Applied Mathematics Seminar

Upcoming seminars

  • François Charton (Meta AI), March 1st at 2pm, Room B211.

    Problem solving as a translation task

    Neural architectures designed for machine translation can be used to solve problems of mathematics, by considering that solving amounts to translating the problem, a sentence in some mathematical language, into its solution, another sentence in mathematical language. Presenting examples from symbolic and numerical mathematics, and theoretical physics, I show how such techniques can be applied to develop AI for Science, and help understand the inner workings of language models.

  • Tobias Grafke (University of Warwick), March 22th at 10:00, Room B211.

    Quantifying extreme events in complex systems via sharp large deviations estimates

    Rare and extreme events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. Large deviation theory provides a classical way of dealing with events of extremely small probability, but generally only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, as well as corresponding algorithms, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single computation and without fitting parameters. The applicability of this method to high-dimensional real-world systems, for example coming from fluid dynamics or molecular dynamics, are discussed.

  • Nathalie Ayi (LJLL Sorbonne Université), March 25th at 10:00, Room B211.

    Title and abstract TBD

  • Francis Nier (Université de Paris 13, délégation MATHERIALS), 28 Mars from 9:30AM to 11:30AM, Room B211.

    Problème de Grushin et autres techniques spectrales basées sur le complément de Schur

    La méthode dite du problème de Grushin, dont le nom et les notations ont été fixés dans les premiers travaux de J. Sjöstrand (autour de 1970), est un outil maintenant très commun pour comprendre et étudier finement des problématiques d’asymptotique spectrale. En tant que méthode s’appuyant sur la formule du complément de Schur ila des liens avec la méthode de Feshbach populaire en physique mathématique ou les techniques dites de Lyapunov-Schmidt en systèmes dynamiques ou EDP non linéaires. L’approche du problème de Grushin est stable par perturbation. Combinée avec du calcul pseudodifférentiel, semiclassique et des découpages (micro)locaux, elle fournit une approche générale pour ramener des problèmes spectraux multi-échelles à des modèles calculables. En ce sens cette approche permet souvent de donner une formalisation mathématique précise de l’intuition des physiciens.

    Je commencerai par une présentation élémentaire dont une application rapide est la théorie de Fredholm et la théorie de Fredholm holomorphe. J’en profiterai pour rappeler des résultats sur quelques exemples symptomatiques de spectres d’opérateurs non auto-adjoints.

    Je montrerai la variante Feshbach sur un cas simple pour voir les différences.

    Pour illustrer l’interaction entre calcul semiclassique et problème de Grushin, je traiterai le cas simple d’une méthode LCAO (Linear Combination of Atomic Orbital) des chimistes pour un problème à2 puits. J’évoquerai ensuite la question du problème des résonancesde formes pour le problème du puits dans une île.

    Enfin, pour préparer une exposé futur, je rappellerai le calculformel de complément de Schur  qui permet de faire le lien entre processus de Langevin et processus de Langevin suramorti dans l’asymptotique des grandes frictions.

Past seminars (2023-2024)

  • Cecilia Pagliantini (TU Eindhoven), October 10th, 14:00pm, Room F102.

    Structure-preserving adaptive model order reduction of parametric Hamiltonian systems

    Model order reduction of parametric differential equations aims at constructing low-complexity high-fidelity surrogate models that allow rapid and accurate solutions under parameter variation. The development of reduced order models for Hamiltonian systems is challenged by several factors: (i) failing to preserve the geometric structure encoding the physical properties of the dynamics might lead to instabilities and unphysical behaviors of the resulting approximate solutions; (ii) the slowly decaying Kolmogorov n-width of transport-dominated and non-dissipative phenomena demands large reduced spaces to achieve sufficiently accurate approximations; and (iii) nonlinear operators require hyper-reduction techniques that preserve the gradient structure of the flow velocity. We will discuss how to address these aspects via structure-preserving nonlinear model order reduction. The gist of the proposed method is to adapt in time an approximate low-dimensional phase space endowed with the geometric structure of the full model and to ensure that the hyper-reduced flow retains the physical properties of the original model.

