Search

In increasingly different contexts, it happens that a human player has to interact with artificial players who make decisions following decision-making algorithms. How should the human player play against these algorithms to maximize his utility? Does anything change if he faces one or more artificial players? The main goal of the paper is to answer these two questions. Consider n-player games in normal form repeated over time, where we call the human player optimizer, and the (n -- 1) artificial players, learners. We assume that learners play no-regret algorithms, a class of algorithms widely used in online learning and decision-making. In these games, we consider the concept of Stackelberg equilibrium. In a recent paper, Deng, Schneider, and Sivan have shown that in a 2-player game the optimizer can always guarantee an expected cumulative utility of at least the Stackelberg value per round. In our first result, we show, with counterexamples, that this result is no longer true if the optimizer has to face more than one player. Therefore, we generalize the definition of Stackelberg equilibrium introducing the concept of correlated Stackelberg equilibrium. Finally, in the main result, we prove that the optimizer can guarantee at least the correlated Stackelberg value per round. Moreover, using a version of the strong law of large numbers, we show that our result is also true almost surely for the optimizer utility instead of the optimizer's expected utility.

Resolvents of operators are the core of many fundamental algorithms used in optimization. However their computation is in general difficult except for very particular operators. In the paper we provide a new simple algorithm with linear convergence rate to compute the resolvents for the class of operators which can be decomposed as a sum of a maximally monotone operator with a computable resolvent and a single-valued locally Lipschitz continuous mapping.

The goal of this paper is to investigate the influence of the waning immunity on the dynamics of Tilapia Lake Virus (TiLV) transmission in wild and farmed tilapia within freshwater. We formulate a model for which susceptible individuals can contract the disease in two ways: (i) direct mode caused by contact with infected individuals; (ii) indirect mode due to the presence of pathogenic agents in the water. We obtain an age-structured model which combines both age since infection and age since recovery. We derive an explicit formula for the reproductive number $ \mathcal {R}_0 $ and show that the disease-free equilibrium is locally asymptotically stable when, $ \mathcal {R}_0 \lt 1 $ . We discuss on the form of the waning immunity parameter and show numerically that a Hopf bifurcation may occur for suitable immunity parameter values, which means that there is a periodic solution around the endemic equilibrium when, $ \mathcal {R}_0 \gt 1 $ .

We optimize a general model of bioprocesses, which is nonconvex due to the microbial growth in the biochemical reactors. We formulate a convex relaxation and give conditions guaranteeing its exactness in both the transient and steady state cases. When the growth kinetics are modeled by the Monod function under constant biomass or the Contois function, the relaxation is a second-order cone program, which can be solved efficiently at large scales. We implement the model on a numerical example based on a wastewater treatment system.

Version v2 après acceptation du document par la conférence de la prépublication hal-03938311; International audience; Mathematical models of cancer growth and evolution of cancer cell characteristics, aka traits or phenotypes, together with optimisation and optimal control methods to contain them, in the framework of adaptive dynamics of cell populations, are presented. They take into account the heterogeneity of cancer cell populations, i.e., their biological variability, and their intrinsic plasticity, i.e., their nongenetic instability that allows them to quickly adapt to changing environments. The presented vision of the cancer disease, which is specific to multicellular organisms, relies on a relatively novel vision, consistent with a billion-year evolutionary perspective. Based on recent contributions from philosophy of cancer, these mathematical models aim at designing theoretical therapeutic strategies to simultaneously contain tumour progression and limit adverse events of drugs to healthy cell populations.

International audience; The coffee berry borer (CBB) Hypothenemus hampei (Coleoptera: Scolytidae) is the most important insect pest affecting coffee production worldwide and generating huge economic losses. As most of its life cycle occurs inside the coffee berry, its control is extremely difficult. To tackle this issue, we solve an optimal control problem based on a berry age-structured dynamical model that describes the infestation dynamics of coffee berries by CBB during a cropping season. This problem consists in applying a bio-insecticide at discrete times in order to maximise the economic profit of healthy coffee berries while minimising the CBB population for the next cropping season. We derive analytically the first-order necessary optimality conditions of the control problem. Numerical simulations are provided to illustrate the effectiveness of the optimal control strategy.

