Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the $\Gamma$-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way)

How mineralogy and geochemistry can improve the significance of Pb isotopes in metal provenance studies

Lead isotopes combined with trace element data represent a powerful tool for non-ferrous metal provenance studies. Nevertheless, unconsidered geological factors and archaeological data, as well as ignored analytical procedures, may substantially modify the interpretation of the isotopic and trace element signature obtained as a potential ore candidate. Three archaeological examples, accompanied by high-resolution lead isotopic measurements (MC–ICP–MS), are presented here to discuss the above-mentioned criticisms and to propose some solutions. The first example deals with prehistoric/historical gold/silver-mining activity from Romania (the Baia Borşa and Roşia Montană ore deposits). The second one regards the lead/silver metallurgical activity from the Mont-Lozère massif (France) during medieval times. The third example focuses on the comparison between two batches of lead isotope data gathered on Roman lead ingots from Saintes-Maries-de-la-Mer, using different SRM 981 Pb values

Oligocene and Miocene continental sedimentation, tectonics, and S-type magmatism in the southeastern Andes of Peru (Crucero basin) : geodynamic implications

Dans le bassin de Crucero, la formation Cayconi est constituée par des dépôts d'origine continentale; des roches volcaniques basiques et acides y sont intercalées. A partir des données de terrain et des analyses isotopiques, les auteurs montrent que les sédiments et les produits volcaniques sont d'âge OLigocène-Miocène. Les données sédimentologiques, structurales et pétrologiques permettent d'illustrer l'évolution géodynamique du bassin et d'avancer des hypothèses concernant l'association, au niveau de la formation Cayconi, de laves acides et basique

A Projection Approach to the Numerical Analysis of Limit Load Problems

International audienceThis paper proposes a numerical scheme to approximate the solution of (vectorial) limit load problems. The method makes use of a strictly convex perturbation of the problem, which corresponds to a projection of the deformation field under bounded deformation and incompressibility constraints. The discretized formulation of this perturbation converges to the solution of the original landslide problem when the amplitude of the perturbation tends to zero. The projection is computed numerically with a multi-steps gradient descent on the dual formation of the problem

The lithosphere of Southern Peru: A result of the accretion of allochthonous blocks during the Mesoproterozoic

Southern Peru exhibits different juxtaposed structural blocks. These blocks have a distinct sedimentary, tectonic and magmatic evolution. They are bounded by complex, mainly NW-SE fault systems, locally marked by Cenozoic and Mesozoic magmatic units. The specific Mesozoic and Cenozoic geologic evolution of each structural block is ascribed to the high heterogeneity of the southern Peruvian depth lithosphere. This lithosphere results from the accretion of different lithospheric blocks during Laurentia-Amazonia collision at around 1000 Ma

A Γ\Gamma-Convergence Result for the Upper Bound Limit Analysis of Plates

Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have proposed to use various finite elements discretizations. We provide in this paper a mathematical analysis which ensures the convergence of the finite element method, even with finite elements with discontinuous derivatives such as the quadratic 6 node Lagrange triangles and the cubic Hermite triangles. More precisely, we prove the $\Gamma$-convergence of the discretized problems towards the continuous limit analysis problem. Numerical results illustrate the relevance of this analysis for the yield design of both homogeneous and non-homogeneous materials

Paleogeographie, structural and magmatic evidences for the existence of different lithospheric blocks in the Central Andes: samples from southern Peru and northern Chile

The Andes are classically considered to represent the type of orogenie chains resulting from the subduction of an oceanic plate beneath a continental plate. In spite of a considerable literature invoJving stratigraphie, paleogeographie, structural, magmatic and geophysical data, any model that convincingly ex plains the modality of the Andean surrection has been emerged. Ali the current models focused on the importance of the subducting plate and very few considerations on the heterogeneity of continental plate. Paleogeographie reconstructions, structural evidences and magmatic and metallogenic evolution of Southern Peruvian Western Cordillera and Northern Chi.lean Domeyko Cordillera are used to identify a structure that delineates two different lithospheric blocks