Publications

2024

• Clustering data for the Optimal Classication Tree Problem
  
  • Ales Zacharie
  • Huré Valentine
  • Lambert Amélie
  , 2024. Solving the optimal classification tree problem enables to compute classifiers which are both interpretable and efficient. Most of the exact methods for this problem are based on on a Mixed Integer Linear Program (MILP) formulation. However, the efficiency of MILP solvers generally does not allow these formulations to be solved directly, once the dataset exceeds a critical size. To address this challenge, we propose in this paper an iterative exact algorithm than handles medium-sized datasets from the state-of-the-art. The basic idea is to start by solving a MILP formulation on a small subset of data points representative of the considered dataset. Then, the subset is iteratively extended until global optimality of the initial problem is reached. A key feature is to compute relevant initial subsets of data points. For this, we introduce the concept of data-partitions and design several algorithms to compute them. We then define two MILP formulations to compute optimal classification trees on data-partitions. We prove that combining our iterative algorithm with our first formulation enables to obtain an optimal solution of the original problem. We also propose an alternative method based on the second formulation which is significantly faster. We present extensive computational experiments to compare our algorithms with state-of-the-art approaches. We show that our methods constitute the best compromise between in-sample accuracy and interpretability.
• Statistical Linearization for Robust Motion Planning
  
  • Leparoux Clara
  • Bonalli Riccardo
  • Hérissé Bruno
  • Jean Frédéric
  Systems and Control Letters, Elsevier, 2024, 189, pp.105825. The goal of robust motion planning consists of designing open-loop controls which optimally steer a system to a specific target region while mitigating uncertainties and disturbances which affect the dynamics. Recently, stochastic optimal control has enabled particularly accurate formulations of the problem. Nevertheless, despite interesting progresses, these problem formulations still require expensive numerical computations. In this paper, we start bridging this gap by leveraging statistical linearization. Specifically, through statistical linearization we reformulate the robust motion planning problem as a simpler deterministic optimal control problem subject to additional constraints. We rigorously justify our method by providing estimates of the approximation error, as well as some controllability results for the new constrained deterministic formulation. Finally, we apply our method to the powered descent of a space vehicle, showcasing the consistency and efficiency of our approach through numerical experiments. (10.1016/j.sysconle.2024.105825)
  DOI : 10.1016/j.sysconle.2024.105825
• Nonlinear Optimization Filters for Stochastic Time-Varying Convex Optimization
  
  • Simonetto Andrea
  • Massioni Paolo
  International Journal of Robust and Nonlinear Control, Wiley, 2024. 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. (10.1002/rnc.7380)
  DOI : 10.1002/rnc.7380
• Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer
  
  • Chesnel Lucas
  • Nazarov Sergei A.
  • Taskinen Jari
  Journal of Spectral Theory, European Mathematical Society, 2024. We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector κ = (κ_1,κ_2) which characterizes the domain at the edges. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to κ. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases κ_1\ge0 and κ_2∈ [−κ_1,κ_1]. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to κ. In particular, we show that for a given κ_1 > 0, there is some h(κ_1) > 0 such that discrete spectrum exists for κ_2 ∈ (−κ_1,0) ∪ (h(κ_1),κ_1) whereas it is empty for κ_2 ∈ [0; h(κ_1)]. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.
• New optimization models for optimal classification trees
  
