A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

Abstract

Abstract. We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge−Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson−Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.Résumé. Nousétudions, dans cet article, une méthode numérique, pour le calcul des solutions convexes du problème de Dirichlet pour l'équation de Monge−Ampère elliptique, dans des domaines bi-dimensionnel convexes. Une méthode de moindres carrés est coupléeà un algorithme de relaxation, conduisantà la résolution d'une suite de problèmes de Poisson−Dirichlet, et d'une suite de problèmes de valeurs propres de petite dimension d'un type nouveau. Une approximation paréléments finis mixtes, coupléeà une méthode de régularisation, est utilisée pour implémenter la méthode de moindres-carrés/relaxation ci-dessus, de sorte que les domaines avec frontière courbe sont traités facilement. Des expériences numériques montrent la convergence des solutions calculées vers la solution convexe du problème continu, lorsqu'une telle solution existe. Par ailleurs, si le problème n'a pas de solution classique, notre méthodologie fournit des solutions au sens des moindres carrés.Mathematics Subject Classification. 65N30, 65K10, 65F30, 49M15, 49K20.

A Least-Squares Method for Optimal Transport Using the Monge--Ampère Equation

“…The minimization over φ with fixed P comes down to solving a biharmonic equation, which is discretized using mixed finite elements. The function φ converges to the convex solution u of (1.5) [5]. Our algorithm differs from the algorithm of Caboussat et al in three ways.…”

“…From this mapping we can also calculate the convex solution of the corresponding Monge-Ampère equation. Our new method is inspired by a least-squares method published recently by Caboussat et al [5]. Their method numerically solves the Dirichlet problem of the elliptic MongeAmpère equation, given by…”

“…Let P : Q denote the Fröbenius inner product of the matrices P and Q, defined by 5) then the Fröbenius norm is defined as ||P || = √ P : P . We minimize this functional over m and P , where m comes from the set 6) which is the set of two-dimensional, twice continuously differentiable vector fields.…”

On Standard Finite Difference Discretizations of the Elliptic Monge–Ampère Equation

“…Finite element discretizations have also been proposed, e.g. [25,8,21,9,31,15,10]. Since we use standard discretizations, the efficient tools developed for computational mathematics such as adaptive mesh refinements and multigrid algorithms can be transferred seamlessly to the Monge-Ampère context.…”

Isogeometric Method for the Elliptic Monge-Ampère Equation

“…The analysis of numerical methods for the Monge-Ampère equation is an active research area. The references [5,11,24,19,9,22,15,39,12,30,37,17,14,21,13,34] cover most of the various approaches.…”