{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.

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.

Physiography of the Ampère Seamount in the Horseshoe Seamount chain off Gibraltar

The Ampère Seamount, 600 km west of Gibraltar, is one of nine inactive volcanoes along a bent chain, the so called Horseshoe Seamounts. All of them ascend from an abyssal plain of 4000 to 4800 m depth up to a few hundred meters below the sea surface, except two, which nearly reach the surface: the Ampère massif on the southern flank of the group and the summit of the Gorringe bank in the north. The horseshoe, serrated like a crown, opens towards Gibraltar and stands in the way of its outflow. These seamounts are part of the Azores-Gibraltar structure, which marks the boundary between two major tectonic plates: the Eurasian and the African plate. The submarine volcanism which formed the Horseshoe Seamounts belongs to the sea floor spread area of the Mid-Atlantic Ridge. The maximum activity was between 17 and 10 Million years ago and terminated thereafter. The volcanoes consist of basalts and tuffs. Most of their flanks and the abyssal plain around are covered by sediments of micro-organic origin. These sediments, in particular their partial absence on the upper flanks are a circumstantial proof and a kind of diary of the initial rise and subsequent subsidence of about 6oo m of these seamounts. The horizons of erosion where the basalt substrate is laid bare indicate the rise above sea level in the past. Since the Ampère summit is 60 m deep today, this volcano must have been an island 500 m high. The stratification of the sediments covering the surrounding abyssal plain reveals discrete events of downslope suspension flows, called turbidites, separated by tens of thousands of years and perhaps induced by changes in climate conditions. The Ampère sea mount of 4800 m height and a base diameter of 50 km exceeds the size of the Mont Blanc massif. Its southern and eastern flanks are steep with basalts cropping out, in parts with nearly vertical walls of some hundred meters. The west and north sides consist of terraces and plateaus covered with sediments at 140 m, 400 m, 2000 m, and 3500 m. The Horseshoe Seamount area is also remarkable as a kind of disturbed crossing of three major oceanic flow systems at different depths and directions with forced upwelling and partial mixing of the water masses. Most prominent is the Mediterranean Outflow Water (MOW) with its higher temperature and salinity between 900 to 1500 m depth. It enters the horseshoe unimpaired from the open eastern side but penetrates the seamount chain through its valleys on the west, thereafter diverging and crossing the entire Atlantic Ocean. Below the MOW is the North Atlantic Deep Water (NADW) between 2000 m to 3000 m depth flowing southward and finally there is the Antarctic Bottom Water (AABW) flowing northward below the two other systems.

The Monge-Ampére equation method in freeform optics design

The Monge-Ampére equation method could be the most advanced point source algorithm of freeform optics design. This paper introduces this method, and outlines two key issues that should be tackles to improve this method.

Exposé des nouvelles découvertes sur l'électricité et le magneétisme, de mm. Œrsted, Arago, Ampère, H. Davy, Biot, Erman, Schweiger, De La Rive, etc.;

Mode of access: Internet.