  • Claire Boyer (LPSM, Sorbonne Université), October 12th, 10:00am, Room F202.

    Some statistical insights into PINNs

    Physics-informed neural networks (PINNs) combine the expressiveness of neural networks with the interpretability of physical modeling. Their good practical performance has been demonstrated both in the context of solving partial differential equations and in the context of hybrid modeling, which consists of combining an imperfect physical model with noisy observations. However, most of their theoretical properties remain to be established. We offer some statistical guidelines into the proper use of PINNs.

  • Harold Berjamin, (University of Galway, Ireland), November 8th, 11:00am, Room B211.

    Recent developments on the propagation of mechanical waves in soft solids

    In this talk, I will give an overview of recent results obtained during my postdoctoral fellowships at the University of Galway (Ireland), covering several topics related to wave propagation in soft solids. The broader context of these works is the study of Traumatic Brain Injury, which is a major cause of death and disability worldwide. First, the nonlinear propagation of shear waves in viscoelastic solids will be addressed, including thermodynamic aspects and shock formation. Then, fluid-saturated porous media will be considered. Ongoing and future developments will also be presented.

  • Silvère Bonnabel (Mines Paristech), November 29th, 10:30am, Room B211.

    Wasserstein Gradient Flows for Variational Inference

    In this talk, we will introduce the article [1] and a few extensions. We propose a new method for approximating a posterior probability distribution in Bayesian inference. To achieve this, we offer an alternative to well-established MCMC methods, based on variational inference. Our goal is to approximate the target distribution with a Gaussian distribution, or a mixture of Gaussians, that come with easy-to-compute summary statistics. This approximation is obtained as the asymptotic limit of a gradient flow in the sense of the 2-Wasserstein distance on the space of Gaussian measures (Bures-Wasserstein space). Akin to MCMC, this approach allows for strong convergence guarantees for log-concave target distributions. We will also briefly discuss low-rank implementations for tractability in higher dimensions.

    [1] Variational inference via Wasserstein gradient flows, Marc Lambert, Sinho Chewi, Francis Bach, Silvère Bonnabel, and Philippe Rigollet, NeurIPS 2022

  • Andrea Bertazzi (École Polytechnique), December 7th at 9:30am, Room B211.

    Sampling with (time transformations of) Piecewise deterministic Markov processes

    Piecewise deterministic Markov processes (PDMPs) received substantial interest in recent years as an alternative to classical Markov chain Monte Carlo (MCMC) algorithms. While theoretical properties of PDMPs have been studied extensively, their exact implementation is only possible when bounds on the gradient of the negative log-target can be derived. In the first part of the talk we discuss how to overcome this limitation taking advantage of approximations of PDMPs obtained using splitting schemes. Focusing on the Zig-Zag sampler (ZZS), we show how to introduce a suitable Metropolis adjustment to eliminate the discretisation error incurred by the splitting scheme. In the second part of the talk we study time transformations as a resource to improve the performance of (PDMP-based) MCMC algorithms in the setting of multimodal distributions. For a suitable time transformation, we argue that the process can explore the state space more freely and jump between modes more frequently. Qualitative properties of time transformed Markov process are derived, with emphasis on uniform ergodicity of the time transformed ZZS. We conclude the talk with a proposal on how to make use of this idea taking advantage of our Metropolis adjusted ZZS.

  • Andrew Stuart (Caltech), December 14th, afternoon, Room B211.

    Learning Solution Operators For PDEs: Algorithms, Analysis and Applications [Slides]

    Neural networks have shown great success at approximating functions between spaces X and Y, in the setting where X is a finite dimensional Euclidean space and where Y is either a finite dimensional Euclidean space (regression) or a set of finite cardinality (classification); the neural networks learn the approximator from N data pairs {x_n, y_n}.