  • Alès Zacharie
  • Huré Valentine
  • Lambert Amélie
  Computers and Operations Research, Elsevier, 2024, 164, pp.106515. Interpretability is a growing concept in Machine Learning. Decision-making algorithms are more and more used in healthcare, finance or other high stakes contexts. Therefore, the need for algorithms whose decisions are understandable is of the utmost importance. Intrinsically interpretable classifiers such as decision trees are often seen as less accurate than black box models such as neural networks. For decision trees, state-of-the-art methods are recursive heuristics (e.g. CART) that may fail to find underlying characteristics in datasets. Recently, linear formulations were introduced to model the problem of the construction of the best decision tree for a given dataset. Notably, a MIO formulation, introduced by Bertsimas and al., has shown better accuracy than CART. However this model does not scale up to datasets with more than 1000 data points. Our work focuses on improvements of MIOs that speed up their resolution in order to handle larger problems. We present a quadratic formulation of the MIP devised by Bertsimas and al. as well as its linearization and another that extends a flow-formulation (from binary dataset to real-value dataset). We prove that our new formulations have stronger continuous relaxation than the MIP introduced by Bertsimas and al.. Finally, our experiments show that they have a significantly smaller resolution time than the MIP of Bertsimas and al. while maintaining or improving performances on test sets. (10.1016/j.cor.2023.106515)
  DOI : 10.1016/j.cor.2023.106515
• Simulation and analysis of sign-changing Maxwell’s equations in cold plasma
  
  • Peillon Etienne
  , 2024. Nowadays, plasmas are mainly used for industrial purpose. One of the most frequently cited examples of industrial use is electric energy production via fusion nuclear reactors. Then, in order to contain plasma properly inside the reactor, a background magnetic field is imposed, and the density and temperature of the plasma must be precisely controlled. This is done by sending electromagnetic waves at specific frequencies and directions depending on the characteristics of the plasma.The first part of this PhD thesis consists in the study of the model of plasma in a strong background magnetic field, which corresponds to a hyperbolic metamaterial. The objective is to extend the existing results in 2D to the 3D-case and to derive a radiation condition. We introduce a splitting of the electric and magnetic fields resembling the usual TE and TM decomposition, then, it gives some results on the two resulting problems. The results are in a very partial state, and constitute a rough draft on the subject.The second part consists in the study of the degenerate PDE associated to the lower-hybrid resonant waves in plasma. The associated boundary-value problem is well-posed within a ``natural'' variational framework. However, this framework does not include the singular behavior presented by the physical solutions obtained via the limiting absorption principle. Notice that this singular behavior is important from the physical point of view since it induces the plasma heating mentioned before. One of the key results of this second part is the definition of a notion of weak jump through the interface inside the domain, which allows to characterize the decomposition of the limiting absorption solution into a regular and a singular parts.
• MAPL: Model Agnostic Peer-to-peer Learning
  
  • Mukherjee Sayak
  • Simonetto Andrea
  • Jamali-Rad Hadi
  , 2024. Effective collaboration among heterogeneous clients in a decentralized setting is a rather unexplored avenue in the literature. To structurally address this, we introduce Model Agnostic Peer-to-peer Learning (coined as MAPL) a novel approach to simultaneously learn heterogeneous personalized models as well as a collaboration graph through peer-to-peer communication among neighboring clients. MAPL is comprised of two main modules: (i) local-level Personalized Model Learning (PML), leveraging a combination of intra- and inter-client contrastive losses; (ii) network-wide decentralized Collaborative Graph Learning (CGL) dynamically refining collaboration weights in a privacy-preserving manner based on local task similarities. Our extensive experimentation demonstrates the efficacy of MAPL and its competitive (or, in most cases, superior) performance compared to its centralized model-agnostic counterparts, without relying on any central server. Our code is available and can be accessed here: https://github.com/SayakMukherjee/MAPL
• ABOUT SEMILINEAR LOW DIMENSION BESSEL PDEs
  
  • Ohashi Alberto
  • Russo Francesco
  • Teixeira Alan
  , 2024. We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator $L^\delta$ with low dimension $0 < \delta < 1$. $L^\delta$ is a local operator, whose drift is the derivative of $x \mapsto \log (\vert x\vert)$: in particular it is a Schwartz distribution, which is not the derivative of a continuous function. The solutions are intended in a duality ("weak") sense with respect to state space $L^2(\R_+, d\mu),$ $\mu$ being an invariant measure for the Bessel semigroup.
• Imagerie d’interface barrage-fondation par inversion de forme d'onde complète
  
