Publications

2022

• Relaxed-inertial proximal point algorithms for problems involving strongly quasiconvex functions
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2022. Introduced in the 1970's by Martinet for minimizing convex functions and extended shortly afterward by Rockafellar towards monotone inclusion problems, the proximal point algorithm turned out to be a viable computational method for solving various classes of optimization problems, in particular with nonconvex objective functions. We propose first a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. The method is then extended to equilibrium problems where the involved bifunction is strongly quasiconvex in the second variable. Possible modifications of the hypotheses that would allow the algorithms to solve similar problems involving quasiconvex functions are discussed, too. Numerical experiments confirming the theoretical results, in particular that the relaxed-inertial algorithms outperform their ``pure'' proximal point counterparts, are provided, too.
• Achievement and Fragility of Long-term Equitability
  
  • Simonetto Andrea
  • Notarnicola Ivano
  , 2022. Equipping current decision-making tools with notions of fairness, equitability, or other ethically motivated outcomes, is one of the top priorities in recent research efforts in machine learning, AI, and optimization. In this paper, we investigate how to allocate limited resources to {locally interacting} communities in a way to maximize a pertinent notion of equitability. In particular, we look at the dynamic setting where the allocation is repeated across multiple periods (e.g., yearly), the local communities evolve in the meantime (driven by the provided allocation), and the allocations are modulated by feedback coming from the communities themselves. We employ recent mathematical tools stemming from data-driven feedback online optimization, by which communities can learn their (possibly unknown) evolution, satisfaction, as well as they can share information with the deciding bodies. We design dynamic policies that converge to an allocation that maximize equitability in the long term. We further demonstrate our model and methodology with realistic examples of healthcare and education subsidies design in Sub-Saharian countries. One of the key empirical takeaways from our setting is that long-term equitability is fragile, in the sense that it can be easily lost when deciding bodies weigh in other factors (e.g., equality in allocation) in the allocation strategy. Moreover, a naive compromise, while not providing significant advantage to the communities, can promote inequality in social outcomes. (10.1145/3514094.3534132)
  DOI : 10.1145/3514094.3534132
• Long time behaviour for electromagnetic waves in dissipative Lorentz media
  
  • Cassier Maxence
  • Joly Patrick
  • Rosas Martinez Luis Alejandro
  , 2022. A very general class of models for describing the propagation of waves in dispersive electromagnetic media is provided by generalized Lorentz models. In this work, we study the long time behaviour of the solutions of the dissipative version of these models.
• Inverse problem for the Helmholtz equation and singular sources in the divergence form
  
  • Baratchart Laurent
  • Haddar Houssem
  • Villalobos Guillén Cristóbal
  , 2022. We shall discuss an inverse problem where the underlying model is related to sources generated by currents on an anisotropic layer. This problem is a generalization of another motivated by the recovering of magnetization distribution in a rock sample from outer measurements of the generated static magnetic field. The original problem can be formulated as inverse source problem for the Laplace equation [1,2] with sources being the divergence of the magnetization whereas the generalization comes from taking the Helmholtz equation. Either inverse problem is non uniquely solvable with a kernel of infinite dimension. We shall present a decomposition of the space of sources that will allow us to discuss constraints that may restore uniqueness and propose regularization schemes adapted to these assumptions. We then present some validating experiments and some related open questions.
• Scattering in a partially open waveguide: the inverse problem
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  , 2022.
• Learning equilibria with personalized incentives in a class of nonmonotone games
  
  • Fabiani Filippo
  • Simonetto Andrea
  • Goulart Paul J.
  , 2022. We consider quadratic, nonmonotone generalized Nash equilibrium problems with symmetric interactions among the agents, which are known to be potential. As may happen in practical cases, we envision a scenario in which an explicit expression of the underlying potential function is not available, and we design a two-layer Nash equilibrium seeking algorithm. In the proposed scheme, a coordinator iteratively integrates the noisy agents' feedback to learn the pseudo-gradients of the agents, and then design personalized incentives for them. On their side, the agents receive those personalized incentives, compute a solution to an extended game, and then return feedback measures to the coordinator. We show that our algorithm returns an equilibrium in case the coordinator is endowed with standard learning policies, and corroborate our results on a numerical instance of a hypomonotone game.
• New optimization models for optimal classification trees
  