The problem of minimizing a polynomial over a set of polynomial inequalities is an NP-hard non-convex problem. Thanks to powerful results from real algebraic geometry, one can convert this problem into a nested sequence of finite-dimensional convex problems. At each step of the associated hierarchy, one needs to solve a fixed size semidefinite program, which can be in turn solved with efficient numerical tools. On the practical side however, there is no-free lunch and such optimization methods usually encompass severe scalability issues. Fortunately, for many applications, we can look at the problem in the eyes and exploit the inherent data structure arising from the cost and constraints describing the problem, for instance sparsity or symmetries. This book presents several research efforts to tackle this scientific challenge with important computational implications, and provides the development of alternative optimization schemes that scale well in terms of computational complexity, at least in some identified class of problems. The presented algorithmic framework in this book mainly exploits the sparsity structure of the input data to solve large-scale polynomial optimization problems. We present sparsity-exploiting hierarchies of relaxations, for either unconstrained or constrained problems. By contrast with the dense hierarchies, they provide faster approximation of the solution in practice but also come with the same theoretical convergence guarantees. Our framework is not restricted to static polynomial optimization, and we expose hierarchies of approximations for values of interest arising from the analysis of dynamical systems. We also present various extensions to problems involving noncommuting variables, e.g., matrices of arbitrary size or quantum physic operators., Comment: 220 pages, to appear in Series on Optimization and Its Applications, World Scientific Press

Technologies for 3D data acquisition and 3D printing have enormously developed in the past few years, and, consequently, the demand for 3D virtual twins of the original scanned objects has increased. In this context, feature-aware denoising, hole filling and context-aware completion are three essential (but far from trivial) tasks. In this work, they are integrated within a geometric framework and realized through a unified variational model aiming at recovering triangulated surfaces from scanned, damaged and possibly incomplete noisy observations. The underlying non-convex optimization problem incorporates two regularisation terms: a discrete approximation of the Willmore energy forcing local sphericity and suited for the recovery of rounded features, and an approximation of the $$\ell _0$$ ℓ 0 pseudo-norm penalty favouring sparsity in the normal variation. The proposed numerical method solving the model is parameterization-free, avoids expensive implicit volume-based computations and based on the efficient use of the Alternating Direction Method of Multipliers. Experiments show how the proposed framework can provide a robust and elegant solution suited for accurate restorations even in the presence of severe random noise and large damaged areas.

The purpose of this article is to underline the links between some no-regret algorithms used in on-line learning, games and convex optimization and to compare the continuous and discrete time versions.

International audience; Denoising, detrending, deconvolution: usual restoration tasks, traditionally decoupled. Coupled formulations entail complex ill-posed inverse problems. We propose PENDANTSS for joint trend removal and blind deconvolution of sparse peak-like signals. It blends a parsimonious prior with the hypothesis that smooth trend and noise can somewhat be separated by low-pass filtering. We combine the generalized quasi-norm ratio SOOT/SPOQ sparse penalties $\ell_p/\ell_q$ with the BEADS ternary assisted source separation algorithm. This results in a both convergent and efficient tool, with a novel Trust-Region block alternating variable metric forward-backward approach. It outperforms comparable methods, when applied to typically peaked analytical chemistry signals. Reproducible code is provided.

We look at a stochastic time-varying optimization problem and we formulate online algorithms to find and track its optimizers in expectation. The algorithms are derived from the intuition that standard prediction and correction steps can be seen as a nonlinear dynamical system and a measurement equation, respectively, yielding the notion of nonlinear filter design. The optimization algorithms are then based on an extended Kalman filter in the unconstrained case, and on a bilinear matrix inequality condition in the constrained case. Some special cases and variations are discussed, notably the case of parametric filters, yielding certificates based on LPV analysis and, if one wishes, matrix sum-of-squares relaxations. Supporting numerical results are presented from real data sets in ride-hailing scenarios. The results are encouraging, especially when predictions are accurate, a case which is often encountered in practice when historical data is abundant., 22 pages

We consider the controllability of a class of systems of n Stokes equations, coupled through terms of order zero and controlled by m distributed controls. Our main result states that such a system is null-controllable if and only if a Kalman type condition is satisfied. This generalizes the case of finite-dimensional systems and the case of systems of coupled linear heat equations. The proof of the main result relies on the use of the Kalman operator introduced in [1] and on a Carleman estimate for a cascade type system of Stokes equations. Using a fixed-point argument, we also obtain that if the Kalman condition is verified, then the corresponding system of Navier-Stokes equations is locally null-controllable.

