Publications

2022

• 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. 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. Computational results confirm the theoretical advances.
• Viscosity theory of first order Hamilton Jacobi equations in some metric spaces
  
  • Jerhaoui Othmane
  , 2022. The main subject of this thesis is the study first order Hamilton Jacobi equations posed in certain classes of metric spaces. Furthermore, the Hamiltonian of these equations can potentially present some structured discontinuities.In the first part of this thesis, we study a discontinuous first order Hamilton Jacobi Bellman equation defined on a stratification of R^N. The latter is a finite and disjoint union of smooth submanifolds of R^N called the the subdomains of R^N. On each subdomain, a continuous Hamiltonian is defined on it, However the global Hamiltonian in R^N presents discontinuities once one goes from one subdomain to the other. We give an optimal control interpretation of this problem and we use nonsmooth analysis techniques to prove that the value function is the unique viscosity solution to the discontinuous Hamilton Jacobi Bellman equation in this setting. The uniqueness of the solution is guaranteed by means of a strong comparison principle valid for any lower semicontinuous supersolution and any upper semicontinuous subsolution. As far as existence of the solution is concerned, we use the dynamic programming principle verified by the value function to prove that it is a viscosity solution of the discontinuous Hamilton Jacobi equation. Moreover, we prove some stability results in the presence of perturbations on the discontinuous Hamiltonian. Finally, by virtue of the comparison principle, we prove a general convergence result of monotone numerical schemes approximating this problem.The second part of this thesis is concerned with defining a novel notion of viscosity for first order Hamilton Jacobi equations defined in proper CAT(0) spaces. A metric space is said to be a CAT(0) space if, roughly speaking, it is a geodesic space and its geodesic triangles are "thinner" than the triangles of the Euclidean plane. They can be seen as a generalization of Hilbert spaces or Hadamard manifolds. Typical examples of CAT(0) spaces include Hilbert spaces, metric trees and networks obtained by gluing a finite number of half-spaces along their common boundary. We exploit the additional structure that these spaces enjoy to study stationary and time-dependent first order Hamilton-Jacobi equation in them. In particular, we want to recover the main features of viscosity theory: the comparison principle and Perron's method}.We define the notion of viscosity using test functions that are Lipschitz and can be represented as a difference of two semiconvex function. We show that this new notion of viscosity coincides with the classical one in R^N by studying the examples of Hamilton Jacobi Bellman and Hamilton Jacobi Isaacs' equations. Furthermore, we prove existence and uniqueness of the solution of Eikonal type equations posed in networks that can result from gluing half-spaces of different Hausdorff dimension.In the third part of this thesis, we study a Mayer optimal control problem on the space of Borel probability measures over a compact Riemannian manifold M. This is motivated by certain situations where a central planner of a deterministic controlled system has only imperfect information on the initial state of the system. The lack of information here is very specific. It is described by a Borel probability measure along which the initial state is distributed. We define the new notion of viscosity in this space in a similar manner as in the previous part by taking test functions that are Lipschitz and can be written as a difference of two semiconvex functions. With this choice of test functions, we extend the notion of viscosity to Hamilton Jacobi Bellman equations in Wasserstein spaces and we establish that the value function is the unique viscosity solution of a Hamilton Jacobi Bellman equation in the Wasserstein space over M.
• AutoExpe.jl: Julia package that automates repetitive tasks in numerical experiments and the generation of their result tables
  
  • Alès Zacharie
  , 2022.
• Inverse Scattering Theory and Transmission Eigenvalues
  