  • Alès Zacharie
  • Huré Valentine
  • Lambert Amélie
  , 2022.
• 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
• Convergence analysis of multi-step one-shot methods for linear inverse problems
  
  • Bonazzoli Marcella
  • Haddar Houssem
  • Vu Tuan Anh
  , 2022. In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.
• OpReg-Boost: Learning to Accelerate Online Algorithms with Operator Regression
  
  • Bastianello Nicola
  • Simonetto Andrea
  • Dall'Anese Emiliano
  , 2022. This paper presents a new regularization approach -- termed OpReg-Boost -- to boost the convergence and lessen the asymptotic error of online optimization and learning algorithms. In particular, the paper considers online algorithms for optimization problems with a time-varying (weakly) convex composite cost. For a given online algorithm, OpReg-Boost learns the closest algorithmic map that yields linear convergence; to this end, the learning procedure hinges on the concept of operator regression. We show how to formalize the operator regression problem and propose a computationally-efficient Peaceman-Rachford solver that exploits a closed-form solution of simple quadratically-constrained quadratic programs (QCQPs). Simulation results showcase the superior properties of OpReg-Boost w.r.t. the more classical forward-backward algorithm, FISTA, and Anderson acceleration.
• A high-order discontinuous Galerkin Method using a mixture of Gauss-Legendre and Gauss-Lobatto quadratures for improved efficiency
  
  • Chaillat Stéphanie
  • Cottereau Régis
  • Sevilla Ruben
  , 2022. In discontinuous Galerkin spectral element methods (DGSEM), the two most common approaches to numerically integrate the terms of the weak form are either using Gauss-Legendre or Gauss-Lobatto quadratures. The former yields more accurate results but at a higher computational cost, so that a priori it is not clear whether one approach is more efficient that the other. In this paper, it is shown (theoretically for a particular case and numerically for the general case) that using Gauss-Lobatto quadrature for the convection matrix actually introduces a negligible error. In contrast, using Gauss-Lobatto quadratures for the evaluation of the jump term in the element faces introduces a sizeable error. This leads to the proposal of a new DG approach, where the convection matrix is evaluated using Gauss-Lobatto quadratures, whereas the face mass matrices are integrated using Gauss-Legendre quadratures. For elements with constant Jacobian and constant coefficients, a formal proof shows that no numerical integration error is actually introduced in the evaluation of the residual, even though both the mass and the convection matrices are not computed exactly with Gauss-Lobatto quadratures. For elements with non-constant Jacobian and/or non-constant coefficients, the impact of numerical integration error on the overall error is evaluated through a series of numerical tests, showing that this is also negligible. In addition, the computational cost associated to the matrix-vector products required to evaluate the residual is evaluated precisely for the different cases considered. The proposed approach is particularly attractive in the most general case, since the use of Gauss-Lobatto quadratures significantly speeds-up the evaluation of the residual.
• Stochastic incremental mirror descent algorithms with Nesterov smoothing
  
  • Grad Sorin-Mihai
  • Bitterlich Sandy
  , 2022. We propose a stochastic incremental mirror descent method constructed by means of the Nesterov smoothing for minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in a Euclidean space. The algorithm can be adapted in order to minimize (in the same setting) a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Another modification of the scheme leads to a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing for minimizing the sum of finitely many proper, convex and lower semicontinuous functions with a prox-friendly proper, convex and lower semicontinuous function in the same framework. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modeled as optimization problems illustrate the theoretical achievements.
• Acoustic passive cloaking using thin outer resonators
  
  • Chesnel Lucas
  • Heleine Jérémy
  • Nazarov Sergei A
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2022, 73 (3). We consider the propagation of acoustic waves in a 2D waveguide unbounded in one direction and containing a compact obstacle. The wavenumber is fixed so that only one mode can propagate. The goal of this work is to propose a method to cloak the obstacle. More precisely, we add to the geometry thin outer resonators of width ε and we explain how to choose their positions as well as their lengths to get a transmission coefficient approximately equal to one as if there were no obstacle. In the process we also investigate several related problems. In particular, we explain how to get zero transmission and how to design phase shifters. The approach is based on asymptotic analysis in presence of thin resonators. An essential point is that we work around resonance lengths of the resonators. This allows us to obtain effects of order one with geometrical perturbations of width ε. Various numerical experiments illustrate the theory. (10.1007/s00033-022-01736-6)
  DOI : 10.1007/s00033-022-01736-6
• Robust treatment of cross points in Optimized Schwarz Methods
  