  • Boukraa Mohamed Aziz
  • Audibert Lorenzo
  • Bonazzoli Marcella
  • Haddar Houssem
  • Vautrin Denis
  , 2024, 504, pp.04002. Dans le cadre de l’étude de la stabilité des barrages, la connaissance de l’interface entre le barrage et la roche revêt une grande importance. Le recours à des techniques géophysiques peut apporter des informations complémentaires par rapport aux mesures géotechniques. Nous proposons ici une méthode de traitement des mesures sismiques, l’objectif étant d'obtenir une image de l'interface entre le béton du barrage et le rocher de la fondation avec une résolution métrique. Il s’agit d’une technique de type « Full Waveform Inversion » avec optimisation de forme. Des résultats numériques utilisant des mesures synthétiques montrent la capacité de la méthode à retrouver l'interface avec une précision satisfaisante, pour un nombre limité de points de mesure et en présence de bruit. (10.1051/e3sconf/202450404002)
  DOI : 10.1051/e3sconf/202450404002
• Proximal Point Type Algorithms with Relaxed and Inertial Effects Beyond Convexity
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  Optimization, Taylor & Francis, 2024, pp.1-18. We show that the recent relaxed-inertial proximal point algorithm due to Attouch and Cabot remains convergent when the function to be minimized is not convex, being only endowed with certain generalized convexity properties. Numerical experiments showcase the improvements brought by the relaxation and inertia features to the standard proximal point method in this setting, too. (10.1080/02331934.2024.2329779)
  DOI : 10.1080/02331934.2024.2329779
• Cell seeding dynamics in a porous scaffold material designed for meniscus tissue regeneration
  
  • Jäger Henry
  • Grosjean Elise
  • Plunder Steffen
  • Redenbach Claudia
  • Keilmann Alex
  • Simeon Bernd
  • Surulescu Christina
  , 2024, 24 (2). We study the dynamics of a seeding experiment where a fibrous scaffold material is colonized by two types of cell populations. The specific application that we have in mind is related to the idea of meniscus tissue regeneration. In order to support the development of a promising replacement material, we discuss certain rate equations for the densities of human mesenchymal stem cells and chondrocytes and for the production of collagen‐containing extracellular matrix. For qualitative studies, we start with a system of ordinary differential equations and refine then the model to include spatial effects of the underlying nonwoven scaffold structure. Numerical experiments as well as a complete set of parameters for future benchmarking are provided. (10.1002/pamm.202400133)
  DOI : 10.1002/pamm.202400133
• Reconstruction of averaging indicators for highly heterogeneous media
  
  • Audibert Lorenzo
  • Haddar Houssem
  • Pourre Fabien
  Inverse Problems, IOP Publishing, 2024, 40 (4), pp.045028. We propose a new imaging algorithm capable of producing quantitative indicator functions for unknown and possibly highly oscillating media from multistatic far field measurements of scattered fields at a fixed frequency. The algorithm exploits the notion of modified transmission eigenvalues and their determination from measurements. We propose in particular the use of a new modified background obtained as the limit of a metamaterial background. It has the specificity of having a unique non trivial eigenvalue, which is particularly suited for the proposed imaging procedure. We show the efficiency of this new algorithm on some 2D experiments and emphasize its superiority with respect to some clasical approaches such as the Linear Sampling Method. (10.1088/1361-6420/ad2f64)
  DOI : 10.1088/1361-6420/ad2f64
• Modélisation hybride modale-éléments finis pour le contrôle ultrasonore d'une plaque élastique. Traitement des intégrales oscillantes de la méthode HSM
  