  • Cakoni Fioralba
  • Colton David
  • Haddar Houssem
  , 2022, 98. In the first edition of this book, we discussed methods for determining the support of an inhomogeneous media from measured far field data as well as an extensive study of the central role played by the transmission eigenvalue problem in the mathematical development of these methods. In particular, we introduced the generalized linear sampling method (GLSM) and showed that this method provides a mathematical explanation of why it is permissible to use Tikhonov regularization to obtain an approximate solution of the far field equation associated with the linear sampling method. In the five years since the first edition of our book appeared, there has been considerable progress in both the development of GLSM as well as the theory of transmission eigenvalues. In this second edition, in addition to correcting typos in the first edition, we have added several highlights taken from these new developments. In particular, we have included new chapters on 1) the use of modified background media in the nondestructive testing of materials and in particular methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data, 2) a study of a subset of transmission eigenvalues, called non-scattering wave numbers, through the use of techniques taken from the theory of free boundary value problems for elliptic partial differential equations and 3) the duality between scattering poles and transmission eigenvalues which, in addition to their intrinsic mathematical interest, leads to new methods for the numerical computation of scattering poles.
• Robust MILP formulations for the two-stage p-Center Problem
  
  • Durán Mateluna Cristian
  • Jorquera-Bravo Natalia
  • Alès Zacharie
  • Elloumi Sourour
  , 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.
• The Satellite Constellation Design Problem via MI(N)LP Boosted with a Genetic Algorithm
  
  • Mencarelli Luca
  • Floquet Julien
  • Georges Frédéric
  • Guillaud Baptiste
  • Mellouki Wissal
  , 2022.
• Edge states in rationally terminated honeycomb structures
  
  • Fefferman Charles L
  • Fliss Sonia
  • Weinstein Michael I
  Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 2022, 119 (47), pp.e2212310119. Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type,” generalizing the classical zigzag and armchair edges. We prove that zero-energy/flat-band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulae for flat-band edge states when they exist. We produce strong evidence for the existence of dispersive (nonflat) edge state curves of nonzero energy for most l. (10.1073/pnas.2212310119)
  DOI : 10.1073/pnas.2212310119
• Algorithmes de placement optimisé de drones pour la conception de réseaux de communication
  
  • Alès Zacharie
  • Elloumi Sourour
  • Pass-Lanneau Adele
  , 2022. L'établissement d'un réseau de télécommunications performant et robuste est un besoin opérationnel clé pour les forces armées en opérations. Dans les prochaines années, des drones haute altitude pourront être utilisés comme noeuds du réseau sur le théâtre. Une question est alors de décider de leur placement spatial. Dans ce travail, une approche de recherche opérationnelle est proposée afin d'obtenir des algorithmes efficaces de placement de drones.
• 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 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 \cite{ILP, LAP}, are provided, too. Then we briefly discuss another generalized convexity notion for functions we called \textit{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.
• Path-dependent SDEs with jumps and irregular drift: well-posedness and Dirichlet properties
  
  • Bandini Elena
  • Russo Francesco
  , 2022. We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.
• Multistage Optimization of a Petroleum Production System with Material Balance Model
  
  • Vessaire Cyrille
  • Chancelier Jean-Philippe
  • de Lara Michel
  • Carpentier Pierre
  • Rodríguez-Martínez Alejandro
  • Robert Anna
  Computers & Chemical Engineering, Elsevier, 2022, 167, pp.108005. In this paper, we propose a mathematical formulation for the management of an oil production network as a multistage optimization problem. The reservoir is modeled as a controlled dynamical system by using material balance equations. We use a dynamic programming algorithm to solve the optimization problem. Two numerical applications illustrate our work: the first one consists in optimizing the production of a gas reservoir, whereas the second one tackles an oil reservoir with water injection. (10.1016/j.compchemeng.2022.108005)
  DOI : 10.1016/j.compchemeng.2022.108005
• Stochastic calculus via regularizations
  
  • Russo Francesco
  • Vallois Pierre
  , 2022, 11. A self-contained contribution to stochastic analysis mixing probability functional analysis and pathwise techniques A comprehensive book starting from elementary concepts in probability Formulates the state of the art of stochastic calculus beyond semimartingales (10.1007/978-3-031-09446-0)
  DOI : 10.1007/978-3-031-09446-0
• 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.
• Windowed Green function method for wave scattering by periodic arrays of 2D obstacles
  
  • Strauszer-Caussade Thomas
  • Faria Luiz
  • Fernandez-Lado Agustín
  • Pérez‐arancibia Carlos
  Studies in Applied Mathematics, Wiley-Blackwell, 2022, 150 (1), pp.277-315. This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second-kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology. (10.1111/sapm.12540)
  DOI : 10.1111/sapm.12540
• Modélisation et simulation numérique de la propagation d'ondes électromagnétiques dans les câbles coaxiaux.
  