  • Claeys Xavier
  • Parolin Emile
  Numerische Mathematik, Springer Verlag, 2022, 151 (2), pp.405-442. In the field of Domain Decomposition (DD), Optimized Schwarz Method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between sub-domains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two sub-domains abut, if no appropriate treatment is provided. In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the performance of the proposed method. (10.1007/s00211-022-01288-x)
  DOI : 10.1007/s00211-022-01288-x
• Convergence d'un couplage élastique-acoustique FEM-BEM itératif, global en temps
  
  • Nassor Alice
  • Chaillat Stéphanie
  • Bonnet Marc
  • Leblé Bruno
  • Barras Guillaume
  , 2022. Une méthode itérative convergente pour le couplage FEM-BEM élastodynamique-acoustique global en temps, permettant de traiter un problème d’interaction fluide-structure est proposée. Les équations structures sont résolues en éléments finis (FEM), tandis que la partie fluide est traitée par éléments de frontière (BEM), formulée en temps discrets par Convolution Quadrature method (CQM). Le couplage présenté se base sur la formulation de conditions de transmission de Robin. La convergence est démontrée et illustrée. Un deuxième couplage itératif en temps à convergence garantie est proposé.
• Méthode des éléments de frontière pour la mécanique des failles et le contrôle sismique
  
  • Bagur Laura
  • Chaillat Stéphanie
  • Semblat Jean-François
  • Stefanou Ioannis
  , 2022. Ce travail consiste à vérifier numériquement des stratégies de contrôle de séismes par injection de fluide dans le sol. Nous étudions les capacités des méthodes d’éléments de frontière (BEMs) à simuler des séquences de glissements sismiques et asismiques en géomécanique. Un algorithme basé sur les BEMs accélérées par FFT est considéré, validé, et des résultats sont présentés pour un problème simple de mécanique des failles. Les challenges en lien avec l’extension des BEMs accélérées pour incorporer les couplages multi-physiques en jeu sont discutés.
• A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation
  
  • Royer Anthony
  • Geuzaine Christophe
  • Béchet Eric
  • Modave Axel
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2022, 395, pp.115006. It is well-known that the convergence rate of non-overlapping domain decomposition methods (DDMs) applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly depends on the transmission condition enforced at the interfaces between the subdomains. Transmission operators based on perfectly matched layers (PMLs) have proved to be well-suited for configurations with layered domain partitions. They are shown to be a good compromise between basic impedance conditions, which lead to suboptimal convergence, and computational expensive conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care. In this work, we present a non-overlapping substructured DDM with PML transmission conditions for checkerboard (Cartesian) decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. Two possible Lagrange multiplier finite element spaces are presented, and the behavior of the corresponding DDM is analyzed on several numerical examples. (10.1016/j.cma.2022.115006)
  DOI : 10.1016/j.cma.2022.115006
• Spectral theory for Maxwell's equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance
  