In this paper we study obstacles immerged in a Stokes flow with Navier boundary conditions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension 1. The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation (SBD). In dimension 2, we prove that the solution is classical.

In this article we study the null controllability of some abstract linear parabolic systems in tensor product spaces. This special structure allows us to reduce our controllability problem to a particular set of equations that looks like a moment problem, but that does not fall into the previous existing results of the literature. We transform this non standard moment problem into an infinite family of more usual moment problems, yet coupled one from each other. This reformulation is done with enough care to ensure that the resulting set of equations can be solved, with suitable estimates, by using the recent "block moment method". This is based on a careful analysis of the spectral structure of the underlying operator. We notably apply our abstract result to show how strong the influence of geometry can be: we provide an example of boundary controlled parabolic system on a rectangle domain which is null controllable in arbitrary small time if two perpendicular faces of the boundary are controlled, whereas it is never null controllable if the control acts on only one face., Dans cet article nous étudions la contrôlabilité à zéro d'une classe de systèmes paraboliques linéaires abstraits dans des produits tensoriels. Cette structure particulière nous permet de réduire la question de la contrôlabilité à un ensemble particulier d'équations qui ressemble à un problème de moments, mais qui ne relève pas des résultats existants de la littérature. Nous transformons ce problème de moments non standard en une famille infinie de problèmes de moments usuels, mais couplés entre eux. Cette reformulation est choisie avec soin pour que le nouveau système obtenu puisse être résolu, avec des bonnes estimations des solutions, en utilisant la méthode des moments par blocs développée récemment. Tout ce travail est basé sur une analyse spectrale minutieuse de l'opérateur sous-jacent. Nous appliquons notamment ce résultat abstrait pour montrer que la position du domaine de contrôle pour un problème de contrôle au bord de deux équations de la chaleur couplées peut être déterminante: nous donnons un exemple explicite d'un tel système posé sur un domaine rectangulaire en 2D qui est contrôlable à zéro en tout temps si le contrôle agit sur deux bords perpendiculaires du domaine mais qui n'est jamais contrôlable à zéro si le contrôle n'agit que sur un seul des bords du domaine.

International audience; Minimizing the weighted number of tardy jobs {on one machine} is a classical and intensively studied scheduling problem. In this paper, we develop a two-stage robust approach, where exact weights are known after accepting to perform the jobs, and before sequencing them on the machine. This assumption allows diverse recourse decisions to be taken in order to better adapt one's mid-term plan.The contribution of this paper is twofold: first, we introduce a new scheduling problem and model it as a min-max-min optimization problem with mixed-integer recourse by extending existing models proposed for the deterministic case. Second, we take advantage of the special structure of the problem to propose twosolution approaches based on results from the recent robust optimization literature: namely the finite adaptability (Bertsimas and Caramanis, 2010) and a convexification-based approach (Arslan and Detienne, 2022). We also study the additional cost of the solutions if the sequenceof jobs has to be decided before the uncertainty is revealed.Computational experiments are reported to analyze the effectiveness of our approaches.

The exact distributed controllability of the semilinear heat equation $\partial_{t}y-\Delta y + g(y)=f \,1_{\omega}$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^\prime$ does not grow faster than $\beta \ln^{3/2}\vert r\vert$ at infinity for $\beta>0$ small enough and that $g^\prime$ is uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations., Comment: arXiv admin note: text overlap with arXiv:2008.12656, arXiv:2101.06446

Consider a time horizon and a set of possible states for a given system. The system must be in exactly one state at a time. In this paper, we generalize classical results on min-up/min-down constraints for a 2-state system to an n-state system. The minimum-time constraints enforce that if the system switches to state i at time t, then it must remain in state i for a minimum number of time steps. The minimum-time polytope is defined as the convex hull of integer solutions satisfying the minimum-time constraints. A variant of minimum-time constraints is also considered, namely the no-spike constraints. They enforce that if state i is switched on at time t, the system must remain on states j ≥ i during a minimum time. Symmetrically, they enforce that if state i is switched off at time t, the system must remain on states j < i during a minimum time. The no-spike polytope is defined as the convex hull of integer solutions satisfying the no-spike constraints. For both the minimum-time polytope and the no-spike polytope, we introduce families of valid inequalities. We prove that these inequalities lead to a complete description of linear size for each polytope.

