Séminaire de mathématiques appliquées (archives)

In this talk we will discuss the approximation to nonlinear dispersive equations, which asks for low-regularity assumptions on the initial data, both for deterministic and random initial data.

We will put forth a novel time discretization to the nonlinear Schrödinger equation, allowing for a low-regularity approximation, while maintaining good long-time preservation of the mass density and energy on the discrete level.

Higher order extensions will be presented, following new techniques based on decorated tree series expansions, inspired by singular SPDEs.

Nous considérons un modèle bidimensionnel d'une marche aléatoire en auto-interaction, soumise à un accrochage le long d'un mur dur horizontal. Lorsque l'intensité de l'auto-interaction est suffisamment grande, le système entre dans une phase effondrée à l'intérieur de laquelle la marche aléatoire se replie sur elle-même pour former une boule compacte comprise entre une enveloppe supérieure et une enveloppe inférieure. Une particularité de ce modèle est que cette phase effondrée peut être divisée en deux régimes séparés par une courbe critique explicite. Ainsi, lorsque l'intensité de l'accrochage est faible, l'enveloppe inférieure s'éloigne du mur dur, tandis qu'elle est accrochée le long de la paroi dès que l'intensité de l'accrochage est suffisamment forte.

We consider the problem of state estimation from $m$ linear measurements, where the state $u$ to recover is an element of the manifold $\mathcal{M}$ of solutions of a parameter-dependent equation. The state is estimated using a prior knowledge on $\mathcal{M}$ coming from model order reduction. Variational approaches based on linear approximation of $\mathcal{M}$ yields a recovery error li-mited by the Kolmogorov $m$-width of $\mathcal{M}$. To overcome this issue, piecewise-affine approximations of $\mathcal{M}$ have also be considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to $\mathcal{M}$.

La marche aléatoire de l'éléphant (ERW) est une marche aléatoire discrète qui a été introduite au début des années 2000 par deux physiciens afin d'étudier l’influence d’un paramètre de mémoire sur le comportement de la marche aléatoire. Dans cet exposé, on présentera plusieurs possibilités pour étudier et obtenir des résultats sur l’ERW. On s’intéressera à l’approche martingale, puis au lien avec les urnes de Polya ou encore avec les arbres aléatoires récursifs. En particulier, on expliquera comment l’utilisation des trois approches est nécessaire pour obtenir des informations sur la variable aléatoire limite qui apparaît dans le régime super-diffusif.

Abstract

In this talk, we investigate sampling strategies for approximation of functions by weighted least-squares. Although a quasilinear sampling budget can already be achieved by iid random draws according to the adapted density [3], further reductions of the needed number of points are possible [2, 4], achieving provable performance estimates thanks to the solution to the Kadison-Singer problem [6]. However this involves a subsampling step, which is not algorithmically tractable. We show how greedy sampling methods [1, 5] can circumvent this defect, while attaining optimal sample sizes.

References

[1] J. Batson, D. A. Spielman, and N. Srivastava. “Twice-ramanujan spar- sifiers”. In: SIAM Journal on Computing 41.6 (2012).

In this talk, we present a posteriori estimates for finite element approximations of nonlinear elliptic problems satisfying strong-monotonicity and Lipschitz-continuity properties. These estimates include, and build on, any iterative linearization method that satisfies a few clearly identified assumptions; this includes the Picard, Newton, and Zarantonello linearizations. The estimates give a guaranteed upper bound on an augmented energy difference reliability with constant one, as well as a lower bound efficiency up to a generic constant.

This talk aims at introducing a new multi-frequency method to reconstruct width defects in waveguides. Different inverse methods already exist. However, those methods are not using some frequencies, called resonant frequencies, where propagation equations are known to be ill-conditioned. Since waves seem very sensible to defects at these particular frequencies, we exploit them instead. After studying the forward problem at these resonant frequencies, we approximate the wavefield and focus on the inverse problem. Given partial wavefield measurements, we reconstruct slowly varying width defects in a stable and precise way and provide numerical validations and comparisons with existing methods.

Abstract

A method for constructing signatures of random reparametrizations of periodic functions is presented.

The proposed signatures are functions, which contain information about the height and order of local extrema of the observation. In contrast to other statistical methods for reparametrized curves, the observations can be of different lengths and the construction does not involve aligning them.

The signature is shown to be stable with respect to changes in the distribution of reparametrizations and to enjoy standard CLT properties, including in the case of dependent observations.

The positioning of a vehicle based on magnetic signals is the industrial application which motivated this work.

Heterogeneous problems that take place at multiple scales are ubiquitous in science and engineering. Examples are wind turbines made from composites or groundwater flow relevant e.g., for the design of flood prevention measures. However, finite element or finite volume methods require an often prohibitively large amount of computational time for such tasks. Multiscale methods that are based on ansatz functions which incorporate the local behavior of the (numerical) solution of the partial differential equations (PDEs) have been developed to tackle these heterogeneous problems.

Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure.