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/2007.11612, https://arxiv.org/abs/2112.12662. We also comment on the practical implications of said results.
Greta Marino (TU Chemnitz)
May 31th 2022, 10 am
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
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
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 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)
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.