    In many problems arising in PDEs it is desirable to learn solution operators: maps between spaces of functions X and Y; here X denotes a function space of inputs to the PDE (such as initial conditions, boundary data, coefficients) and Y denotes the function space of PDE solutions. Such a learned map can provide a cheap surrogate model to accelerate computations.

    The talk overviews the methodology being developed in this field of operator learning and describes analysis of the associated approximation theory. Applications are described to the learning of homogenized constitutive models in mechanics.

  • Luca Nenna (Université Paris-Saclay, délégation MATHERIALS)

    Introduction au transport optimal [Lecture notes]

    All the lectures of this short course will take place in the CERMICS seminar room (B211)

    • Wed 17 Jan from 9h30 to 11h30: Monge and Kantorovich problems
    • Thu 18 Jan from 9:00 to 11:00: Dual problem, optimality conditions, optimal transport maps [Slides]
    • Wed 24 Jan from 9h30 to 11h30: Entropic optimal transport and Sinkhorn algorithm [Slides]
    • Fri 2 Feb from 9h30 to 11h30: A glimpse of multi-marginal OT and applications [Slides]
  • Emma Horton (Unviersity of Warwick), February 1st at 10:00, Room B211. [cancelled]

    Monte Carlo methods for branching processes

    Branching processes naturally arise as pertinent models in a variety of situations such as cell division, population dynamics and nuclear fission. For a wide class of branching processes, it is common that their first moment exhibits a Perron Frobenius-type decomposition. That is, the first order asymptotic behaviour is described by a triple $(\lambda, \varphi, \eta)$, where $\lambda$ is the leading eigenvalue of the system and $\varphi$ and $\eta$ are the corresponding right eigenfunction and left eigenmeasure respectively. Thus, obtaining good estimates of these quantities is imperative for understanding the long-time behaviour of these processes. In this talk, we discuss various Monte Carlo methods for estimating this triple. This talk is based on joint work with Alex Cox (University of Bath) and Denis Villemonais (Université de Lorraine).


 

 

 

Past seminars (2022-2023)

Shengquan Xiang (Peking University) – Amaury Hayat (Ecole des Ponts), Colloquium du Labex Bézout, June 28th, 4:00pm, Room B211.

Stabilization of evolution equations

Control theory is about asking oneself: “if I can act on a system, what can I make it do ?”. One of the main branches of this theory is about understanding how to act on a system as a function of what we measure. This is called stabilization. Stabilization combines many theoretical questions in mathematics (e.g. well-posedness of the system, spectral inequalities, optimal control theory, propagation of singularities) with many practical applications (autonomous vehicles, robotics, space industry, etc.). One of the great difficulties of this subject is that it is very difficult to find generic results for infinite dimensional systems (e.g. models of fluid mechanics, electromagnetism, flux propagation, etc.). In this talk, we’ll set out the basics of this subject, as well as a new promising approach to find generic results: the F-equivalence (also known as generalized backstepping).

Aline Lefebvre-Lepot (Ecole Polytechnique) – June 22nd, 10:30am, Room B211.

TBD

TBD

Helge Dietert (Univ. Paris Diderot) – March 30th (NEW DATE), 10am, Room B211.

Quantitative Geometric Control in Linear Kinetic Theory

(joint work with Frédéric Hérau, Harsha Hutridurga, Clément Mouhot)

We consider kinetic equations where the dissipation is only acting in a
region of the spatial domain. E. Bernard and F. Salvaran (2013) gave a
first result on exponential decay for bounded collision operators and
identified a geometric control condition. I will present our work on a
general framework covering many cases in a quantitive way.

Gianluca Ceruti (EPFL) – February 9, 2pm, Room B211

Robust numerical integrators for dynamical low-rank approximation. 

