Multilevel proximal methods for Image Restoration

• Guillaume Lauga

Room: Salle René Baire (4th floor) Solving large scale optimization problems is a challenging task and exploiting their structure can alleviate its computational cost. This idea is at the core of multilevel optimization methods. They leverage the definition of coarse approximations of the objective function to minimize it. In this talk, we present a multilevel proximal algorithm IML FISTA that draws ideas from the multilevel optimization setting for smooth optimization to tackle non-smooth optimization. In the proposed method we combine the classical accelerations techniques of inertial algorithm such as FISTA with the multilevel acceleration. IML FISTA is able to handle state-of-the-art regularization techniques such as total variation and non-local total-variation, while providing a relatively simple construction of coarse approximations. The convergence guarantees of this approach are equivalent to those of FISTA. Finally we demonstrate the effectiveness of the approach on color images reconstruction problems and on hyperspectral images reconstruction problems.

Generalized conditional gradient method for potential mean field games

• Laurent Pfeiffer

Room: Salle René Baire (4th floor) Mean field games (MFGs) are a class of problems modeling Nash equilibria for a very large number of small agents in evolution, interacting through coupling terms depending on their distribution. We will describe in detail an MFG model consisting of two coupled second-order PDEs, equivalent to the optimality conditions for an optimal control problem of the Fokker-Planck equation. We will investigate a numerical method, called fictitious play, in which the agents play at each iteration a "best-response", corresponding to a predicted value of the coupling terms. We will show that this procedure is equivalent to the generalized conditional gradient method, which will allow us to establish convergence. Joint work with Pierre Lavigne (Institut Louis Bachelier).

3D Optimal control problems constrained on surfaces

• Ugo Boscain

Room: Salle René Baire (4th floor) In this talk I consider a surface embedded in a 3D contact sub-Riemannian manifold (i.e., an optimal control problem in dimension 3 with 2 controls which is linear w.r.t. the controls and with quadratic cost; we will also make a natural controllability assumption). Such a surface inherits a field of direction (with norm) from the ambient space. This field of directions is singular at characteristic points (i.e., where the surface is tangent to the contact distribution). Generically singularities are either of elliptic type (nodes and foci) or of hyberbolic type (saddles). In this talk we will study when the normed field of directions permits to give to the surface the structure of metric space (of ``SNCF'' type). We will also study how to define the heat and the Schroedinger equation on such a structure and if the singular points are ``accessible'' or not by the evolution.

Moment-Constrained Approximation of the Lieb functional

• Luca Nenna

Room: Salle René Baire (4th floor) The aim of this talk is to present new sparsity results about the so-called Lieb functional, which is a key quantity in Density Functional Theory for electronic structure calculations for molecules. The Lieb functional was actually shown by Lieb to be a convexification of the so-called Lévy-Lieb functional. Given an electronic density for a system of N electrons, which may be seen as a probability density on R^3, the value of the Lieb functional for this density is defined as the solution of a quantum multi-marginal optimal transport problem, which reads as a minimization problem defined on the set of trace-class operators acting on the space of electronic wavefunctions that are antisymmetric L^2 functions of R^{3N}, with partial trace equal to the prescribed electronic density. We introduce a relaxation of this quantum optimal transport where the full partial trace constraint is replaced by a finite number of moment constraints on the partial trace of the set of operators. We show that, under mild assumptions on the electronic density, there exist sparse minimizers to the moment-constrained approximation of the Lieb (MCAL) functional that read as operators with rank at most equal to the number of moment constraints. We also prove under appropriate assumptions on the set of moment functions that the value of the MCAL functional converges to the value of the exact Lieb functional as the number of moments go to infinity. Finally, we show that a semi-classical limit holds, namely MCAL \Gamma-converges to the moment constraints multi-marginal optimal transport.This is a joint work with Virginie Ehrlacher.

A dynamical system perspective of optimization in data science

• Jalal Fadili

Room: Salle René Baire (4th floor) In this talk, I will discuss and introduce deep insight from  the dynamical system perspective to understand the convergence guarantees of first-order algorithms involving inertial features for convex optimization in a Hilbert space setting. Such algorithms are widely popular in various areas of data science (data processing, machine learning, inverse problems, etc.). They can be viewed discrete as time versions of an inertial second-order dynamical system involving different types of dampings (viscous  damping,  Hessian-driven geometric damping). The dynamical system perspective offers not only a powerful way to understand the geometry underlying the dynamic, but also offers a versatile framework to obtain fast, scalable and new algorithms enjoying  nice convergence guarantees (including fast rates). In addition, this framework encompasses known algorithms and dynamics such as the Nesterov-type accelerated gradient methods, and the introduction of time scale factors makes it possible to further accelerate these algorithms. The framework is versatile enough to handle non-smooth and non-convex objectives that are ubiquituous in various applications.

Mean Field Games planning problems with general initial and final measures

• Daniela Tonon

Room: Salle René Baire (4th floor) The planning problem in Mean Field Games (MFG) was introduced by P.-L. Lions in his lessons, to describe models in which a central planner would like to steer a population to a predetermined final configuration while still allowing individuals to choose their own strategies. In a recent variational approach, see Graber, Mészáros, Silva and Tonon (2019) and Orrieri, Porretta and Savaré (2019) the authors studied the well-posedness of this problem in case of merely summable initial and final measures, using techniques, coming from optimal transport, introduced by Benamou and Brenier in 2000, extended to the congestion case in Carlier, Cardaliaguet and Nazaret (2013), and already used to show the existence and uniqueness of weak solutions for classical MFGs by Cardaliaguet and collaborators. The case of less regular initial and final measures is now studied via techniques introduced by Jimenez in 2008, for the analogous problem in optimal transport.

