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Concept

Calcul des variations

Le calcul des variations (ou calcul variationnel) est, en mathématiques et plus précisément en analyse fonctionnelle, un ensemble de méthodes permettant de minimiser une fonctionnelle. Celle-ci, qui est à valeurs réelles, dépend d'une fonction qui est l'inconnue du problème. Il s'agit donc d'un problème de minimisation dans un espace fonctionnel de dimension infinie. Le calcul des variations s'est développé depuis le milieu du jusqu'aujourd'hui ; son dernier avatar est la théorie de la commande optimale, datant de la fin des années 1950. Le calcul des variations a des applications dans de nombreux domaines :

L'inconnue étant une courbe paramétrée, on recherche une courbe de longueur minimale (ou extrémale), autrement dit une géodésique ; c'est une question fondamentale en géométrie différentielle ;

L'inconnue étant une surface, on recherche, pour un périmètre donné, la surface d'aire maximale (problème d'isopérimétrie) ;

En physique, le principe de moindre action affirme que le

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Séances de cours associées (112)

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Chargement

Multiscale algorithm with patches of finite elements and applications

Vittoria Rezzonico

We develop a discretisation and solution technique for elliptic problems whose solutions may present strong variations, singularities, boundary layers and oscillations in localised regions. We start with a coarse finite element discretisation with a mesh size H, and we superpose to it local patches of finite elements with finer mesh size h < H to capture local behaviours of the solution. The two meshes (coarse and patch) are not necessarily compatible. The algorithm used to compute the finite element solution on the coarse mesh and patch falls in the class of subspace correction methods [50, 48]. This technique has been introduced in [23]. Similarly to mesh adaptation methods, the location of the fine patches is identified by an a posteriori error estimator. Unlike mesh adaptation, no re-meshing is involved. We discuss the implementation and illustrate the method on an industrial problem. Moreover we generalise the algorithm to several patches which leads to further applications in multi-scale problems.

EPFL2007

Chargement

Regularity in metric spaces

Serguei Timochine

Using arguments developed by De Giorgi in the 1950's, it is possible to prove the regularity of the solutions to a vast class of variational problems in the Euclidean space. The main goal of the present thesis is to extend these results to the more abstract context of metric spaces with a measure. In particular, working in the axiomatic framework of Gol'dshtein – Troyanov, we establish both the interior and the boundary regularity of quasi-minimizers of the p-Dirichlet energy. Our proof works for quite general domains, assuming some natural hypotheses on the (axiomatic) D-structure. Furthermore, we prove analogous results for extremal functions lying in the class of Sobolev functions in the sense of Hajłasz – Koskela, i.e. functions characterized by the single condition that a Poincaré inequality be satisfied. Our strategy to prove these regularity results is first to show that, in a very general setting, the (Hölder) continuity of a function is a consequence of three specific technical hypotheses. This part of the argument is the essence of the De Giorgi method. Then, we verify that for a function u which is a quasi-minimizer in an axiomatic Sobolev space or an extremal Sobolev function in the sense of Hajłasz – Koskela, these technical hypotheses are indeed satisfied and u is thus (Hölder) continuous. In addition to that, we establish the Harnack's inequality for these extremal functions, and we show that the Dirichlet semi-norm of a piecewise-extremal function is equivalent to the sum of the Dirichlet semi-norms of its components.

EPFL2006

Chargement

Exploring stabilisation techniques for the reduced basis approximation of avection-diffusion PDEs

Christèle Marie Zbinden

In this thesis, we explore possible stabilisation methods for the reduce basis approximation of advection-diffusion problems, for which the advection term is dominating. The options we consider are mainly inspired by the Variational Multiscale method (VMS), which decomposes the solution of a variational problem into its coarse scale component, from a coarse scale space, and a fine scale component, from a fine scale space. Our stabilisation proposals are divided into three classes. The first one groups methods that rely on a stabilisation parameter. The second class uses VMS at the algebraic level to attempt stabilisation. Finally the third class is also inspired by VMS at the algebraic level, but with the additional constraint that the fine scale space is orthogonal to the coarse scale space. Numericals tests reported in this thesis show that the methods of the first class is not viable options as the best stabilisation parameter among those tested is the stabilisation parameter that is used at the high fidelity level. Although the stabilisation methods of the second class give accurate results when applied to stable problems, they were also dismissed by the numerical tests, as they did not improve the accuracy of the already stabilised problem. The third class also performs well when applied to stable problems. It has been shown in [7] one of those methods can improve accuracy. However in the current implementation, this result was not achieved here.

2016

Chargement

Joseph-Louis Lagrange

Joseph Louis de Lagrange (en italien Giuseppe Luigi Lagrangia ou aussi Giuseppe Ludovico De la Grange Tournier), né à Turin le de parents français descendants de Descartes et mort à Paris le , est un

Analyse (mathématiques)

L'analyse (du grec , délier, examiner en détail, résoudre) a pour point de départ la formulation rigoureuse du calcul infinitésimal. C'est la branche des mathématiques qui traite explicitement de la

Équations de Lagrange

vignette|Joseph-Louis Lagrange Les équations de Lagrange, découvertes en 1788 par le mathématicien Joseph-Louis Lagrange, sont une reformulation de la mécanique classique. Équations de première

PHYS-216: Mathematical methods for physicists

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