In the present seminar, we begin with a recapitulation of the dynamical low-rank approximation for matrices of fixed rank-r together with the derivation of the so-called matrix projector-splitting integrator. We show that the matrix projector splitting integrator satisfies two remarkable properties: It reproduces rank-r time-dependent matrices exactly, and it is robust with respect to the presence of small singular values in the approximation or the solution. Furthermore, two major built-in drawbacks of the matrix projector-splitting integrator are discussed: The integrator contains a backward sub step, which is a major source of instability for dissipative problems, and it does not allow for an adaptive choice of the rank. Therefore, a novel unconventional integrator is introduced. The so-called unconventional integrator is shown to maintain the remarkable properties of the original matrix projector-splitting integrator, meanwhile overcoming the aforementioned drawbacks. Finally, the extension to a continuous L2-setting of the dynamical low-rank approximation together with the matrix projector-splitting integrator for matrices is introduced. In the spirit of the Von Neumann stability analysis, a stability analysis of dynamical low-rank approaches for linear hyperbolic problems is discussed and the application to Burgers’ equation with 2 uncertainties is illustrated. This seminar is based upon recent joint collaborations with Christian Lubich, Hanna Walach, Jonas Kusch, Lukas Einkemmer, and Martin Frank.

Ludovick Gagnon (INRIA Nancy) – February 13, 10:30am, Room B211

Stabilisation rapide des water waves linéarisées et backstepping de type Fredholm pour opérateurs critiques

Dans cet exposé, on présente un résultat récent de stabilité rapide de l’équation des water waves linéarisée grâce à la méthode du backstepping de type Fredholm. Initialement introduite avec une transformation de Volterra, la méthode du backstepping avec une transformation de Fredholm permet de montrer la stabilisation rapide pour une grande classe d’EDP grâce à des propriétés de contrôlabilité. L’équation des water waves linéarisée représente un cas critique pour cette méthode, puisque les techniques classiques ne permettent pas de traiter des opérateurs de type i|D_x|^a, avec 1 < a \leq 3/2. Nous introduisons un nouvel argument de compacité/dualité permettant de franchir le seuil a=3/2 et nous montrons que la méthode du backstepping de type Fredholm s’applique pour des opérateurs anti-adjoints du type i|D_x|^a, avec a >1. Il s’agit d’un travail en collaboration avec Amaury Hayat, Shengquan Xiang et Christophe Zhang.

 

Alexander Keimer (FAU Erlangen) – February 16, 4pm – Room F107

Nonlocal conservation laws

We will give an overview on nonlocal conservation law and show their
applicability in a variety of research areas. We then discuss recent
results and advances ranging from fundamental questions of existence and
uniqueness of solutions to control problems as well as the singular
limit problem, i.e., whether we can recover the solution to the
corresponding local problem when the nonlocal range collapses to a
single point.

Yves Achdou (Université de Paris et projet Matherials)

Introduction aux jeux à champ moyen (cours)

La théorie des jeux à champ moyen a été introduite en 2006 par JM. Lasry et PL. Lions pour décrire des jeux différentiels (équilibres de Nash) dans la limite où le nombre de joueurs tend vers l’infini. Cette théorie a depuis connu un essor considérable. Elle constitue un point de rencontre de plusieurs domaines des mathématiques appliquées: théorie des jeux, contrôle optimal déterministe ou stochastique, calcul des variations, transport optimal,  analyse des EDPs, méthodes numériques. Les applications sont nombreuses: économie, étude des comportements collectifs avec anticipations rationnelles, etc…
On donnera un aperçu de la théorie, en insistant sur plusieurs aspects : les fondements théoriques, la master equation, les systèmes d’EDP forward-backward, et l’analyse de ces systèmes. Comme il est crucial, si on veut utiliser les jeux à champ moyen à des fins prédictives, de disposer de méthodes numériques robustes, on présentera aussi les schémas aux différences finies utilisés, des résultats de convergence,  ainsi que des exemples de simulations dans des modèles de trafic ou de mouvement de piétons.

