{C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

Precise solutions of the one-dimensional Monge-Kantorovich problem

Plakhov, A.Y., (2004) 'Precise solutions of the one-dimensional Monge-Kantorovich problem', Sbornik: Mathematics 195(9) pp.1291-1307 RAE2008

Recuerdo de viejos abogados : Robert, Löewenfeld, Gross y Monge Bernal.

Estudio sobre cuatro célebres abogados de mediados de siglo XX, Henri Robert, Guillermo Löewenfeld, Federico L. Gross y José Monge Bernal, a través de escritos sobre su vida que ellos mismos publican, en los que definen el ejercicio de la profesión y las moralinas de las experiencias de cada uno en el ámbito jurídico.

Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.

The Complex Monge-Ampère Equation

In questa trattazione si studia la regolarità delle soluzioni viscose plurisubarmoniche dell’equazione di Monge-Ampère complessa. Si tratta di un’equazione alle derivate parziali del secondo ordine completamente non lineare il cui termine del secondo ordine è il determinante della matrice hessiana complessa di una funzione incognita a valori reali u. Il principale risultato della tesi è un nuovo controesempio di tipo Pogorelov per questa equazione. Si prova cioè l’esistenza di soluzioni viscose plurisubarmoniche e non classiche per un equazione di Monge-Ampère complessa.

La misura di Monge-Ampère

In questo documento si costruisce la misura di Monge-Ampère partendo da una funzione continua convessa u a n variabili a valori reali. Si studiano le proprietà fondamentali di questa misura. Si enuncia la definizione di soluzioni generalizzate e soluzioni di viscosità dell'equazione di Monge-Ampère e si mostrano alcuni risultati importanti riguardo queste soluzioni. Utilizzando la nozione di misura di Monge-Ampère si dimostra il principio di massimo di Aleksandrov-Bakelman-Pucci.

Fitogeneica (Mejora de plantas), por Enrique Sánchez-Monge y colaboradores. Colección Agrícola Salvat. 1955.

Fil: González, F. F..

Initial design with L2 Monge-Kantorovich theory for the Monge–Ampère equation method of freeform optics

The Monge–Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L2 Monge-Kantorovich (LMK) theory, and introduce an efficient approach for finding the optimal mapping of the LMK problem. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.