Algèbre multilinéaire

En mathématiques, l’algèbre multilinéaire étend les méthodes de l’algèbre linéaire. Tout comme l’algèbre linéaire est bâtie sur le concept de vecteur et développe la théorie des espaces vectoriels, l’algèbre multilinéaire est bâtie sur le concept de tenseur et développe la théorie des espaces tensoriels. Dans les applications, de nombreux types de tenseurs surviennent. La théorie se veut exhaustive et comprend l'étude d'un certain nombre d'espaces et l'exposé de leurs relations. Histoire L'algèbre multilinéaire a des racines variées plongeant dans ce qui a été appelé au l’analyse tensorielle ou le « calcul tensoriel des champs tensoriels ». Elle s’est développée à partir de l’utilisation des tenseurs en géométrie différentielle, en relativité générale, et dans de nombreuses branches des mathématiques appliquées. Vers le milieu du , l’étude des tenseurs est reformulée plus abstraitement. Le traité d’algèbre multilinéaire du groupe Bourbaki (le chapitre 3 du livre d'Algè

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L'équation de rappel

Olivier Kneuss

In this thesis we are interested in the following problem : given two differential k–forms g and f, most of the time they will be assumed closed, on what conditions can we pullback g to f by a map φ ? In other words we ask when it is possible to solve the pullback equation φ*(g) = f. This equation originates from differential geometry. This work is divided into five main parts. In Part I we focus on exterior forms. The study of these forms, which is part of the domain of multilinear algebra, will allow us to understand the pullback equation at the algebraic level and will turn out to be a main first step for the study of the problem. Part II deals with the study of differential forms. A special attention will be brought to different versions of Poincaré lemma which will be constantly used all the way through this thesis. In Part III we solve the problem for volume forms k = n. This part, which will use very seldom the machinery of differential geometry, can be read independently of the previous two parts. Part IV sheds light on the case 0 ≤ k ≤ n – 1. Emphasis will be given on the case k = 2. The best results of this present work work will concern 2–forms and volume forms. The cases k = 0, 1, n – 1 are the easiest and will be discussed briefly while for the case 3 ≤ k ≤ n – 2, which will turn much more difficult even already at the algebraic level, we will only obtain results for k-forms with a special structure. Finally, in Part V, we gather numerous useful properties of Hödler spaces. These spaces will be very suitable for the pullback equation ; indeed there are an algebra (contrary to Sobolev spaces in general) and allow to get results with sharp regularity.

EPFL2011

Chargement

Non-smooth solutions in incompressible fluid dynamics

Luigi De Rosa

This work is devoted to the study of the main models which describe the motion of incompressible fluids, namely the Navier-Stokes, together with their hypodissipative version, and the Euler equations. We will mainly focus on the analysis of non-smooth weak solutions to those equations. Most of the results have been obtained by using the convex integration techniques introduced by Camillo De Lellis and László Székelyhidi in the context of the Euler equations, which recently led to the proof of the Onsager's conjecture on the anomalous dissipation of the kinetic energy. With various refinements of those iterative schemes we prove ill-posedness of Leray-Hopf weak solutions of the hypodissipative Navier-Stokes equations, sharpness of the kinetic energy regularity for Euler, typicality results in the sense of Baire's category for both Euler and Navier-Stokes, estimate on the dimension of the singular set in time of non-conservative Hölder weak solutions of the Euler equations. Moreover, building on different techniques, we also address some regularizing effects of those equations in various classes of weak solutions with some fractional differentiability in terms of Hölder, Sobolev and Besov regularity. The latter make use of new abstract interpolation results for multilinear operators which we developed for our specific context but which may also have independent interests.

EPFL2021

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Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

Fabio Nobile

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak’s algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.

2018

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MATH-110(a): Advanced linear algebra I

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux de ce sujet.

PHYS-216: Mathematical methods for physicists

This course complements the Analysis and Linear Algebra courses by providing further mathematical background and practice required for 3rd year physics courses, in particular electrodynamics and quantum mechanics.

MATH-111(en): Linear algebra (english)

The purpose of the course is to introduce the basic notions of linear algebra and its applications.

Algèbre extérieure

En mathématiques, et plus précisément en algèbre et en analyse vectorielle, l'algèbre extérieure d'un espace vectoriel E est une algèbre associative graduée, notée \Lambda E. La mul

Tenseur

En mathématiques, plus précisément en algèbre multilinéaire et en géométrie différentielle, un tenseur est un objet très général, dont la valeur s'exprime dans un espace vectoriel. On peut l'utiliser

Espace vectoriel

vignette|Dans un espace vectoriel, on peut additionner deux vecteurs. Par exemple, la somme du vecteur v (en bleu) et w (en rouge) est v + w. On peut aussi multiplier un vecteur, comme le vecteur w qu