  • Beni Hamad Akram
  , 2022. Dans cette thèse, nous nous intéressons à la propagation des ondes électromagnétiques dans un réseau de câbles coaxiaux minces (constitués d'un matériau diélectrique qui entoure un fil intérieur métallique) avec sections transverses hétérogènes. Le premier objectif, atteint dans la thèse de G. Beck, était de réduire les équations de Maxwell 3D à un graphe quantique dans lequel on se ramène au calcul du potentiel et du courant électriques en résolvant des modèles 1D simplifiés. Ainsi, l'objectif principal de cette thèse est la validation numérique de ces modèles 1D.Dans un premier temps, nous avons proposé, analysé et mis en œuvre des méthodes numériques efficaces pour résoudre les modèles simplifiés 1D. Afin de réaliser la comparaison 1D/3D, un défi majeur est de concevoir des méthodes numériques pour résoudre les équations de Maxwell 3D qui sont adaptées à la spécificité des câbles électriques fins. Une procédure de discrétisation naïve basée sur un schéma explicite saute-mouton peut être vraiment coûteuse en raison de la finesse du câble. Nous avons alors proposé une approche originale consistant à adapter les éléments d'arête "Nedelec" à des mailles prismatiques allongées et à proposer une procédure de discrétisation temporelle hybride, explicite dans les directions longitudinales et implicite dans les directions transversales. En particulier, la condition de stabilité de la CFL qui en résulte n'est pas affectée par l'épaisseur du câble.Cependant, la méthode ci-dessus n'est efficace que pour des câbles parfaitement cylindriques : son extension naïve aux câbles déformés génère un recouplage longitudinal-transversal qui détruit l'efficacité de la méthode. En présence de déformations, la méthode doit donc être modifiée. En conséquence, afin de préserver le découplage longitudinal-transversal, nous proposons une méthode hybride combinant une discrétisation conforme dans les variables longitudinales et une méthode Galerkin discontinue dans les variables transversales. Cette méthode coïncide avec la précédente dans les parties cylindriques du câble.
• Stabilisation des équations des ondes
  
  • Kafnemer Meryem
  , 2022. Cette thèse traite trois problèmes liés à la stabilisation des équations des ondes. Nous considérons différents cadres fonctionnels et nous utilisons des techniques de démonstration basées sur la méthode des multiplicateurs. Tout d'abord, nous étudions la stabilité de l'équation des ondes avec un amortissement non-linéaire et localisé dans un cadre hilbertien standard en dimension deux. La preuve est basée sur des travaux existants dans le cas d'un amortissement non-localisé. Rajoutons une localisation ainsi que des perturbations. Nous démontrons la stabilité exponentielle le long des solutions fortes en l'absence de perturbation ainsi qu'une sorte stabilité au sens Input-To-State par rapport aux perturbations considérées. Dans un deuxième travail, nous considérons un cadre fonctionnel plus général non forcément hilbertien, c-a-d un cadre L^p avec p appartient (1,infini). Nous étudions la stabilité L^p de l'équation des ondes avec un amortissement linéaire et localisé. Cette étude n'est effectuée qu'en dimension un car il n'est pas toujours possible de définir l'opérateur des ondes en dimensions supérieures lorsque p différent 2. Nous démontrons la stabilité exponentielle du problème en généralisant les multiplicateurs du cadre hilbertien dans notre cadre plus général, avec des preuves différentes suivant que 1<p> 2. Nous démontrons également dans le même problème mais avec des cas particuliers d'un amortissement global et constant, une stabilité exponentielle dans le cas p=1 et p=infini. Dans un troisième travail, nous considérons la version non-linéaire du problème précédent: en nous basant sur une technique de linéarisation, nous nous ramenons à la preuve du problème linéaire pour démontrer la stabilité exponentielle du non-linéaire.</p>
• 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 afterwards 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.
• Sketching the Best Approximate Quantum Compiling Problem
  