We consider the problem of jointly optimizing the daily production planning and energy supply management of an industrial complex, with manufacturing processes, renewable energies and energy storage system. It is naturally formulated as a mixed-integer multistage stochastic problem. This problem is challenging for three main reasons: there is a large number of time steps (typically 24), renewable energies are uncertain and uncontrollable, and we need binary variables modeling hard constraints. We discuss various solution strategies, in particular Model Predictive Control, Dynamic Programming, and heuristics based on the Stochastic Dual Dynamic Programming algorithm. We compare these strategies on two variants of the problem: with or without day-ahead energy purchases.

Let M be a σ-compact C^∞ manifold of dimension n ≥ 2 and consider a single-input control system: ẋ(t) = X (x(t)) + u(t) Y (x(t)), where X , Y are C^∞ vector fields on M. We prove that there exist an open set of pairs (X , Y ) for the C^∞ –Whitney topology such that they admit singular abnormal rays so that the spectrum of the projective singular Hamiltonian dynamics is feedback invariant. It is applied to controlled Lotka–Volterra dynamics where such rays are related to shifted equilibria of the free dynamics.

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision / consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [12]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman-Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.

Assessing non-negativity of multivariate polynomials over the reals, through the computation of {\em certificates of non-negativity}, is a topical issue in polynomial optimization. This is usually tackled through the computation of {\em sums-of-squares decompositions} which rely on efficient numerical solvers for semi-definite programming. This method faces two difficulties. The first one is that the certificates obtained this way are {\em approximate} and then non-exact. The second one is due to the fact that not all non-negative polynomials are sums-of-squares. In this paper, we build on previous works by Parrilo, Nie, Demmel and Sturmfels who introduced certificates of non-negativity modulo {\em gradient ideals}. We prove that, actually, such certificates can be obtained {\em exactly}, over the rationals if the polynomial under consideration has rational coefficients and we provide {\em exact} algorithms to compute them. We analyze the bit complexity of these algorithms and deduce bit size bounds of such certificates., Comment: 24 pages, 2 tables

We analyze the performance of Liquidity Providers (LPs) providing liquidity to different types of Automated Market Makers (AMMs). This analysis is carried out using a mean / standard deviation viewpoint \`a la Markowitz, though based on the PnL of LPs compared to that of agents holding coins outside of AMMs. We show that LPs tend to perform poorly in a wide variety of CFMMs under realistic market conditions. We then explore an alternative AMM design in which an oracle feeds the current market exchange rate to the AMM which then quotes a bid/ask spread. This allows us to define an efficient frontier for the performance of LPs in an idealized world with perfect information and to show that the smart use of oracles greatly improves LPs' risk / return profile, even in the case of a lagged oracle.

In the last couple of years, performances of light soaring aircraft (para gliders, hang gliders, or light sailplanes) have increased significantly, allowing pilots to fly great distances using only the convective energy of the atmosphere. This activity, called "Cross-Country flying" or "soaring", requires topological and aerological knowledge and a lot of pre-flight preparation in order to make the right decisions, thus maximizing the flying distance or minimizing the flying time in a race. To optimize these flights, a pre-tactical decision support tool has been developed. This tool results from the adaptation of the Fast Marching Tree Star algorithm. It is extended in such a way it can deal with differential constraints associated with light soaring aircraft, operating in a convective atmosphere, in a field of wind, and close to reliefs. The method has been tested and validated on a real competition light soaring aircraft.

International audience; The study of synchronization in coupled genetic networks is a very challenging topic that is usually analyzed on a case-by-case basis. Here we consider a general model of genetic networks and examine two forms of interconnection, either homogeneous or heterogeneous coupling, corresponding to coupling functions that are either equal or different from those governing the individual dynamics. In the case of individual subsystems having unique but different steady states, we prove that the homogeneous coupled system has a unique globally asymptotically stable steady state. Moreover, in the case of large coupling strength, we show that under suitable assumptions the network achieves weak synchronization in the sense that the individual steady states become arbitrarily close. In the heterogeneous case, stability conditions are more intricate and some stronger assumptions on the individual dynamics have to be made, under which we prove a similar weak synchronization result in the case of large coupling strength. We apply the results to the synchronization of damped oscillators and to the control of multistable systems.

