OPTIMIZATION APPROACH FOR THE MONGE-AMPÈRE EQUATION

doi:10.1016/S0252-9602(18)30814-2

Abstract

Abstract:

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.

Key words: elliptic Monge-Ampère equation, gradient conjugate method, finite element Galerkin method

Fethi BEN BELGACEM. OPTIMIZATION APPROACH FOR THE MONGE-AMPÈRE EQUATION[J].Acta mathematica scientia,Series B, 2018, 38(4): 1285-1295.

Trendmd

References 39

[1]	Armijo L. Minimization of functions having Lipschits continuous partial derivatives. Pacific J Math, 1966, 16:1-3
[2]	Aubin T. Nonlinear Analysis on Manifolds, Monge-Ampère Equations. Berlin:Springer-Verlag, 1982
[3]	Benamou J D, Brenier Y. The Monge-Kantorovich mass transfer and its computational fluid mechanics formulation. Int J Numer Meth Fluids, 2002
[4]	Benamou J-D, Frose B D, Oberman A M. Two numerical methods for the elliptic Monge-Ampere equation. ESSAIM:M2AN, 2010, 44:737-758
[5]	Bokanowski O, Grébert B. Deformations of density functions in molecular quantum chemistry. J Math Phys, 1996, 37(4):1553-1573
[6]	Böhmer K. On finite element methods for fully nonlinear elliptic equations of second order. SIAM J Numer Anal, 2008, 46(3):1212-1249
[7]	Brenier Y. Some geometric PDEs related to hydrodynamics and electrodynamics//Proceedings of the International Congress of Mathematicians, Beijing 2002, August 20-28, Vol Ⅲ. Beijing:Higher Eduction Press, 2002:761-771
[8]	Cabré X. Topics in regularity and qualitatives properties of solutions of non linear elliptic equations. Discrete Contin Dyn Syst, 2002, 8:331-359
[9]	Caffarelli L. Non linear elliptic theory and the Monge-Ampère equation//Proceedings of the International Congress of Mathematicians, Beijing 2002, August 20-28, Vol I. Beijing:Higher Eduction Press, 2002:179-187
[10]	Caffarelli L, Cabré X. Fully Nonlinear Elliptic Equations. Providence, RI:American Mathematical Society, 1995
[11]	Caffarelli L, Kochengin S A, Oliker V I. On the Numerical solution of the problem of reflector design with given far-field scattering data. Contemporary Mathematics, 1999, 226:13-32
[12]	Caffarelli L, Li Y Y. A liouville theorem for solutions of the Monge-Ampère equation with periodic data. Ann Inst H Poincarè Anal Non Linéaire, 2004, 21:97-120
[13]	Caffarelli L, Nirenberg L, Spruck J. The Dirichlet problem for nonlinear second-order elliptic equation I. Monge-Ampère equation. Comm Pure Appl Math, 1984, 17:396-402
[14]	Caffarelli L, Nirenberg L, Spruck J. The Dirichlet problem for the degenerate Monge-Ampère equation. Rev Mat Ibero, 1985, 2:19-27
[15]	Chang S-Y A, Yang P C. Nonlinear partial differential equations in conformal geometry//Proceedings of the International Congress of Mathematicians, Beijing 2002, August 20-28, Vol I. Beijing:Higher Eduction Press, 2002:189-207
[16]	Ciarlet P G. The Finite Element Method for Elliptic Problems. Amsterdam:North-Holland, 1978
[17]	Courant R, Hilbert D. Methods of Mathematical Physics. Vol Ⅱ. New York:Wiley, 1962
[18]	Cullen M J P, Douglas R J. Applications of the Monge-Ampère equation and Monge transport problem to metorology and oceanog-raphy. Contemporary Mathematics, 1999, 226:33-53
[19]	Dean E J, Glowinski R. Numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions:an augmented Lagrangian approach. C R Acad Sci Paris, Ser I, 2003, 336:779-784
[20]	Dean E J, Glowinski R. Numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions:a least-squares approach. C R Acad Sci Paris, Ser I, 2004, 339:887-892
[21]	Dean E J, Glowinski R. Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput Methods Appl Mech Engrg, 2006, 195(13/16):1344-1386
[22]	Dean E J, Glowinski R. On the numerical solution of the elliptic Monge-Ampère equation in dimension two:a least-squares approach//Partial Differential Equations. Volume 16 of Comput Methods Appl Sci, Dordrecht:Springer, 2008:43-63
[23]	Feng X, Neilan M. Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. J Scient Comp, 2009, 38(1):74-98
[24]	Goldstein A A. On steepest descent. SIAM J Control, 1965, 3:147-151
[25]	Haltiner G J. Numerical Weather Prediction. New York:Wiley, 1971
[26]	Kochengin S A, Oliker V L. Determination of reflector surfaces from near-filed scattering data. Inverse Problems, 1997, 13(2):363-373
[27]	Le Dimet F X, Ouberdous M. Retrieval of balanced fields:an optimal control method. Tellus, 1993, 45A:449-461
[28]	Lions P L. Une méthode nouvelle pour l'existense de solutions réulière de l'équation de Monge-Ampère réelle. C R Acad Sc Paris, Serie I, 1981, 293(30):589-592
[29]	Michael Neilan. A nonconforming Moreley finite element method for the fully nonlineair Monge-Ampere equation. Numer Math, 2010, 115:371-394
[30]	Newman E, Oliker V I. Differential-geometric methods in the design of reflector antennas. Sympos Math, 1992, 35:205-223
[31]	Newman, Pamela Cook L. A generalized Monge-Ampère equation arising in compressible flow. Contemporary Mathematics, 1999, 226:149-156
[32]	Oberman A M. Wide stencil finite difference schemes for elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discr Cont Dynam Sys B, 2008, 10(1):221-238
[33]	Oliker V. On the linearized Monge-Ampère equations related to the boundary value Minkowski problem and its generalizations//Gherardelli F, ed. Monge-Ampère Equations and Related Topics. Proceedings of a Seminar Help in Firenze, 1980. Roma, 1980:79-112
[34]	Oliker V I, Prussner L D. On the numerical solution of th equation zxzy-zxy2=f and its discretization. Numer Math, 1988, 54:271-293
[35]	Oliker V, Waltman P. Radilly symmetric solutions of a Monge-Ampère eqaution arising in a reflector mapping problem. Lecture Notes in Mathematics 1285. Springer, 1987
[36]	Polak E, Ribiè G. Note sur la convergence de directions conjuguées. Rev Francaise Infomat Recherche Operationnelle, 3e Année, 1969, 16:35-43
[37]	Polyak B T. The conjugate gradient method in extreme problems. USSR Comput Math Math Phys, 1969, 9:94-112
[38]	Wolfe P. Convergence conditions for ascent methods. SIAM Rev, 1969, 11:226-235
[39]	Zheligovsky V, Podvigina O, Frish U. The Monge-Ampère equation:Various forms and numerical solution. J Comput Phys, 2010, 229(13):5043-5061