Dates du cours : Le cours a lieu les mardi et jeudi du 27 septembre au 13 octobre de 14h à 16h.

mardi 27 sept
jeudi 29 sept
mardi 4 oct (initialement prévu le 5, le cours aura bien lieu le mardi 4)
jeudi 6 oct
mardi 11 oct
jeudi 13 oct

 

Danny Perez (Los Alamos National Lab) – October 11, 4:30pm and October 13, 9am

October 11, 4:30pm: introduction aux méthodes de dynamique moléculaire accélérées

October 13, 9am: aspects récents de ces algorithmes.

Teemu Pennanen  (King’s College London) – October 19, 3pm

Convex Stochastic Optimization

Frederic Barbaresco (Thales) – October 20, 10:30am

Théorie Symplectique de la chaleur et de l’Information pour les modèles non-dissipatifs et dissipatifs et application à l’équation de Lindblad

Jean-Marie Souriau a étendu la notion classique d’ensemble canonique de Gibbs au cas d’une variété symplectique sur laquelle un groupe de Lie possède une action hamiltonienne. Le modèle de Souriau est une nouvelle théorie symplectique de la chaleur et de la Géométrie de l’Information, appelée « Thermodynamique des Groupes de Lie ». Ce modèle donne une caractérisation archétypale, et purement géométrique, à l’Entropie, qui apparaît comme une fonction de Casimir invariante en représentation coadjointe, dont on déduit une équation géométrique de la chaleur. L’approche permet également de généraliser la métrique de Fisher de la géométrie de l’information grâce à la 2 forme KKS (Kirillov, Kostant, Souriau) dans le cas de l’action affine de l’opérateur coadjoint via le cocyle symplectique de Souriau. Les feuilles symplectiques des orbites coadjointes sont également les ensembles de niveaux de l’entropie. Cette thermodynamique des groupes de Lie s’interprète dans le cadre de la thermodynamique par le fait que la dynamique décrite sur ces feuilles symplectiques caractérise les phénomènes non dissipatifs, alors que la dynamique transverse à ces feuilles symplectiques explique ceux qui sont dissipatifs. Dans une deuxième partie, nous aborderons la théorie symplectique dissipative de la chaleur et de l’Information en considérant la structure de Poisson transverse aux feuilles symplectiques, munie de sa structure de Poisson canonique, basée sur le crochet métriplectique. Récemment, Baptiste Coquinot a introduit un point de vue général à partir de la thermodynamique hors d’équilibre pour dériver une nouvelle théorie fondamentale des crochets dissipatifs, en considérant un ensemble de variables qui semblent plus naturelles pour construire le crochet métriplectique, puis en présentant une manière systématique de dériver des crochets dissipatifs généraux. La construction de Baptiste Coquinot montre que les crochets dissipatifs métriplectiques sont tout à fait naturels pour la thermodynamique hors équilibre, tout comme les crochets de Poisson sont naturels pour la dynamique hamiltonienne, dérivant pour la première fois un crochet dissipatif général à partir des premiers principes de base de la thermodynamique et des relations d’Onsager-Casimir. Nous explorerons les liens avec la structure transverse de la théorie de Dirac des crochets dissipatifs pour les systèmes hamiltoniens contraints. Nous explorerons une autre approche que l’approche métriplectique, basée sur la structure de Poisson transverse qui a été étudiée par Michel Saint-Germain dans sa thèse inspirée des travaux fondateurs de Fokko du Cloux. Dans la continuité, Hervé Sabourin a étudié la nature polynomiale des structures de Poisson transverses aux orbites adjointes nilpotentes et a prouvé que la restriction à la tranche transverse des fonctions de Casimir de la structure de Lie-Poisson sont des fonctions de Casimir indépendantes de la structure de Poisson transverse, et a également montré qu’il existe deux structures de Poisson polynomiales sur la tranche transversale à la feuille symplectique, qui ont des fonctions de Casimir, à savoir la structure de Poisson transverse et la structure déterminantale, construites en utilisant ces Casimir. Pour illustrer les modèles précédents, nous considérerons pour application l’équation de Lindblad décrivant les versions dissipatives de l’équation hamiltonienne de Liouville de la matrice densité en quantique, en utilisant que la partie dissipative de l’opérateur de Lindblab peut être décrit comme un gradient de l’entropie relative, et que des équilibres peuvent être déduits par le principe d’entropie maximale.