  • Allouko Amond
  , 2024. Cette thèse porte sur la méthode Half-Space Matching (HSM) pour la résolution de problèmes de diffraction dans une plaque élastique non-bornée, en vue de la simulation du contrôle non-destructif de plaques composites. La méthode HSM est une approche hybride qui couple un calcul éléments finis dans une boite contenant les défauts, avec des représentations semi-analytiques dans quatre demi-plaques qui recouvrent la partie saine de la plaque. Les représentations semi-analytiques de demi-plaques font intervenir des tenseurs de Green, exprimés à l'aide d'intégrales de Fourier et de séries modales. Or ces expressions peuvent être délicates à évaluer en pratique (coût et précision), rendant la méthode HSM inexploitable industriellement. Les difficultés sont d'abord analysées dans un cas scalaire bidimensionnel (acoustique). Deux méthodes sont proposées pour une évaluation efficace des intégrales de Fourier : la première exploite une approximation de type champ lointain et la seconde repose sur une déformation du chemin d'intégration dans le plan complexe (méthode de la complexification). Ces deux méthodes sont validées dans les cas scalaires isotrope et anisotrope où l'on dispose des valeurs exactes des intégrales de Fourier exprimées à l'aide de fonctions de Hankel. Elles sont ensuite généralisées au cas tridimensionnel de la plaque élastique. Dans ce cas, la formule de représentation est obtenue en faisant une transformée de Fourier suivant une direction parallèle à la plaque, puis, pour chaque valeur de la variable de Fourier ξ, une décomposition modale dans l'épaisseur. Les modes mis en jeu, appelés ξ-modes, sont étudiés en détail et comparés aux modes classiques (Lamb et SH dans le cas isotrope). Afin d'exploiter la bi-orthogonalité des ξ-modes, la formule de demi-plaque requiert la connaissance à la fois du déplacement et de la contrainte normale sur la frontière. Dans le cas isotrope, les propriétés d'analyticité des ξ-modes permettent de justifier et d'étendre la méthode de la complexification, y compris en présence de modes inverses. Ceci réduit les effets de couplage modal parasite induits par la discrétisation des intégrales de Fourier. La méthode de la complexification est ensuite utilisée pour le calcul des opérateurs intervenant dans la méthode HSM, qui dérivent tous de la formule de demiplaque. Différentes validations de la méthode HSM sont ainsi effectuées dans le cas isotrope. Des résultats préliminaires encourageants sont également obtenus pour une plaque orthotrope. Les améliorations réalisées ont permis à la fois de réduire significativement le temps de calcul et d'assurer une plus grande précision de la méthode HSM, permettant d'envisager son exploitation systématique dans un cadre de simulation industrielle.
• Introduction aux équations aux dérivées partielles hyperboliques et à leur approximation numérique
  
  • Fliss Sonia
  • Bonnet-Ben Dhia Anne-Sophie
  • Joly Patrick
  • Moireau Philippe
  , 2024.
• Flexible Optimization for Cyber-Physical and Human Systems
  
  • Simonetto Andrea
  IEEE Control Systems Letters, IEEE, 2024, 8, pp.1475-1480. Can we allow humans to pick among different, yet reasonably similar, decisions? Are we able to construct optimization problems whose outcome are sets of feasible, close-to-optimal decisions for human users to pick from, instead of a single, hardly explainable, do-as-I-say ``optimal'' directive? In this paper, we explore two complementary ways to render optimization problems stemming from cyber-physical applications flexible. In doing so, the optimization outcome is a trade off between engineering best and flexibility for the users to decide to do something slightly different. The first method is based on robust optimization and convex reformulations. The second method is stochastic and inspired from stochastic optimization with decision-dependent distributions. (10.1109/LCSYS.2024.3411931)
  DOI : 10.1109/LCSYS.2024.3411931
• Fast Imaging of Local Perturbations in a Unknown Bi-Periodic Layered Medium
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Nguyen Thi-Phong
  Journal of Computational Physics, Elsevier, 2024, 501, pp.112773. We discuss a novel approach for imaging local faults inside an infinite bi-periodic layered medium in R 3 using acoustic measurements of scattered fields at the bottom or the top of the layer. The faulted area is represented by compactly supported perturbations with erroneous material properties. Our method reconstructs the support of perturbations without knowing or reconstructing the constitutive material parameters of healthy or faulty bi-period layer; only the size of the period is needed. This approach falls under the class of non-iterative imaging methods, known as the generalized linear sampling method with differential measurements, first introduced in [2] and adapted to periodic layers in [25]. The advantage of applying differential measurements to our inverse problem is that instead of comparing the measured data against measurements due to healthy structures, one makes use of periodicity of the layer where the data operator restricted to single Floquet-Bloch modes plays the role of the one corresponding to healthy material. This leads to a computationally efficient and mathematically rigorous reconstruction algorithm. We present numerical experiments that confirm the viability of the approach for various configurations of defects. (10.1016/j.jcp.2024.112773)
  DOI : 10.1016/j.jcp.2024.112773
• The scattering phase: seen at last
  