This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca's friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular the friction threshold involved in the Tresca's friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally numerical simulations are provided in order to illustrate our main result.

International audience; Attractivity specifications consist in driving the state of a system to a target region and to keep it in that region afterwards. In this paper, we develop an approach to controller synthesis for attractivity specifications based on iterative refinements of symbolic abstractions. We compute iteratively sequences of least-violating attractivity controllers and of associated attractors. The current controller is used to restrict possible control actions at the next iteration where a new abstraction is used following a local refinement applied inside the current attractor. The approach results in a nested sequence of attractors which are closer to the target region at each iteration. To illustrate the effectiveness of our approach, we show an application to a marine vessel system.

International audience; Although Model Predictive Control (MPC) has proven its e ciency in the process operation and it is known for its high performance, it also su ers from design limitation. One of the fundamental issues for such an optimization-based technique is the di culty to guarantee recursive feasibility in the absence of terminal constraints, or alternatively, the complexity of design when a certain basin of a raction or a controlled invariant set needs to be certi ed for the closed loop in the most general nonlinear se ing. Based on symbolic control techniques, this paper proposes a simple and guaranteed solution for such problems. As a main result, a Symbolically guided Model Predictive Control scheme is developed. is controller is an improved version of the generic MPC approach such that the recursive feasibility is guaranteed through appending time-varying terminal constraints, carefully designed using the symbolic control approach, to the optimization problem of the original MPC formulation.

We consider a smooth control system that is subject to loss of control in the sense that the state space is partitioned into several disjoint regions and, in each region, either the system can be controlled, as usual, in a permanent way (that is, one can change the value of the control at any real time), or, on the contrary, the control has to remain constant from the entry time into the region until the exit time. The latter case corresponds to a non-control region. The objective of this paper is to state the necessary optimality conditions for a Mayer optimal control problem in such a setting of loss of control. Our main result is based on a hybrid maximum principle that takes into account a regionally switching parameter.

A quasi-boundary method is used to study an ill-posed, time-fractional diffusion equation involving the fractional Riemann-Liouville derivative. In particular, we consider an ill-posed problem for a family of well-posed problems, and prove, by means of eigenfunction expansions, that the solutions of the latter problems converge to the solutions associated with the former problem. The analysis presented includes providing conditions for the rate of the convergence.

In [14] Duca and Nersesyan proved a small-time controllability property of nonlinear Schrödinger equations on a d-dimensional torus $\mathbb{T}^d$. In this paper we study a similar property, in the linear setting, starting from a closed Riemannian manifold. We then focus on the 2-dimensional sphere $S^2$, which models the bilinear control of a rotating linear top: as a corollary, we obtain the approximate controllability in arbitrarily small times among particular eigenfunctions of the Laplacian of $S^2$., 10 pages, 1 figure

The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation H ( x , ∇ u ) = 0 , H(x,\nabla u)=0, coupled with non-zero Dirichlet condition u = g u=g on ∂ Ω \partial \Omega . In this article, the Hamiltonian H H is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.

We investigate inexact proximity operators for weakly convex functions. To this aim, we derive sum rules for proximal ε-subdifferentials, by incorporating the moduli of weak convexity of the functions into the respective formulas. This allows us to investigate inexact proximity operators for weakly convex functions in terms of proximal ε-subdifferentials.

