] Yu, Uniqueness of values of Aronsson operators and running costs in ”tug-of-war” games, Ann. Inst. H. Poincaré Non Linéaire 26 (4) pp 1299– (2009)
[47] Yu, Maximal and minimal solutions of an Aronsson equation: L variational problems versus the game theory, Calc. Var. Partial Differential Equations 37 (1-2) pp 73– (2010) · Zbl 1184.35145
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.