  • Galkowski Jeffrey
  • Marchand Pierre
  • Wang Jian
  • Zworski Maciej
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024, 84 (1), pp.246-261. The scattering phase, defined as $ \log \det S ( \lambda ) / 2\pi i $ where $ S ( \lambda ) $ is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Krein's spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. Perhaps more importantly, we provide the first numerical calculations of scattering phases for non-radial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM. (10.1137/23M1547147)
  DOI : 10.1137/23M1547147
• Design and Dimensioning of Natural Gas Pipelines with Hydrogen Injection
  
  • Jorquera-Bravo Natalia
  • Elloumi Sourour
  • Plateau Agnès
  , 2024. The global focus on reducing air pollution and dependence on fossil fuels has led to efforts to shift to renewable energy sources. Hydrogen is a promising alternative due to its high energy capacity and ability to regulate electricity production through electrolysis. In this context, the problem of designing and sizing natural gas pipelines with hydrogen injection is presented. The objective is to establish the network topology and diameter dimensions of each pipeline section for hydrogen distribution, in order to cover the demand at a minimum cost. To address the proposed problem, we consider the dimensioning as the selection of a diameter from a set of available measures, i.e., a discrete diameter approach, and we compare it with a continuous diameter approach from the literature, including a mixed integer nonlinear programming (MINLP) formulation of degree six. In our discrete diameter approach, we propose a non-convex quadratic (MIQLP) model, and we derive a mixed-integer quadratic convex relaxation (MIQCP). Finally, we adapt a Delta Change heuristic to this context. We implement several solution methods for a real case study in France. These include solving the dimensioning problem on a fixed Minimum Spanning Tree topology, considering both continuous and discrete diameters, employing the Delta Change heuristic for both cases, continuous and discrete, and solving the MIQCP relaxation problem. The strengths and weaknesses of each of these proposals are demonstrated through the study.
• Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raúl
  Journal of Optimization Theory and Applications, Springer Verlag, 2024. We propose a relaxed-inertial proximal point algorithm for solving equilibrium problems involving bifunctions which satisfy in the second variable a generalized convexity notion called strong quasiconvexity, introduced by Polyak in 1966. The method is suitable for solving mixed variational inequalities and inverse mixed variational inequalities involving strongly quasiconvex functions, as these can be written as special cases of equilibrium problems. Numerical experiments where the performance of the proposed algorithm outperforms the one of the standard proximal point methods are provided, too.
• Radial perfectly matched layers and infinite elements for the anisotropic wave equation
  
  • Halla Martin
  • Kachanovska Maryna
  • Wess Markus
  , 2024. We consider the scalar anisotropic wave equation. Recently a convergence analysis for radial perfectly matched layers (PML) in the frequency domain was reported and in the present article we continue this approach into the time domain. First we explain why there is a good hope that radial complex scalings can overcome the instabilities of PML methods caused by anisotropic materials. Next we discuss some sensitive details, which seem like a paradox at the first glance: if the absorbing layer and the inhomogeneities are sufficiently separated, then the solution is indeed stable. However, for more general data the problem becomes unstable. In numerical computations we observe instabilities regardless of the position of the inhomogeneities, although the instabilities arise only for fine enough discretizations. As a remedy we propose a complex frequency shifted scaling and discretizations by Hardy space infinite elements or truncation-free PMLs. We show numerical experiments which confirm the stability and convergence of these methods.
• Exact solution methods for large-scale discrete p-facility location problems
  