  • Madden Liam
  • Akhriev Albert
  • Simonetto Andrea
  , 2022. This paper considers the problem of quantum compilation from an optimization perspective by fixing a circuit structure of CNOTs and rotation gates then optimizing over the rotation angles. We solve the optimization problem classically and consider algorithmic tools to scale it to higher numbers of qubits. We investigate stochastic gradient descent and two sketch-and-solve algorithms. For all three algorithms, we compute the gradient efficiently using matrix-vector instead of matrix-matrix computations. Allowing for a runtime on the order of one hour, our implementation using either sketch-and-solve algorithm is able to compile 9 qubit, 27 CNOT circuits; 12 qubit, 24 CNOT circuits; and 15 qubit, 15 CNOT circuits. Without our algorithmic tools, standard optimization does not scale beyond 9 qubit, 9 CNOT circuits, and, beyond that, is theoretically dominated by barren plateaus. (10.1109/QCE53715.2022.00071)
  DOI : 10.1109/QCE53715.2022.00071
• 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 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 problems involving such functions. We also discuss possible modifications of the hypotheses in order to deal with quasiconvex functions. Numerical experiments confirm the theoretical results, in particular, that the relaxed-inertial algorithms outperform their ``pure'' proximal point counterparts.
• Optimizing through change for cyber-physical and social systems
  
  • Simonetto Andrea
  , 2022.
• An Outer Approximation Algorithm for 0-1 Polynomial Programming
  
  • Mencarelli Luca
  , 2022.
• Global Solution of Quadratic Problems by Interval Methods and Convex Reformulation
  
  • Elloumi Sourour
  • Lambert Amélie
  • Neveu Bertrand
  • Trombettoni Gilles
  , 2022.
• An accelerated level-set method for inverse scattering problems
  
  • Audibert Lorenzo
  • Haddar Houssem
  • Liu Xiaoli
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2022, 15 (3), pp.1576-1600. We propose a rapid and robust iterative algorithm to solve inverse acoustic scattering problems formulated as a PDE constrained shape optimization problem. We use a level-set method to represent the obstacle geometry and propose a new scheme for updating the geometry based on an adaptation of accelerated gradient descent methods. The resulting algorithm aims at reducing the number of iterations and improving the accuracy of reconstructions. To cope with regularization issues, we propose a smoothing to the shape gradient using a single layer potential associated with ik where k is the wave number. Numerical experiments are given for several data types (full aperture, backscattering, phaseless, multiple frequencies) and show that our method outperforms a non accelerated approach in terms of convergence speed, accuracy and sensitivity to initial guesses. (10.1137/21M1457783)
  DOI : 10.1137/21M1457783
• The stochastic Auxiliary Problem Principle in Banach spaces: measurability and convergence
  
  • Bittar Thomas
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • Lonchampt Jéro͂me
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2022, 32 (3), pp.1871-1900. The stochastic Auxiliary Problem Principle (APP) algorithm is a general Stochastic Approximation (SA) scheme that turns the resolution of an original optimization problem into the iterative resolution of a sequence of auxiliary problems. This framework has been introduced to design decomposition-coordination schemes but also encompasses many well-known SA algorithms such as stochastic gradient descent or stochastic mirror descent. We study the stochastic APP in the case where the iterates lie in a Banach space and we consider an additive error on the computation of the subgradient of the objective. In order to derive convergence results or efficiency estimates for a SA scheme, the iterates must be random variables. This is why we prove the measurability of the iterates of the stochastic APP algorithm. Then, we extend convergence results from the Hilbert space case to the Banach space case. Finally, we derive efficiency estimates for the function values taken at the averaged sequence of iterates or at the last iterate, the latter being obtained by adapting the concept of modified Fejér monotonicity to our framework. (10.1137/21M1402467)
  DOI : 10.1137/21M1402467