Résumé : This paper is focused on the statistical analysis of probability measures $\bnu_1,\ldots,\bnu_n$ on $\R$ that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures $\bnu_i$ are absolutely continuous with densities $\bfun_i$ that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables $(X_i,j)_1 \leq i \leq n ; \ ; 1 \leq j \leq p_i $ organized in the form of $n$ experimental units, such that $X_i,1,\ldots,X_i,p_i$ are iid observations sampled from a random measure $\bnu_i$ for each $1 \leq i \leq n$. In this setting, we focus on first-order statistics methods for estimating, from such data, a meaningful structural mean measure. For the purpose of taking into account phase and amplitude variations in the observations, we argue that the notion of Wasserstein barycenter is a relevant tool. The main contribution of this paper is to characterize the rate of convergence of a (possibly smoothed) empirical Wasserstein barycenter towards its population counterpart in the asymptotic setting where both $n$ and $\min_1 \leq i \leq n p_i$ may go to infinity. The optimality of this procedure is discussed from the minimax point of view with respect to the Wasserstein metric. We also highlight the connection between our approach and the curve registration problem in statistics. Some numerical experiments are used to illustrate the results of the paper on the convergence rate of empirical Wasserstein barycenters.—
Lieu : Salle 106 Bat 1R1
Résumé : We are interested in the numerical solution of hyperbolic conservation laws on the most local
compact stencil consisting of only nearest neighbors. In the Finite Volume setting, in order
to obtain higher order methods, the main challenge is the reconstruction of the interface
values. These are crucial for the de-nition of the numerical
ux functions, also referred to
as the Riemann solver of the scheme.
Often, the functions of interest contain smooth parts as well as discontinuities. Treating
such functions with high-order schemes may lead to undesired oscillations. However, what
is required is a solution with sharp discontinuities while maintaining high-order accuracy
in smooth regions. One possible way of achieving this is the use of limiter functions in the
MUSCL framework which switch the reconstruction to lower order when necessary. Another
possibility is the third-order variant of the WENO family, called WENO3.
In this work, we will recast both methods in the same framework to demonstrate the relation
between Finite Volume limiter functions and the way WENO3 performs limiting. We present
a new limiter function, which contains a decision criterion that is able to distinguish between
discontinuities and smooth extrema. Our newly-developed limiter function does not require
an arti-cial parameter, instead, it uses only information of the initial condition.
We compare our insights with the formulation of the weight-functions in WENO3. The
weights contain a parameter ", which was originally introduced to avoid the division by
zero. However, we will show that " has a signi-cant in
uence on the behavior of the
reconstruction and relating the WENO3 weights to our decision criterion allows us to give
a clarifying interpretation.
In a second part, we will review some well-known Riemann solvers and introduce a family
of incomplete Riemann solvers which avoid solving the eigensystem. Nevertheless, these
solvers still reproduce all waves with less dissipation than other methods such as HLL and
FORCE, requiring only an estimate of the globally fastest wave speeds in both directions.
Therefore, the new family of Riemann solvers is particularly e-cient for large systems of
conservation laws when no explicit expression for the eigensystem is available.
Résumé : The Penrose tiling is a central example of many interesting mathematical properties. With the discovery of quasicrystals it was also shown to have physical relevance. This talk will discuss several of these properties, in particular the relationship between the construction using a substitution rule where tiles are inflated and subdivided and the projection method that views the tiling as a slice of a lattice in five dimensions. It will conclude with a characterisation of all tilings that can be generated by both methods.
Lieu : Salle Pellos (1R1, 207)
Résumé : Quasi-smooth schemes (or stacks) arise in several geometric situations : an important example is the stack of G-local systems on a Riemann surface, but there are many other simpler examples. The degree of non-smoothness of a quasi-smooth scheme is captured nicely by the difference between quasi-coherent sheaves and ind-coherent sheaves on it.
I will explain how, while studying a quasi-smooth scheme, one is led to go to the next level : the level of "quasi-quasi-smooth scheme" (pun intended). By definition, if quasi-smooth means that the tangent complex lives in amplitude [0,1], quasi-quasi-smooth is the next best thing : amplitude [0,2].
In this situation, the relation between QCoh and IndCoh gets unwieldy. This wildness has a positive consequence though : it helps define a new theory of left D-modules on derived schemes that is of use in geometric Langlands (for instance, in the description of Hecke eigensheaves for the trivial local system).