Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation

Journal of Computational Physics

Volumn 230, Issue 3, 2011, Pages 818-834

  Froese, B D ^a   Oberman, A M ^a  

^a SIMON FRASER UNIVERSITY   (Canada)

Author keywords

Convexity constraints; Fully nonlinear elliptic Partial Differential Equations; Monge Amp re equations; Monotone schemes; Nonlinear finite difference methods; Viscosity solutions

Indexed keywords

COMPUTATION THEORY; FINITE DIFFERENCE METHOD; IMAGE REGISTRATION; NONLINEAR EQUATIONS; OPTICAL DATA PROCESSING; PARTIAL DIFFERENTIAL EQUATIONS;

CONVEXITY CONSTRAINT; ELLIPTIC PARTIAL DIFFERENTIAL EQUATION; FINITE-DIFFERENCE METHODS; FULLY NONLINEAR; FULLY NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION; MONGE-AMPÈRE EQUATION; MONOTONE SCHEME; NONLINEAR FINITE DIFFERENCE; NONLINEAR FINITE DIFFERENCE METHOD; VISCOSITY SOLUTIONS;

NEWTON-RAPHSON METHOD;

EID: 78649322125     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.10.020     Document Type: Article

References (44)

1. Evans L.C. Partial differential equations and Monge-Kantorovich mass transfer. Current Developments in Mathematics 1997, Int. Press, Boston, MA, Cambridge, MA.
2. Bakelman I. Convex Analysis and Nonlinear Geometric Elliptic Equations 1994, Springer-Verlag.
3. A.V. Pogorelov, Monge-Ampère equations of elliptic type, Translated from the first Russian edition by F. Leo Boron with the assistance of Albert L. Rabenstein, C. Richard, and P. Bollinger, Noordhoff Ltd., Groningen, 1964.
4. 2=f and its discretizations I. Numer. Math. 1988, 54:271-293.
5. Haltiner G.J. Numerical Weather Prediction 1971, Wiley, New York.
6. Kasahara A. Significance of non-elliptic regions in balanced flows of the tropical atmosphere. Mon. Weather Rev. 1982, 110.
7. Stoker J.J. Nonlinear Elasticity 1968, Gordon and Breach Science Publishers, New York.
8. Westcott B.S. Shaped Reflector Antenna Design 1983, Research Studies Press, New York.
9. Stojanovic S.D. Risk premium and fair option prices under stochastic volatility: the hara solution. C. R. Math. 2005, 340:551-556.
10. Ambrosio L. Lecture notes on optimal transport problems. Lecture Notes in Math 2003, 1-52. Springer, Berlin, vol. 1812.
11. Villani C. Topics in optimal transportation. Graduate Studies in Mathematics 2003, 58. American Mathematical Society, Providence, RI.
12. Haker S., Tannenbaum A., Kikinis R. Mass preserving mappings and image registration. MICCAI '01: Proceedings of the 4th International Conference on Medical Image Computing and Computer-Assisted Intervention 2001, Springer-Verlag, London, UK.
13. Haker S., Zhu L., Tannenbaum A., Angenent S. Optimal mass transport for registration and warping. Int. J. Comput. Vision 2004, 60:225-240.
14. ur Rehman T., Haber E., Pryor G., Melonakos J., Tannenbaum A. 3D nonrigid registration via optimal mass transport on the GPU. Med. Image. Anal. 2009, 13:931-940.
15. Delzanno G.L., Chacón L., Finn J.M., Chung Y., Lapenta G. An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization. J. Comput. Phys. 2008, 227:9841-9864.
16. J.M. Finn, G.L. Delzanno, L. Chacón, Grid generation and adaptation by Monge-Kantorovich optimization in two and three-dimensions, in: Proceedings of the 17th International Meshing Roundtable, pp. 551-568.
17. Budd C.J., Williams J.F. Moving mesh generation using the parabolic Monge-Ampère equation. SIAM J. Sci. Comput. 2009, 31:3438-3465.
18. Glimm T., Oliker V. Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem. J. Math. Sci. (N.Y.) 2003, 117:4096-4108. Nonlinear problems and function theory.
