The Monge-Ampère equation: various forms and numerical solution. (English) Zbl 1194.65141

Summary: We present three novel forms of the Monge-Ampère equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge-Ampère problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge-Ampère equation is a sum of such distributions.


65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J96 Monge-Ampère equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text: DOI arXiv


[1] S. Haker, A. Tannenbaum, R. Kikinis. Mass preserving mappings and image registration, in: Proceedings of the Fourth International Conference on Medical Image Computing and Computer-Assisted Intervention, Lectures Notes in Computer Science, vol. 2208, Springer-Verlag, London, 2001, pp. 120-127. · Zbl 1041.68621
[2] Haker, S.; Zhu, L.; Tannenbaum, A.; Angement, S., Optimal mass transport for registration and warping, Int. J. comput. vis., 60, 225-240, (2004)
[3] Hurtut, T.; Gousseau, Y.; Schmitt, F., Adaptive image retrieval based on the spatial organization of colors, Comput. vis. image understand., 112, 101-113, (2008)
[4] Frisch, U.; Matarrese, S.; Mohayaee, R.; Sobolevski, A., A reconstruction of the initial conditions of the universe by optimal mass transportation, Nature, 417, 260-262, (2002), arXiv:astro-ph/0109483
[5] Brenier, Y.; Frisch, U.; Hénon, M.; Loeper, G.; Matarrese, S.; Mohayaee, R.; Sobolevskii, A., Reconstruction of the early universe as a convex optimization problem, Mon. not. R. astron. soc., 346, 501-524, (2003), arXiv:astro-ph/0304214
[6] Mohayaee, R.; Sobolevskii, A., The monge – ampère – kantorovich approach to reconstruction in cosmology, Physica D, 237, 2145-2150, (2008), arXiv:0712.2561 · Zbl 1143.76606
[7] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag Berlin · Zbl 0691.35001
[8] Bakelman, I.J., Convex analysis and nonlinear geometric elliptic equations, (1994), Springer-Verlag · Zbl 0721.35017
[9] L.A. Caffarelli, X. Cabré, Fully nonlinear elliptic equations, Amer. Math. Soc., Providence Rhode Island, vol. 43, American Mathematical Society Colloquium Publications, 1995.
[10] Oliker, V.I.; Prussner, L.D., On the numerical solution of the equation \(\frac{\partial^2 z}{\partial x^2} \frac{\partial^2 z}{\partial y^2} - \left(\frac{\partial^2 z}{\partial x \partial y}\right)^2 = f\) and its discretizations, I. numer. math., 54, 271-293, (1988) · Zbl 0659.65116
[11] D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, A. Bräuer, Incoherent beam shaping with freeform mirror. Nonimaging optics and efficient illumination systems V, in: R. Winston, R.J. Koshel (Eds.), Proceedings of the SPIE, vol. 7059, 2008, p. 705905.
[12] Glimm, T.; Oliker, V., Optical design of single reflector systems and the monge – kantorovich mass transfer problem, J. math. sci., 117, 4096-4108, (2003)
[13] A.V. Pogorelov, The Minkowski Multidimensional Problem, Halsted Press, Washington, DC, 1978 (Translation from Russian: A.V. Pogorelov, The Minkowski Multidimensional Problem, Nauka, Moscow, 1975.)
[14] Benamou, J.-D.; Brenier, Y., A computational fluid mechanics solution to the monge – kantorovich mass transfer problem, Numer. math., 84, 375-393, (2000) · Zbl 0968.76069
[15] 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, 336, 779-784, (2003) · Zbl 1028.65120
[16] 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, 339, 887-892, (2004) · Zbl 1063.65121
[17] X. Feng, M. Neilan, Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation (<arXiv:0712.1240>).
[18] X. Feng, M. Neilan, Mixed Finite Element Methods for the Fully Nonlinear Monge-Ampère Equation Based on the Vanishing Moment Method (<arXiv:0712.1241>). · Zbl 1195.65170
[19] Loeper, G.; Rapetti, F., Numerical solution of the monge – ampère equation by a newton’s algorithm, C.R. acad. sci. Paris, ser. I, 340, 319-324, (2005) · Zbl 1067.65119
[20] J.-D. Benamou, B.D. Froese, A.M. Oberman, Two Numerical Methods for the Elliptic Monge-Ampère Equation, Preprint, 2009. <www.divbyzero.ca/froese/w/images/4/40/MA.pdf>. · Zbl 1192.65138
[21] 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., 227, 9841-9864, (2008) · Zbl 1155.65394
[22] Finn, J.M.; Delzanno, G.L.; Chacon, L., Grid generation and adaptation by monge – kantorovich optimization in two and three dimensions, Proc. 17th int. mesh. roundtable, 551-568, (2008)
[23] Gutiérrez, C.E., The Monge-Ampère equation. progress in nonlinear differential equations and their applications, (2001), Birkhäuser Boston, vol. 44
[24] Ampère, A.-M., Mémoire concernant …l’intégration des équations aux différentielles partielles du premier et du second ordre, J. de L’école royale polytech., 11, 1-188, (1820)
[25] Dean, E.J.; Glowinski, R., Numerical methods for fully nonlinear elliptic equations of the monge – ampère type, Comput. methods appl. mech. eng., 195, 1344-1386, (2006) · Zbl 1119.65116
[26] Kostrikin, A.I., Introduction to algebra, (1977), Nauka Moscow, (in Russian) · Zbl 0464.00007
[27] Peebles, P.J.E., Tracing galaxy orbits back in time, Astrophys. J., 344, L53-L56, (1989)
[28] Zel’dovich, Ya.B., Gravitational instability: an approximate theory for large density perturbations, Astron. astrophys., 5, 84-89, (1970)
[29] Moutarde, F.; Alimi, J.-M.; Bouchet, F.R.; Pellat, R.; Ramani, A., Precollapse scale invariance in gravitational instability, Astrophys. J., 382, 377-381, (1991)
[30] Brenier, Y., Décomposition polaire et réarrangement monotone des champs de vecteur, C.R. acad. sci. Paris, ser. I, 305, 805-808, (1987) · Zbl 0652.26017
[31] Axelsson, O., Iterative solution methods, (1996), Cambridge University Press
[32] Podvigina, O.M.; Zheligovsky, V.A., An optimized iterative method for numerical solution of large systems of equations based on the extremal property of zeroes of Chebyshev polynomials, J. sci. comput., 12, 433-464, (1997) · Zbl 0908.65036
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