Géométrie non euclidienne | EPFL Graph Search

La géométrie non euclidienne (GNE) est, en mathématiques, une théorie géométrique ayant recours aux axiomes et postulats posés par Euclide dans les Éléments, sauf le postulat des parallèles. Les différentes géométries non euclidiennes sont issues initialement de la volonté de démontrer la proposition du cinquième postulat, qui apparaissait peu satisfaisant en tant que postulat car trop complexe et peut-être redondant avec les autres postulats). Préambule Dans les Éléments d'Euclide, l'axiome des parallèles ressemble à la conclusion d'un théorème, mais qui ne comporterait pas de démonstration : qu'on peut comprendre comme : Durant plusieurs siècles, la géométrie euclidienne a été utilisée sans que l'on mette en doute sa validité. Elle a même été longtemps considérée comme l'archétype du raisonnement logico-déductif. Elle présentait en effet l'avantage de définir les propriétés intuitives des objets géométriques dans une construction mathématique rigoureuse. Approc

Finite Element Method on Riemann Surfaces and Applications to the Laplacian Spectrum

Marc Maintrot

The objective of this PhD thesis is the approximate computation of the solutions of the Spectral Problem associated with the Laplace operator on a compact Riemann surface without boundaries. A Riemann surface can be seen as a gluing of portions of the Hyperbolic Plane made with suitable conditions to obtain a 2 dimensional manifold. The solutions of the Spectral Problem associated with the Laplace operator are to be understood as the eigenfunctions defined on the surface and their corresponding eigenvalues. This work is separated into two parts: the first part describes the method used to approximate the eigenvalues and eigenfunctions, the second focuses on the design of a program to compute these approximations. The approximation method is inspired by the Finite Element Method (FEM), in that it relies on the variational expression of the Spectral Problem and the definition of a finite subspace of functions in which the approximated eigenvalues and eigenfunctions are computed. However, it differs from the FEM in that it removes the euclidian basis of the FEM and is invariant under the isometries of the Hyperbolic Plane. To ful fill this objective, we begin by geodesically triangulating the surface as regularly as possible. This hyperbolic triangulation allows us to de ne the finite subspace of functions by using the concept of barycentric coordinates associated with each vertex of the triangulation (idea introduced by Whitney and taken up by Dodziuk). We then prove that the approximated solutions convergence to the exact ones when the diameter of the triangulation decreases, as well as the order of convergence. The program is a practical application of the theoretical work and allows the computation of the approximated eigenfunctions and eigenvalues.

EPFL2012

An Accelerated First-Order Method for Non-convex Optimization on Manifolds

Nicolas Boumal, Christopher Arnold Criscitiello

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

SPRINGER2022

Images des microcosmes flottants

Olivier Nicolas Meystre

This research focuses on the premise that depiction is quintessential to the conception of an architectural project. The representation also provides insights regarding how the architects design the architecture. This thesis is intended towards understanding new forms and paradigms of design mediation that have emerged over the last two decades in Japan. It unravels and highlights the historical, theoretical aspects and analyses the evolution, convergence and major trends in the field of architectural representation. Based on bibliographical sources and exclusive, unpublished interviews with the prominent Japanese architects of the period under consideration, this work presents six cross-functional themes with the objective to identify and examine the critical tools that could be used in the conception and representation of the contemporary Japanese architecture. The understanding of the underlying techniques reveal a very hybridised landscape of digital and analog tools with a paradoxical objective of imparting a manual effect to the entire expression. The treatment of light in the images of the project is characterised by overexposure and dispersion that adds an elusory, unseizable dimension to the objects and spatial limits. In a similar fashion, the graphical limits, are either diminished or diluted giving an impression of a space with evolving outlines. The pseudo-naive expression of multiple images, achieved by the profusion of familiar objects, obliterates the orthodox hierarchy among the architectural elements and opens up the possibility of evoking and imagining tiny, soothing universes in an architecture which is otherwise radical to the beholder. The point of view used in images transcend the Euclidean realism, producing an imagery that seems to float in free space. Finally, the genealogy of relationship between masters and students permits the tracing of the lines of influence, how the craftsmanship changed hands and percolated down the professional world. The specificities in the Japanese representation are a testimony to the complex approach of the architects towards an architectural project, where all the attributes of space are questioned and redefined but still the undercurrent of tradition continues to have a pivotal influence.

EPFL2015

Images des microcosmes flottants

Olivier Nicolas Meystre

EPFL2015