Convergence rate of general entropic optimal transport costs

• Guillaume Carlier

Room: Salle René Baire (4th floor) In recent years, the entropic approximation of the optimal transport problem has received a lot of attention because of its connection with the popular Sinkhorn scaling algorithm. In this talk, I will give a sharp convergence rate of convergence for the value of the entropic  cost to the optimal transport cost as the entropic parameter becomes small. The class of transport costs considered is quite general and covers situations where there is no Monge solution. Upper bounds will be obtained by a block approximation strategy and a refinement of Alexandrov’s theorem. The (matching) lower bound will be obtained, under a nondegeneracy condition, by a quadratic detachment bound derived from Minty’s trick. This is a joint work with Paul Pegon and Luca Tamanini.

Sur la commandabilité des systèmes à dérive périodique

• Jean-Baptiste Caillau

Room: Salle René Baire (4th floor) En 1981, B. Bonnard publiait dans une note aux CRAS un résultat remarquable sur la commandabilité des systèmes non-linéaires en dimension finie, affine en le contrôle et à dérive positivement Poisson stable : sous l'hypothèse de crochets habituelle (rang maximal de l'algèbre de Lie engendrée par les champs de vecteurs contrôlés et la dérive), et à condition que le convexifié de l'ensemble des valeurs admissibles pour le contrôle soit un voisinage de l'origine, le système est commandable. L'objet de cette présentation est l'extension de ce résultat au cas où la dérive est périodique mais où l'ensemble des contraintes pour le contrôle est un convexe dont l'intérieur ne contient plus l'origine. On propose une condition suffisante pour préserver la commandabilité entre fibres de l'espace d'état. Ce travail est motivé par l'étude des voiles solaires (projet avec l'ESA). Travail en collaboration avec A. Herasimenka, L. Dell'Elce et J.-B. Pomet. Références. Bonnard, B. Contrôlabilité des systèmes non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 535–5. Caillau, J.-B.; Dell'Elce, L.; Herasimenka, A.; Pomet, J.-B. On the controllability of nonlinear systems with a periodic drift. HAL preprint no. 03779482 (2023). Herasimenka, A.; Dell'Elce, L.; Caillau, J.-B.; Pomet, J.-B. Controllability properties of solar sails. J. Guidance Control Dyn. 46 (2023), no. 5, 900-909.

Pattern Recovery in Penalized Estimation and its Geometry

• Ulrike Schneider

Room: Salle René Baire (4th floor) We consider the framework of penalized least-squares estimation where the penalty term is given by a real-valued polyhedral gauge, which encompasses methods such as LASSO (and many variants thereof), SLOPE, OSCAR, PACS and others. Each of these estimators can uncover a different structure or ``pattern'' of the unknown parameter vector. We define a general notion of patterns for a penalized procedure based on subdifferentials and provide a necessary condition for a particular pattern to be detected with positive probability, the so-called accessibility condition. We also introduce the stronger noiseless recovery condition which is shown to be necessary for pattern recovery with probability larger than $1/2$, thereby generalizing the irrepresentability condition of the LASSO to a general framework. We show that the noiseless recovery condition can be relaxed when turning to thresholded penalized estimators, generalizing the idea of the thresholded LASSO: we prove that the accessibility condition is already sufficient for sure pattern recovery by thresholded penalized estimation provided that the signal of the pattern is large enough. We also discuss how our findings can be interpreted through a geometrical lens.

Fast continuous and discrete time methods for convex optimization problems with linear constraints

• Radu Ioan Bot

Room: Salle René Baire (4th floor) In this talk we address the minimization of a continuously differentiable convex function under linear equality constraints. We consider a second-order dynamical system with asymptotically vanishing damping term formulated in terms of the Augmented Lagrangian associated with the minimization problem. Time discretization leads to an inertial algorithm with a general rule for the inertial parameters that covers the classical ones by Nesterov, Chambolle-Dossal and Attouch-Cabot used in the context of fast gradient methods. In both settings we prove fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectory/iterates, and also weak convergence of the primal-dual trajectory/iterates to a primal-dual optimal solution. For the unconstrained minimization of a convex differentiable function we rediscover all convergence statements obtained in the literature for Nesterov’s accelerated gradient method. In addition, an alternative approach relying on the fast solving of monotone equations will be presented.

Control and machine learning

• Enrique Zuazua

Room: Salle René Baire (4th floor) We present some recent results on the interplay between control and Machine Learning. We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets) and present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies. This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. The bridge towards optimal transport will also be discussed. This lecture is inspired in joint work, among others, with Borjan Geshkovski (MIT), Carlos Esteve (Cambridge), Domènec Ruiz-Balet (IC, London), Dario Pighin (Sherpa.ai) and Martin Hernández (FAU).

Numerical reconstruction of the fluid flow from local measurements of the velocity

• Muriel Boulakia

Room: Salle René Baire (4th floor) In this talk, we will consider different PDE models in fluid mechanics and present a numerical method for the reconstruction of the velocity and the pressure from local measurements of the velocity. This method which has consistency properties is based on the stabilization of the discretized Finite Element formulation of the equation. We will present results on the analysis of the reconstruction error which rely on the quantification of the unique continuation property and illustrate this method for the reconstruction of the blood flow in a vessel.

Statistical insights on PINNs

• Claire Boyer

Room: Salle René Baire (4th floor) 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 food for thought and statistical insight into the proper use of PINNs.