Lihan Wang (Carnegie Mellon university – MPI Leipzig),

Sept. 22nd, 10am.

Quantitative convergence analysis of hypocoercive sampling dynamics

In this talk, we will discuss some results on quantitative analysis of convergence rates of hypocoercive sampling dynamics, including underdamped Langevin dynamics, randomized Hamiltonian Monte Carlo, zigzag process and bouncy particle sampler. The analysis is based on the Armstrong-Mourrat variational framework for hypocoercivity which combines a Poincare-type inequality in time-augmented state space and an L^2 energy estimate. Joint works with Yu Cao (NYU) and Jianfeng Lu (Duke).

Past seminars (2021-2022)

Matthew Zhang (Uni. Toronto)

June 8th 2022, 10 am

Convergence of Langevin Monte Carlo: Strong Metrics and Weak Tails

In this talk we will discuss Langevin flows and its discretizations. The law of this algorithm evolves according to the parabolic PDE known as the Fokker-Planck equation. The Langevin algorithm has been successfully applied in many practical applications to approximate target distributions with arbitrary accuracy. However, beyond the well-explored cases of strong log-concavity, theoretical guarantees have only recently been made known. In this talk, we will review classical convergence analysis for the Langevin Algorithm and explore a range of recent results. In particular, we will simultaneously discuss convergence in strong metrics under functional inequalities beyond the Poincaré inequality, as seen in https://arxiv.org/abs/2007.11612https://arxiv.org/abs/2112.12662. We also comment on the practical implications of said results.

 Greta Marino (TU Chemnitz)

May 31th 2022, 10 am

Cross-diffusion systems with Cahn-Hilliard terms: new perspectives

In this talk we study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment. If time allows, we will also discuss about some possible further extensions. This is a joint work with V. Ehrlacher and J.-F. Pietschmann.

Fenna Müller (Uni. Leipzig)

May 24th 2022, 10 am

Adiabatic Approximation of Coarse Grained Second Order Response

Response theory has been used in physics to predict the response of a system to an external perturbations in several context. This relies on observation of the system in question. In many systems only a subsystem is available for measurement. Can response theory still be used ot predict the response to an external perturbation? We propose estimators for dynamic second order response theory in coarse grained variables for driven out-of-equilibrium subsystems. The error is controlled through the the notion of subsystem spectral gap for the convergence of coarse grained observables. This is a joint work with Max von Renesse (arxiv arXiv:2204.10217).

Gül Varol (ENPC – Imagine)

May 10th 2022 10am

Action-conditioned 3D human motion synthesis

This talk will summarize our recent works on 3D human motion synthesis. In contrast to methods that complete, or extend, motion sequences, we specifically focus on generating motions without requiring an initial pose or sequence. Our primary focus is controllable synthesis — which means the user specifies the desired action. In our work, this can be specified in the form of (i) a symbolic action category among a pre-defined set of action classes such as “jumping”, “waving”, or (ii) a textual description using natural language such as “a human walks in a circle”. We make use of recent progress in sequence-level modelling using Transformers to design generative variational autoencoder (VAE) models and show promising results for realistic and diverse human motion synthesis.

Shengquan Xiang (EPFL)

May 5th 2022 10am

Quantitative rapid stabilization of some PDE models

Quantitative stabilization is an active research topic in PDEs’ control theory, namely to construct explicit feedback laws as control to make the closed-loop system stable with quantitative estimates. We will talk about some recent developments in this topic based on important models including Frequency Lyapunov method for the heat equation and Navier-Stokes equations, as well as Fredholm backstepping method for water waves.