  • Durán Mateluna Cristian
  , 2024. This thesis focuses on the exact solution of the NP-hard problems p-median and p-center, combinatorial optimization problems that quickly become difficult to solve as the instance size increases. These discrete location problems involve opening a defined number p of facilities and then allocating to them a set of clients according to an objective function to be minimized.First, we study the p-median problem, which seeks to minimize the sum of distances between clients and the open facilities to which they are allocated. We develop an algorithm based on Benders decomposition that outperforms state-of-the-art exact methods. The algorithm considers a two-stage approach and an efficient algorithm for separating Benders cuts. The method has been evaluated on over 230 benchmark instances with up to 238025 clients and sites. Many instances are solved to optimality for the first time or have their best known solution improved.Secondly, we explore the p-center problem, which seeks to minimize the largest distance between a client and its nearest open facility. We first compare the five main MILP formulations in the literature. We study the Benders decomposition and also propose an exact algorithm based on a client clustering procedure based on the structure of the problem. All the proposed methods are compared with the state-of-the-art on benchmark instances. The results obtained are analyzed, highlighting the advantages and disadvantages of each method.Finally, we study a robust two-stage p-center problem with uncertainty on node demands and distances. We introduce the robust reformulation of the problem based on the five main deterministic MILP formulations in the literature. We prove that only a finite subset of scenarios from the infinite uncertainty set can be considered without losing optimality. We also propose a column and constraint generation algorithm and a branch-and-cut algorithm to efficiently solve this problem. We show how these algorithms can also be adapted to solve the robust single-stage problem. The different proposed formulations are tested on randomly generated instances and on a case study drawn from the literature.
• How does the partition of unity influence SORAS preconditioner?
  
  • Bonazzoli Marcella
  • Claeys Xavier
  • Nataf Frédéric
  • Tournier Pierre-Henri
  , 2024, 149, pp.61-68. We investigate the influence of the choice of the partition of unity on the convergence of the Symmetrized Optimized Restricted Additive Schwarz (SORAS) preconditioner for the reaction-convection-diffusion equation. We focus on two kinds of partitions of unity, and study the dependence on the overlap and on the number of subdomains. In particular, the second kind of partition of unity, which is non-zero in the interior of the whole overlapping region, gives more favorable convergence properties, especially when increasing the overlap width, in comparison with the first kind of partition of unity, whose gradient is zero on the subdomain interfaces and which would be the natural choice for ORAS solver instead. (10.1007/978-3-031-50769-4_6)
  DOI : 10.1007/978-3-031-50769-4_6
• Commande des Systèmes
  
  • Jean Frédéric
  , 2024.
• Fair Energy Allocation for Collective Self-Consumption
  
  • Jorquera-Bravo Natalia
  • Elloumi Sourour
  • Kedad-Sidhoum Safia
  • Plateau Agnès
  , 2024. This study explores a collective self-consumption community with several houses, a shared distributed energy resource (DER), and a common energy storage system, as a battery. Each house has an energy demand over a discrete planning horizon, met by using the DER, the battery, or purchasing electricity from the main power grid. Excess energy can be stored in the battery or sold back to the main grid. The objective is to determine a supply plan ensuring a fair allocation of renewable energy while minimizing the overall microgrid cost. We investigate and discuss the formulation of these optimization problems using mixed integer linear programming. We show some dominance properties that allow to reformulate the model into a linear program. We study some fairness metrics like the proportional allocation rule and max-min fairness. Finally, we illustrate our proposal in a real case study in France with up to seven houses and a one-day time horizon with 15minute intervals.
• Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces
  
  • Issoglio Elena
  • Pagliarani Stefano
  • Russo Francesco
  • Trevisani Davide
  , 2024. The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak Hörmander condition. The main result is of existence and uniqueness of a solution in law for the McKean-Vlasov equation, which is formulated as a suitable martingale problem. All analytical tools needed are derived in the paper, such as the well-posedness of the Fokker-Planck and Kolmogorov PDEs with distributional drift, as well as continuity dependence on the coefficients. The solutions to these PDEs naturally live in anisotropic Besov spaces, for which we developed suitable analytical inequalities, such as Schauder estimates.