19. Frisch U., Matarrese S., Mohayaee R., Sobolevski A. A reconstruction of the initial conditions of the universe by optimal mass transportation. Nature 2002, 417.
20. E. Haber, R. Horesh, J. Modersitski, Numerical optimization for constrained image registration, 2010, preprint.
21. Cohen-Or D. Space deformations, surface deformations and the opportunities in-between. J. Comput. Sci. Technol. 2009, 24:2-5.
22. Benamou J.-D., Brenier Y. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 2000, 84:375-393.
23. Dean E.J., Glowinski R. On the numerical solution of the elliptic Monge-Ampère equation in dimension two: a least-squares approach. Computational Methods in Applied Science 2008, vol. 16:43-63. Springer, Dordrecht.
24. Dean E.J., Glowinski R. An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two-dimensions. Electron. Trans. Numer. Anal. 2006, 22:71-96. electronic.
25. R. Glowinski, Numerical methods for fully nonlinear elliptic equations, in: R. Jeltsch, G. Wanner (Eds.), 6th International Congress on Industrial and Applied Mathermatics, ICIAM 07, Invited Lectures, pp. 155-192.
26. Feng X., Neilan M. Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 2009, 47:1226-1250.
27. Feng X., Neilan M. Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 2009, 38:74-98.
28. Loeper G., Rapetti F. Numerical solution of the Monge-Ampére equation by a Newton's algorithm. C.R. Math. Acad. Sci. Paris 2005, 340:319-324.
29. Benamou J.-D., Froese B.D., Oberman A.M. Two numerical methods for the elliptic Monge-Ampère equation. ESAIM: Math. Model. Numer. Anal. 2010, 44.
30. Zheligovsky V., Podvigina O., Frisch U. The Monge-Ampère equation: Various forms and numerical solution. J. Comput. Phys. 2010, 229:5043-5061.
31. Oberman A.M. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete Cont. Dyn. Syst. Ser. B 2008, 10:221-238.
32. Froese B.D., Oberman A.M. Convergent finite difference solvers for viscosity solutions of the Monge-Ampère equation in dimensions two and higher 2010, .
33. Caffarelli L., Nirenberg L., Spruck J. The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Commun. Pure Appl. Math. 1984, 37:369-402.
34. Urbas J.I.E. The generalized Dirichlet problem for equations of Monge-Ampère type. Ann. Inst. H. Poincaré Anal. Non Linéaire 1986, 3:209-228.
35. 2,p estimates for solutions of the Monge-Ampère equation. Ann. Math. 1990, 131(2):135-150.
36. Gutiérrez C.E. The Monge-Ampère equation. Progress in Nonlinear Differential Equations and their Applications 2001, vol. 44. Birkhäuser Boston Inc., Boston, MA.
37. A.V. Pogorelov, The Minkowski multidimensional problem, in: V.H. Winston & Sons, Washington, D.C., Translated from the Russian by Vladimir Oliker, Introduction by Louis Nirenberg, Scripta Series in Mathematics, 1978.
38. Oberman A.M. Computing the convex envelope using a nonlinear partial differential equation. Math. Models Methods Appl. Sci. 2008, 18:759-780.
39. A.M. Oberman, L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., in press (). http://arxiv.org/abs/1007.0773.
40. Pogorelov A.V. The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation. Dokl. Akad. Nauk SSSR 1971, 201:790-793.
41. Crandall M.G., Ishii H., Lions P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 1992, 27:1-67.
42. Strang G. Linear Algebra and its Applications 1980, Academic Press [Harcourt Brace Jovanovich Publishers], New York. second ed.
43. Oberman A.M. Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 2006, 44:879-895. electronic.
44. D.P. Bertsekas, Convex analysis and optimization, Athena Scientific, Belmont, MA, 2003. With Angelia Nedić and Asuman E. Ozdaglar.