Aurélien Velleret (UGE)

April 6th 2022 – 2pm

Distribution quasi-stationnaire et notions associées pour mieux appréhender l’adaptation des populations soumises à la transformation de leur environnement

Dans tout modèle d’éco-évolution qui prend en compte la stochasticité de la taille de la population, on doit faire face à des événements d’extinction. Pour des populations bien adaptées, il est généralement admis que l’extinction est suffisamment négligeable pour qu’on puisse décrire un équilibre probabiliste sans en tenir compte. Pour évaluer les limites de stabilité de tels systèmes, je me suis néanmoins intéressé à la manière de prendre compte de l’extinction et d’évaluer son importance.

Dans un premier temps, je vais vous présenter certains critères généraux garantissant une telle analyse des fluctuations à l’équilibre. En pratique, ces conditions assurent l’existence et l’unicité d’une distribution quasi-stationnaire (DQS). Cette loi de probabilité est laissée invariante par la dynamique temporelle du processus, en tenant compte de la décroissance exponentielle due à l’extinction. L’amplitude des fluctuations est alors directement reliée à la dispersion de cette DQS. De plus, ces conditions impliquent pour la loi du processus qu’il converge à un taux exponentiel vers la DQS. Ce taux décrit en quelque sorte la vitesse à laquelle le processus perd la mémoire de son état ancestral. Une dynamique viable apparaît alors caractérisée par le fait que le taux de convergence est beaucoup plus rapide que le taux d’extinction.

Je compte illustrer ces aspects théoriques par le modèle d’adaptation qui a motivé ces critères. Je considérerai une population asexuée soumise à un changement progressif de ses conditions environnementales. On suppose que cet effet est compensé par l’apparition de mutations génétiques qui envahissent régulièrement la population. La question est alors d’étudier avec quelle efficacité l’advenue de ces mutations parvient à maintenir la population adaptée dans un régime viable. Puisque seules les populations survivantes sont observées, ce biais ne conduirait-il pas à une surestimation de leur capacité d’adaptation future ?

Samuel Daudin (Paris – Dauphine)

March 24th 2022 – 10am

Optimal control of the Fokker-Planck equation with state constraint in the Wasserstein space.

We consider an optimal control problem for the Fokker-Planck equation subject to a state constraint in the space of probability measures equipped with the L^2-Wasserstein distance. The first order necessary optimality conditions associated to this optimization problem give rise to a second order mean field game (MFG) system which has a potential structure. Since the problem is subject to the state constraint, additional unknowns (beside the usual density and value function variables) appear in the MFG system, which can be seen as the corresponding Lagrange multipliers. We will explain how to derive the optimality conditions and what we can deduce for the corresponding solutions. In particular we will show that -despite the presence of the state constraint- optimal controls are Lipschitz continuous in time.

Pierre Jacob (ESSEC)

March 17th 2022 – 3:30pm

Some methods based on couplings of Markov chain Monte Carlo algorithms

Markov chain Monte Carlo algorithms are commonly used to approximate a variety of probability distributions, such as posterior distributions arising in Bayesian analysis. I will review the idea of coupling in the context of Markov chains, and how this idea not only leads to theoretical analyses of Markov chains but also to new Monte Carlo methods. In particular, the talk will describe how coupled Markov chains can be used to obtain 1) unbiased estimators of expectations and of normalizing constants, 2) non-asymptotic convergence diagnostics for Markov chains, and 3) unbiased estimators of the asymptotic variance of MCMC ergodic averages.

Doghonay Arjmand (Mälardalen University)

Thursday, December 9th, 10h at INRIA Paris Room Flajolet (Building C)

Comparison of elliptic local problems for homogenization problems

Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in the model. These effective data need to be computed by running local microscale simulations followed by a local averaging of the microscopic information. Motivated by the classical homogenization theory, it is a common practice to use local elliptic cell problems for computing the missing homogenized coefficients in the macro model. Such a consideration results in a first order error O(ε/δ), where ε represents the wavelength of the microscale variations and δ is the size of the microscopic simulation boxes. This error, called “resonance error”, originates from the boundary conditions used in the micro-problem and typically dominates all other errors in a multiscale numerical method. In this talk, we will discuss a few efficient ways of computing homogenized coefficients, and we will compare their computational complexity in deterministic as well as stationary ergodic random media.

