Lorsque l'on évoque Darwin et la théorie de l'évolution, on ne pense pas aux mathématiques. Pourtant dès que l'on s'intéresse aux mécanismes de la sélection naturelle, au hasard de la reproduction et au rôle des mutations, il est indispensable de les utiliser.
Après une introduction historique aux idées de Darwin sur l'évolution des espèces, nous expliquons l'impact de sa théorie et de ses réflexions sur la communauté scientifique et l'influence qu'il a eue sur la modélisation mathématique des dynamiques de population ou de la génétique des populations. Nous développons quelques exemples d'objets mathématiques, tels les processus de branchement, qui permettent de prédire le futur d'une population (son extinction, sa diversité…) ou au contraire d'en connaître le passé biologique (l'ancêtre commun d'un groupe d'individus par exemple). L'introduction du hasard dans la modélisation des questions liées à la biodiversité et à l'évolution est fondamentale. Elle permet de prendre en compte les variabilités individuelles et de mieux comprendre l'impact des facteurs écologiques et génétiques sur l'évolution des espèces.
Ces idées seront illustrées par des exemples issus de travaux récents développés entre mathématiciens et biologistes.[-]

Tensor methods have emerged as an indispensable tool for the numerical solution of high-dimensional problems in computational science, and in particular problems arising in stochastic and parametric analyses. In many practical situations, the approximation of functions of multiple parameters (or random variables) is made computationally tractable by using low-rank tensor formats. Here, we present some results on rank-structured approximations and we discuss the connection between best approximation problems in tree-based low-rank formats and the problem of finding optimal low-dimensional subspaces for the projection of a tensor. Then, we present constructive algorithms that adopt a subspace point of view for the computation of sub-optimal low-rank approximations with respect to a given norm. These algorithms are based on the construction of sequences of suboptimal but nested subspaces.

Keywords: high dimensional problems - tensor numerical methods - projection-based model order reduction - low-rank tensor formats - greedy algorithms - proper generalized decomposition - uncertainty quantification - parametric equations[-]

Many problems in computational and data science require the approximation of high-dimensional functions. Examples of such problems can be found in physics, stochastic analysis, statistics, machine learning or uncertainty quantification. The approximation of high-dimensional functions requires the introduction of approximation tools that capture specific features of these functions.
In this lecture, we will give an introduction to tree tensor networks (TNs), or tree-based tensor formats. In part I, we will present some general notions about tensors, tensor ranks, tensor formats and tensorization of vectors and functions. Then in part II, we will introduce approximation tools based on TNs, present results on the approximation power (or expressivity) of TNs and discuss the role of tensorization and architecture of TNs. Finally in part III, we will present algorithms for computing with TNs.
This includes algorithms for tensor truncation, for the solution of optimization problems, for learning functions from samples...[-]

The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical approximation of the corrector function (and therefore of the homogenized coefficient) is a challenging computational task.
In practice, the corrector problem is solved on a truncated domain, and the exact homogenized coefficient is recovered only in the limit of infinitely large domains. As a consequence of this truncation, the approximated homogenized coefficient turns out to be stochastic, even though the exact homogenized coefficient is deterministic. One then has to resort to Monte-Carlo methods, in order to compute the expectation of the (approximated, apparent) homogenized coefficient within a good accuracy. Variance reduction questions thus naturally come into play, in order to increase the accuracy (e.g. reduce the size of the confidence interval) for a fixed computational cost. In this talk, we will present some variance reduction approaches to address this question.[-]

Many problems in computational and data science require the approximation of high-dimensional functions. Examples of such problems can be found in physics, stochastic analysis, statistics, machine learning or uncertainty quantification. The approximation of high-dimensional functions requires the introduction of approximation tools that capture specific features of these functions.
In this lecture, we will give an introduction to tree tensor networks (TNs), or tree-based tensor formats. In part I, we will present some general notions about tensors, tensor ranks, tensor formats and tensorization of vectors and functions. Then in part II, we will introduce approximation tools based on TNs, present results on the approximation power (or expressivity) of TNs and discuss the role of tensorization and architecture of TNs. Finally in part III, we will present algorithms for computing with TNs. This includes algorithms for tensor truncation, for the solution of optimization problems, for learning functions from samples...[-]