The study of the optimal transport problem allows to define metrics on spaces of probability measures that come with strong geometrical and physical interpretations. This problem and its metrics (called Wasserstein distances) have found many applications, ranging from quantum physics, fluid dynamics, optics design, economics and statistics, to machine learning. The profound physical rooting of the optimal transport problem calls for the study of its well-posedness.While the existence and uniqueness of solutions to the optimal transport problem have been extensively studied in previous works, the investigation of its stability is less advanced. General and abstract stability results ensure that optimal transport solutions change continuously with the problem data. Yet these results are hardly ever quantitative, which is problematic in applications, where the data is oten approximated. This thesis works towards closing this gap. Focusing in a Euclidean context on the quadratic optimal transport problem, we derive in a first part strong convexity estimates for the Kantorovich dual problem relying on well-known geometric and functional inequalities.We then collect in a second part quantitative stability estimates for optimal transport maps with respect to their target measure, for Wasserstein barycenters with respect to their marginals and for Schrödinger potentials with respect to the temperature parameter in semi-discrete optimal transport. These estimates all suggest natural applications in computational and statistical optimal transport. They also give new insights on the embeddability of the 2-Wasserstein space in a Hilbert space with a controlled distorsion.In a last part, we leverage this last idea in machine learning applications and propose approaches to approximately solve K-means and dictionary learning problems in the 2-Wasserstein space., L'étude du problème de transport optimal permet de définir des métriques sur des espaces de mesures de probabilité qui ont de fortes interprétations physiques et géométriques. Ce problème et ses métriques (appelées distances de Wasserstein) ont trouvé de nombreuses applications, allant de la physique quantique, la mécanique des fluides, la conception optique, l'économie et les statistiques, à l'apprentissage automatique. Le profond ancrage physique du problème de transport optimal appelle à l'étude de son caractère bien posé.Si l'existence et l'unicité des solutions du problème de transport optimal ont été largement étudiées dans des travaux antérieurs, l'étude de sa stabilité est moins avancée. Des résultats de stabilité généraux et abstraits garantissent que les solutions de transport optimal dépendent continûment des données du problème qui les définit. Cependant, ces résultats ne sont presque jamais quantitatifs, ce qui est problématique dans les applications, où les données sont souvent disponibles de manière approximative. Cette thèse vise à combler cette lacune. En se concentrant dans un contexte euclidien sur le problème de transport optimal quadratique, nous donnons dans une première partie des estimations de la forte convexité du problème dual de Kantorovich en nous appuyant sur des inégalités géométriques et fonctionnelles bien connues.Nous rassemblons ensuite dans une deuxième partie des estimations quantitatives de la stabilité des applications de transport optimal par rapport à leur mesure cible, des barycentres de Wasserstein par rapport à leurs mesures marginales et des potentiels de Schrödinger par rapport au paramètre de température dans le transport optimal semi-discret. Ces estimations suggèrent toutes des applications naturelles en transport optimal numérique et statistique. Elles donnent également de nouvelles indications sur la plongeabilité de l'espace de Wasserstein-2 dans un espace de Hilbert avec une distorsion de la métrique contrôlée.Dans une dernière partie, nous exploitons cette dernière idée dans des applications d'apprentissage automatique et proposons des approches pour résoudre approximativement des problèmes de K-moyennes et d'apprentissage de dictionnaire dans l'espace Wasserstein-2.

International audience; The simultaneous proliferation of electric vehicles and intermittent renewable energy sources promises to expedite decarbonization of two sectors with highest emissions. However, both these developments threaten to endanger power system stability, which may hinder their widespread adoption. Totackle these challenges, there is a need for joint modelling of electric vehicle traffic flows, together with their battery dynamics. We propose a macroscopic electromobility model, augmenting the LWR model, describing the traffic dynamics, with an inhomogeneous advection equation, describing the evolution of vehicles’ State of Charge (SoC). The Riemann problem for the joint model is solved for the case of triangular fundamental diagram, and the solutions are used to formulate a Godunov-like scheme for model discretization. Additionally, we propose an advection-based charging station model, discretize it, and link it with the rest of the traffic and SoC model. We demonstrate the capabilities and use of the full coupled model by proposing a pedagogic example where a simple control law regulates the average SoC of all vehicles on a ring road by controlling the traffic flow entering the charging station.

We present a new method to invert variable stress changes of fractures from InSAR ground displacements. Fractures can be either faults or magma intrusions, embeded in a 3D heterogeneous crust with prominent topographies. The method is based on a fictituous domain approach using a finite element discretization of XFEM type. A cost function involves the misfit between the solution of the physical problem and the observed data together with the smoothing terms. Regularization parameters are determined by using L-curves. The method is then reformulated to be applied to InSAR data (masked and noisy), projected in Earth-Satellite directions. Synthetic tests confirm the efficiency and effectiveness of our method.

