Séminaire de mathématiques appliquées (archives)

Hidden Markov models (HMMs) are flexible tools for clustering dependent data coming from unknown populations, allowing nonparametric modelling of the population densities. Identifiability fails when data are in fact independent, and we study the frontier between learnable and unlearnable two-state nonparametric HMMs. Interesting new phenomena emerge when the cluster distributions are modelled via density functions (the `emission' densities) belonging to standard smoothness classes compared to the multinomial setting [1]. Notably, in contrast to the multinomial setting previously considered, the identification of a direction separating the two emission densities becomes a critical, and challenging, issue. Surprisingly, it is possible to "borrow strength" from estimators of the smoothest density to improve estimation of the other. We conduct precise analysis of minimax rates, showing a transition depending on compared smoothnesses of the emission densities.

Dans ce séminaire, nous parlerons d'un extension d'un modèle répulsif. Un polymère est placé dans un champ magnétique faiblement répulsif. Nous étudierons son interaction en fonction de la distance entre deux interfaces répulsives - pensez à une distance d'interfrange, https://www.f-legrand.fr/scidoc/docmml/sciphys/optique/young/young.html en contient quelques unes -, et de la pénalité qu'il reçoit à toucher une interface. Nous présenterons nos résultats sur ce modèle.

References: Francesco Caravenna and Nicolas Pétrélis. Depinning of a polymer in a multi-interface medium." Electron. J. Probab. 14 2038 - 2067, 2009. https://doi.org/10.1214/EJP.v14-698

Nous présenterons la notion de noyau de Stein, qui permet de généraliser la formule d'intégration par partie pour la loi normale (qui possède un noyau de Stein constant, égal à sa covariance). Nous nous focaliserons dans un premier temps sur la dimension 1 où, sous de bonnes conditions, le noyau de Stein possède une formule explicite. Nous verrons que le noyau de Stein apparaît naturellement comme pondération d'une inégalité de type Poincaré et qu'il permet d'obtenir des inégalités de concentration précises, de type ratio de Mills. Dans une seconde partie, nous travaillerons en dimension supérieure, en étudiant notamment comment la notion de noyau de Stein permet de décrire la performance de l'estimateur de shrinkage, sans l'hypothèse classique de données Gaussiennes. Cette présentation se base notamment sur des travaux en collaboration avec Max Fathi, Larry Goldstein, Gesine Reinert et Jon Wellner.

Pour se préparer : https://arxiv.org/pdf/2004.01378.pdf (Relaxing the Gaussian assumption in shrinkage and sure in high dimension) https://arxiv.org/pdf/1804.03926.pdf (Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one)

Abstract

I will talk about several properties of stationary solutions of fractional SDEs. I will first recall some seminal results by Hairer (2005) on the construction of stationary solutions and associated ergodic results. Then, I will focus on a recent paper with Xue-Mei Li and Julian Sieber where we prove smoothness and Gaussian bounds for the density of the related invariant distribution (under appropriate assumptions) in the additive case. The proofs are based on a novel representation of the stationary density in terms of a Wiener-Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the non-stationary density, which extend previously known results in the additive setting. Avoiding any use of Malliavin calculus in our arguments, our results are obtained under minimal regularity requirements.

References

Li, X. M., Panloup, F., & Sieber, J. (2022). On the (Non-) Stationary Density of Fractional-Driven Stochastic Differential Equations. arXiv preprint arXiv:2204.06329.

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. Dans le cadre de cet exposé, nous nous concentrerons sur cette dernière transition.

References:

• https://projecteuclid.org/journals/annals-of-probability/volume-50/issue-4/Surface-transition-in-the-collapsed-phase-of-a-self-interacting/10.1214/22-AOP1567.full
• https://arxiv.org/abs/2007.11313

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 $\mathcal{M}$. In this work, we propose a state estimation method relying on dictionary-based model reduction, where a space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from the path of a $\ell_1$-regularized least-squares problem. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parameterizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.

References: https://arxiv.org/abs/2303.10771

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).

[2] A. Cohen and M. Dolbeault. “Optimal pointwise sampling for L2 ap- proximation”. In: Journal of Complexity 68 (2022), p. 101602.

[3] A. Cohen and G. Migliorati. “Optimal weighted least-squares methods”. In: SMAI J. Comput. Math. 3 (2017), pp. 181–203.

[4] C. Haberstich, A. Nouy, and G. Perrin. “Boosted optimal weighted least- squares”. In: Math. Comp. 91.335 (2022), pp. 1281–1315.

[5] Y. T. Lee and H. Sun. “Constructing linear-sized spectral sparsification in almost-linear time”. In: SIAM J. Comput. 47.6 (2018), pp. 2315–2336.

[6] A. W. Marcus, D. A. Spielman, and N. Srivastava. “Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem”. In: Ann. of Math. (2) 182.1 (2015), pp. 327–350.

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. We prove that for the Zarantonello linearization, this generic constant only depends on the space dimension, the mesh shape regularity, and possibly the approximation polynomial degree in four or more space dimensions, making the estimates robust with respect to the strength of the nonlinearity. For the other linearizations, there is only a local and computable dependence on the nonlinearity. Numerical experiments illustrate and validate the theoretical results, for both smooth and singular solutions.