 

Evan Camrud (Iowa State University)

Thursday, December 2nd, 10h

Stochastic Lorenz 96: predictions on equilibrium and convergence

The Lorenz 96 system, originally introduced by Edward Lorenz in “Predictability: a problem partially solved”, was constructed to model the behavior of a meteorological quantity advected by the atmosphere. It is based on a truncation of a dyadic model for the Navier-Stokes interactions. We approach a stochastically forced version of this system, with degenerate noise and dissipation, to try to understand the spread of dissipation through the system. In particular, our goal is to understand the interaction of energy conserving nonlinearities and dissipation in arbitrary energy conserving systems. Preliminary results on positive recurrence of this stochastic Lorenz 96 system will be introduced.

Christoph Strössner (EPFL)

Thursday, November 18th, 10h

Functional low-rank approximations for trivariate functions based on tensorised Chebyshev polynomials

The approximation of multivariate functions on tensorproduct domains in terms of truncated series expansions using tensorised basis functions suffers from the curse of dimensionality. For certain functions the curse can be mitigated by only storing a low-rank approximation of the coefficient tensor, which leads to so called functional low-rank approximations. In this talk, we focus on approximations of trivariate functions on cubes using tensorised Chebyshev polynomials as basis functions.  In a first part, we study the computation of such low-rank approximations from function evaluations, without evaluating the full coefficient tensor arising in tensorised Chebyshev interpolation. We propose a novel algorithm which for most functions reduces the required number of function evaluations compared to the algorithm used in [Hashemi and Trefethen, SIAM J. Sci. Comput., 39 (2017)]. In a second part, we generalize the ideas of [Townsend and Olver, J. Comput. Phys., 299 (2015)] to develop a spectral method for solving linear PDEs. We treat the application of a linear partial differential operator as function operating on the coefficient tensor. This mapping can be inverted to solve the PDE, which leads to a tensor-valued linear equation. For certain PDEs a Laplace-like structure arises, which we can exploit in the solver.

 

Tommaso Taddei (INRIA Bordeaux)

Thursday, October 14th, 11h — INRIA Paris, A415

Localized model reduction for nonlinear elliptic PDEs.

We present work towards the development of a rapid and robust component-based model reduction procedure for parameterized nonlinear elliptic PDEs. For large-scale systems, traditional (monolithic) projection-based model order reduction methods fail due to prohibitive computational offline costs; if the system of interest can be represented as the union of a set of instantiated archetype components, localized model reduction methods based on localized training are particularly promising. In this contribution, we shall focus on the problem of localized training, which refers to the task of constructing low-dimensional approximation spaces for each archetype component. We review an optimal algorithm for a class of linear elliptic PDEs and we investigate its properties; then, we present a general randomized localized training technique for nonlinear PDEs; numerical investigations for a nonlinear diffusion equation illustrate several features of the problem and justify the proposed randomized approach.

Mohamad Rachid (Nantes)

Thursday, February 11th, 10h — Online seminar

Incompressible Navier-Stokes-Fourier limit from the Landau equation

In this talk, we are interested in the Landau equation which is a kinetic model in plasma physics that describes the evolution of the density function \( f = f(t, x, v) \) representing at time \( t\in \mathbb{R}^{+} \), the density of particles at position \( x\in T^{3} \) (the 3-dimensional unit periodic box) and velocity \( v\in \mathbb{R}^{3} \). We study the Landau equation, depending on the Knudsen Number and its limit to the incompressible Navier-Stokes-Fourier equation on the torus. We prove uniform estimate of some adapted Sobolev norm and get existence and uniqueness of solution for small data. These estimates are uniform in the Knudsen number and allow to derive the incompressible Navier-Stokes-Fourier equation when the Knudsen number tends to 0.

 

 

 

Archive of past seminars before 2021: here

Organizers: Amaury Hayat, Urbain Vaes.