International audience; We consider a control problem for a heterogeneous population composed of customers able to switch at any time between different contracts, depending not only on the tariff conditions but also on the characteristics of each individual. A provider aims to maximize an average gain per time unit, supposing that the population is of infinite size. This leads to an ergodic control problem for a "mean-field" MDP in which the state space is a product of simplices, and the population evolves according to a controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we show that the ergodic eigenproblem admits a solution. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. We illustrate this approach on examples from electricity pricing, and show in particular that the optimal policies may be cyclic-alternating between discount and profit taking stages.

International audience; Contract theory is a powerful tool to reason on systems that are interacting with an external environment, possibly made of other systems. Formally, a contract is usually given by assumptions and guarantees, which specify the expected behavior of the system (the guarantees) in a certain context (the assumptions). In this work, we present a verification framework for discrete-time dynamical systems with assume-guarantee contracts. We first introduce a class of assume-guarantee contracts with their satisfaction semantics parameterized by a time-horizon over which assumptions are evaluated. We then show that the problem of verifying whether such contracts are satisfied is equivalent to show the existence of a positive invariant set for an auxiliary system. This allows us to leverage the extensive literature on invariant set computation. A simple illustrative example is provided to show the effectiveness of our approach.

This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. After stating the Max-Min-Max or Max-Singular-Max form of the optimal control deduced from the Pontryagin Maximum Principle, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc.

This article revolves around a recent numerical framework for shape and topology optimization, which features an exact mesh of the shape at each iteration of the process, while still leaving the room for an arbitrary evolution of the latter (including changes in its topology). In a nutshell, two complementary representations of the shape are combined: on the one hand, it is meshed exactly, which allows for precise mechanical calculations based on the finite element method; on the other hand, it is described implicitly, using the level set method, which makes it possible to track its evolution in a robust way. In the first part of this work, we overview the main aspects of this numerical strategy. After a brief presentation of some necessary background material-related to shape optimization and meshing, among others-we describe the numerical schemes involved, notably when it comes to the practice of the level set method, the remeshing algorithms, and the considered optimization solver. This strategy is illustrated with 2d and 3d numerical examples in various physical contexts. In the second part of this article, we propose a simple albeit efficient python-based implementation of this framework. The code is described with a fair amount of details, and it is expected that the reader can easily elaborate upon the presented examples to tackle his own problems.

In this paper, we present a new strategy to prove the convergence of deep learning architectures to a zero training (or even testing) loss by gradient flow. Our analysis is centered on the notion of Rayleigh quotients in order to prove Kurdyka-{\L}ojasiewicz inequalities for a broader set of neural network architectures and loss functions. We show that Rayleigh quotients provide a unified view for several convergence analysis techniques in the literature. Our strategy produces a proof of convergence for various examples of parametric learning. In particular, our analysis does not require the number of parameters to tend to infinity, nor the number of samples to be finite, thus extending to test loss minimization and beyond the over-parameterized regime., Comment: Published at the 36th conference on Neural Information Processing Systems (NeurIPS 2022)

International audience; Contact dynamics relies on the simultaneous satisfaction of constraints at the robot body level and at the contact level. At both levels, various formulations can be chosen that all must lead to the same results, given the same hypothesis, hence the little importance of their details. Yet when using it in an optimal control problem, a particular formulation is often imposed by the task to be performed by the robot. In this paper, we detail the formulation of the contact quantities (force, movement) in an arbitrary frame imposed by the task. In that case, we will show that we are typically not interested in working in the local frame (attached to the robot contact point), nor in the world frame, but in a user-defined frame centered at the contact location with a fixed orientation in the world. The derivations can then be used for 6D, 3D or normal (pure-sliding) contact. We implemented the corresponding derivatives on top of the contact dynamics of Pinocchio in the optimal control solver Crocoddyl. We show that a unique formulation is able to handle several operational orientations by achieving several surfacing tasks in model predictive control with the robot Talos.

