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.
Accepted:
Published online:
Grégoire Loeper 1; Francesca Rapetti 2
@article{CRMATH_2005__340_4_319_0, author = {Gr\'egoire Loeper and Francesca Rapetti}, title = {Numerical solution of the {Monge{\textendash}Amp\`ere} equation by a {Newton's} algorithm}, journal = {Comptes Rendus. Math\'ematique}, pages = {319--324}, publisher = {Elsevier}, volume = {340}, number = {4}, year = {2005}, doi = {10.1016/j.crma.2004.12.018}, language = {en}, }
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] A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., Volume 84 (2000) no. 3, pp. 375-393
[2] Reconstruction of the early Universe as a convex optimization problem, Mon. Not. R. Astron. Soc., Volume 346 (2003) no. 2, pp. 501-524
[3] Interior estimates for solutions of Monge–Ampère equation, Ann. Math. (2), Volume 131 (1990) no. 1, pp. 135-150
[4] Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Coll. Publ., vol. 43, American Mathematical Society, Providence, RI, 1995
[5] 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] 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] Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. [Fund. Princ. Math. Sci.], vol. 224, Springer-Verlag, Berlin, 1983
[8] On the numerical solution of the equation and its discretization. I, Numer. Math., Volume 54 (1988), pp. 271-293
[9] Numerical Approximation of Partial Differential Equations, Comput. Math., vol. 23, Springer-Verlag, Berlin, 1994
[10] Topics in Optimal Transportation, Graduate Ser. in Math., American Mathematical Society, 2003
Cited by Sources:
Comments - Policy