Comptes Rendus
Numerical Analysis/Partial Differential Equations
Numerical solution of the Monge–Ampère equation by a Newton's algorithm
Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 319-324.

We solve numerically the Monge–Ampère equation with periodic boundary condition using a Newton's algorithm. We prove convergence of the algorithm, and present some numerical examples, for which a good approximation is obtained in 10 iterations.

Nous résolvons numériquement l'équation de Monge–Ampère avec donnée au bord périodique en utilisant un algorithme de Newton. Nous prouvons la convergence de l'algorithme, et présentons quelques exemples numériques, pour lesquels une bonne approximation de la solution est obtenue en 10 itérations.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.12.018
Grégoire Loeper 1; Francesca Rapetti 2

1 Département de mathématiques, École polytechnique fédérale de Lausanne, CH-1015 Lausanne, Switzerland
2 Laboratoire J.-A. Dieudonné, CNRS & université de Nice et Sophia-Antipolis, parc Valrose, 06108 Nice cedex 02, France
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     author = {Gr\'egoire Loeper and Francesca Rapetti},
     title = {Numerical solution of the {Monge{\textendash}Amp\`ere} equation by a {Newton's} algorithm},
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Grégoire Loeper; Francesca Rapetti. Numerical solution of the Monge–Ampère equation by a Newton's algorithm. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 319-324. doi : 10.1016/j.crma.2004.12.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.018/

[1] J.-D. Benamou; Y. Brenier A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., Volume 84 (2000) no. 3, pp. 375-393

[2] Y. Brenier; U. Frisch; M. Henon; G. Loeper; S. Matarrese; R. Mohayaee; A. Sobolevskii˘ Reconstruction of the early Universe as a convex optimization problem, Mon. Not. R. Astron. Soc., Volume 346 (2003) no. 2, pp. 501-524

[3] L. Caffarelli Interior W2,p estimates for solutions of Monge–Ampère equation, Ann. Math. (2), Volume 131 (1990) no. 1, pp. 135-150

[4] L. Caffarelli; X. Cabre Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Coll. Publ., vol. 43, American Mathematical Society, Providence, RI, 1995

[5] E. Dean; R. Glowinski 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, Volume 336 (2003) no. 9, pp. 779-784

[6] E. Dean; R. Glowinski Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least square approach, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004) no. 12, pp. 887-892

[7] D. Gilbarg; N. Trudinger Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. [Fund. Princ. Math. Sci.], vol. 224, Springer-Verlag, Berlin, 1983

[8] V.I. Ollicker; L.D. Prussner On the numerical solution of the equation zxxzyyzxy2=f and its discretization. I, Numer. Math., Volume 54 (1988), pp. 271-293

[9] Q. Quarteroni; A. Valli Numerical Approximation of Partial Differential Equations, Comput. Math., vol. 23, Springer-Verlag, Berlin, 1994

[10] C. Villani Topics in Optimal Transportation, Graduate Ser. in Math., American Mathematical Society, 2003

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