In this article, the Lotka-Volterra model is analyzed to reduce the infection of a complex microbiote. The problem is set as an optimal control problem, where controls are associated to antibiotic or probiotic agents, or transplantations and bactericides. Candidates as minimizers are selected using the Maximum Principle and the closed loop optimal solution is discussed. In particular a 2d-model is constructed with 4 parameters to compute the optimal synthesis using homotopies on the parameters.

Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds. With the aim of achieving such a unification program, in this paper we initiate the study of gauge metric measure spaces., 148 pages

This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate discrete convex shapes and is easily implementable using standard optimization software. The framework can handle various objective functions ranging from geometric quantities to functionals depending on partial differential equations. Width or diameter constraints are handled using the support function. Functionals depending on a convex body and its polar body can be handled using a unified framework.

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: ∂ t u − ∑ i , j = 1 N a i , j ( t , x ) ∂ i j u − ∑ i = 1 N q i ( t , x ) ∂ i u = f ( t , x , u ) . \begin{equation*} \partial _{t} u - \sum _{i,j=1}^N a_{i,j}(t,x)\partial _{ij}u-\sum _{i=1}^N q_i(t,x)\partial _i u=f(t,x,u). \end{equation*} These are concerned with the dynamics of the solution starting from initial data with compact support. The nonlinearity f f is of Fisher-KPP type, and admits 0 0 as an unstable steady state and 1 1 as a globally attractive one (or, more generally, admits entire solutions p ± ( t , x ) p^\pm (t,x) , where p − p^- is unstable and p + p^+ is globally attractive). Here, the coefficients a i , j , q i , f a_{i,j}, q_i, f are only assumed to be uniformly elliptic, continuous and bounded in ( t , x ) (t,x) . To describe the spreading dynamics, we construct two non-empty star-shaped compact sets S _ ⊂ S ¯ ⊂ R N \underline {\mathcal {S}}\subset \overline {\mathcal {S}} \subset \mathbb {R}^N such that for all compact set K ⊂ i n t ( S _ ) K\subset \mathrm {int}(\underline {\mathcal {S}}) (resp. all closed set F ⊂ R N ∖ S ¯ F\subset \mathbb {R}^N\backslash \overline {\mathcal {S}} ), one has lim t → + ∞ sup x ∈ t K | u ( t , x ) − 1 | = 0 \lim _{t\to +\infty } \sup _{x\in tK} |u(t,x)-1| = 0 (resp. lim t → + ∞ sup x ∈ t F | u ( t , x ) | = 0 \lim _{t\to +\infty } \sup _{x\in tF} |u(t,x)| =0 ). The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {S}} and to establish an exact asymptotic speed of propagation in various frameworks. These include: almost periodic, asymptotically almost periodic, uniquely ergodic, slowly varying, radially periodic and random stationary ergodic equations. In dimension N N , if the coefficients converge in radial segments, again we show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {S}} and this set is characterized using some geometric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets.

International audience; Optical metasurfaces are becoming ubiquitous optical components to mold the amplitude, the phase and the polarization properties of light beams. So far, most of these devices are passive in essence, that is, they cannot be arbitrarily reconfigured or optimized according to the user's interest and/or change in their surrounding environment. Here we propose an innovative design strategy relying on the position of topological singularities, namely zeros and poles of the reflection coefficient, to address full phase modulation of light reflecting off an active metasurface with almost unity efficiency. The active metasurface unit cells, consisting of asymmetric Gires-Tournois resonators filled with electro-optics materials comprising either of Liquid crystal or multiple quantum wells, are able to modulate the reflected field from 0 to 2π associated with 100% reflection amplitude despite dealing with extremely low refractive index change on the order of 0.01. The second ingredient to achieve high efficiency modulation consists in optimizing the arrangement of the sub-wavelength unit-cells so as to account for local imperfection and near-field coupling between neighboring structures. Consequently, an advanced optimization method accompanied with high-fidelity full-wave solver are required to identify the refractive index distribution. In our case we rely on a statistical learning optimization technique. It is based on a surrogate model representation which provides a quick estimation to approximate the true function of interest to select the best candidates for the next simulation. Accordingly, programable beam steering configurations have been designed with ultimate performance in reflection. The realization of active beam splitter and active beam steering devices operating at GHz deflection frequency would open important applications in imaging microscopy, high resolution image projection, optical communication and 3D light detection and ranging (LiDAR).

International audience