  • Cassier Maxence
  • Hazard Christophe
  • Joly Patrick
  Communications in Partial Differential Equations, Taylor & Francis, 2022, 47 (6), pp.1217-1295. This paper is concerned with the time-dependent Maxwell's equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long time response of the medium to a time-harmonic source of prescribed frequency. This paper also underlines the existence of an interface resonance which occurs when there exists a particular frequency characterized by a ratio of permittivities and permeabilities equal to −1 across the interface. At this frequency, the response of the system to a harmonic forcing term blows up linearly in time. Such a resonance is unusual for wave problem in unbounded domains and corresponds to a non-zero embedded eigenvalue of infinite multiplicity of the underlying operator. This is the time counterpart of the ill-posdness of the corresponding harmonic problem. (10.1080/03605302.2022.2051188)
  DOI : 10.1080/03605302.2022.2051188
• Modélisation semi-analytique du bruit large bande produit par l’interaction entre un écoulement turbulent et un obstacle rigide de forme complexe
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotté Benjamin
  • Mercier Jean-François
  , 2022. L’interaction entre un écoulement turbulent et un obstacle rigide produit un rayonnement acoustique large bande qui peut avoir un impact significatif dans de nombreuses problématiques industrielles. Les méthodes de prédiction existantes ne sont pour la plupart pas adaptées au contexte des applications navales qui impose trois contraintes : les systèmes considérés sont de formes complexes et les écoulements sont généralement à très bas nombre de Mach et à haut Reynolds. Dans ces conditions, les méthodes de calcul directs du bruit qui reposent sur l’utilisation d’une simulation compressible de l’écoulement sont trop coûteuses. D’autres approches doivent être utilisées. Basées sur les analogies acoustiques, elles reposent sur l’idée de séparer les mécanismes de production et les mécanismes de propagation du bruit. Dans cette étude, l’équation d’onde de Lighthill est résolue grâce à une fonction de Green adaptée qui peut être analytique, pour des géométries canoniques, ou déterminée numériquement grâce à la méthode des éléments de frontière. De plus, un modèle pour l’interspectre des fluctuations turbulentes de vitesse, exprimé en espace-fréquence est introduit. Il peut être construit soit à partir d’une estimation des paramètres de couche limite, soit à partir d’une simulation de l’écoulement moyen. Les prédictions obtenues pour le bruit de bord d’attaque et le bruit de bord de fuite sont validées grâce à des mesures effectuées sur un profil NACA 0012, en air.
• Extending the proximal point algorithm beyond convexity
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2022. Introduced in the 1970's by Martinet for minimizing convex functions and extended shortly afterwards by Rockafellar towards monotone inclusion problems, the proximal point algorithm turned out to be a viable computational method for solving various classes of (structured) optimization problems even beyond the convex framework. In this talk we discuss some extensions of proximal point type algorithms beyond convexity. First we propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces, that can be extended to equilibrium functions involving such functions. Then we briefly discuss another generalized convexity notion for functions we called prox-convexity for which the proximity operator is single-valued and firmly nonexpansive, and see that the standard proximal point algorithm and Malitsky’s Golden Ratio Algorithm (originally proposed for solving convex mixed variational inequalities) remain convergent when the involved functions are taken prox-convex, too.
• Géométrie Différentielle et Application au Contrôle Géométrique
  
  • Jean Frédéric
  , 2022.
• Multidirectional sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems
  
  • Dai Ruiyang
  • Modave Axel
  • Remacle Jean-François
  • Geuzaine Christophe
  Journal of Computational Physics, Elsevier, 2022 (453), pp.110887. This paper explores a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. Several two-dimensional finite element results are proposed to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity. (10.1016/j.jcp.2021.110887)
  DOI : 10.1016/j.jcp.2021.110887
• Maxwell's equations with hypersingularities at a conical plasmonic tip
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Rihani Mahran
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2022, 161, pp.70-110. In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical $L^2$ framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle. (10.1016/j.matpur.2022.03.001)
  DOI : 10.1016/j.matpur.2022.03.001
• Recent trends in vector optimization
  
  • Grad Sorin-Mihai
  , 2022. We discuss about some current developments in Vector Optimization. Emphasis will be placed on recent contributions on algorithmic methods for solving vector optimization problems. In particular we will talk about the existing extensions of the proximal point methods towards Vector Optimization and about some open questions in this direction.
• Mathematical Analysis of Goldstein's Model for time harmonic acoustics in flow
  
  • Bensalah Antoine
  • Joly Patrick
  • Mercier Jean-François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022, 56 (2), pp.451-483. Goldstein’s equations have been introduced in 1978 as an alternative model to linearized Euler equations to model acoustic waves in moving fluids. This new model is particularly attractive since it appears as a perturbation of a simple scalar model: the potential model. In this work we propose a mathematical analysis of boundary value problems associated with Goldstein’s equations in the time-harmonic regime. (10.1051/m2an/2022007)
  DOI : 10.1051/m2an/2022007