[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 |