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1. Compact MILP formulations for the $p$-center problem

Author: Sourour Elloumi, Zacharie Alès, Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Centre d'Etude et De Recherche en Informatique du Cnam ( CEDRIC ), Conservatoire National des Arts et Métiers [CNAM] ( CNAM ), CEDRIC. Optimisation Combinatoire (CEDRIC - OC), Centre d'études et de recherche en informatique et communications (CEDRIC), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM)-Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Jon Lee, Giovanni Rinaldi, A. Ridha Mahjoub, Ales, Zacharie, and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)
Subjects: Discrete mathematics, 021103 operations research, p-center, equivalent formulations, 0211 other engineering and technologies, discrete location, 02 engineering and technology, [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], Optimization and Control (math.OC), [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, 020201 artificial intelligence & image processing, Center (algebra and category theory), Cover (algebra), Integer programming, integer programming, Mathematics - Optimization and Control, ComputingMilieux_MISCELLANEOUS, Mathematics, Integer (computer science), [ INFO.INFO-RO ] Computer Science [cs]/Operations Research [cs.RO]
Abstract: The p-center problem consists in selecting p centers among M to cover N clients, such that the maximal distance between a client and its closest selected center is minimized. For this problem we propose two new and compact integer formulations. Our first formulation is an improvement of a previous formulation. It significantly decreases the number of constraints while preserving the optimal value of the linear relaxation. Our second formulation contains less variables and constraints but it has a weaker linear relaxation bound. We besides introduce an algorithm which enables us to compute strong bounds and significantly reduce the size of our formulations. Finally, the efficiency of the algorithm and the proposed formulations are compared in terms of quality of the linear relaxation and computation time over instances from OR-Library., Lecture Notes in Computer Science 2018
Published: 2023

2. An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients

Author: Ciarlet, Patrick, Lassounon, David, Mahran Rihani, Ciarlet, Patrick, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Centre National de la Recherche Scientifique (CNRS)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-INSTITUT AGRO Agrocampus Ouest
Subjects: optimal control, transmission problem, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH] Mathematics [math], [MATH]Mathematics [math], sign-changing coefficients, fictitious domain methods
Abstract: In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments.
Published: 2023

3. DISCRETE HONEYCOMBS, RATIONAL EDGES AND EDGE STATES

Author: Fefferman, Charles Louis, Fliss, Sonia, Weinstein, Michael, Department of Mathematics - Princeton University, Princeton University, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Department of Applied Physics and Applied Mathematics [New York], and Columbia University [New York]
Subjects: Condensed Matter::Materials Science, honeycomb structure, tight binding model, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], edge states, discrete operator on a graph
Abstract: International audience; 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 formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l.
Published: 2022

4. The Morozov's principle applied to data assimilation problems

Author: Laurent Bourgeois, Jérémi Dardé, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Optimization and Control (math.OC), FOS: Mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control, 35J05, 35R25, 35R30, 65M60
Abstract: This paper is focused on the Morozov’s principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov’s choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov’s principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation.
Published: 2022
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5. Fourier representation of the diffusion MRI signal using layer potentials

Author: Chengran Fang, Demian Wassermann, Jing-Rebecca Li, Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École polytechnique (X)-EDF (EDF), Modèles et inférence pour les données de Neuroimagerie (MIND), IFR49 - Neurospin - CEA, Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), French National Institute for Research in Computer Science and Automation (Inria) for providing the PhD scholarship of Chengran Fang, and Li, Jing-Rebecca
Subjects: Applied Mathematics, Bloch-Torrey equation, Fourier transform, [MATH] Mathematics [math], [MATH]Mathematics [math], simulation, layer potential, diffusion magnetic resonance imaging
Abstract: The diffusion magnetic resonance imaging signal arising from biological tissues can be numerically simulated by solving the Bloch-Torrey partial differential equation. Numerical simulations can facilitate the investigation of the relationship between the diffusion MRI signals and cellular structures. With the rapid advance of available computing power, the diffusion MRI community has begun to employ numerical simulations for model formulation and validation, as well as for imaging sequence optimization.Existing simulation frameworks use the finite difference method, the finite element method, or the Matrix Formalism method to solve the Bloch-Torrey partial differential equation. We propose a new method based on the efficient evaluation of layer potentials. In this paper, the mathematical framework and the numerical implementation of the new method are described. We demonstrate the convergence of our method via numerical experiments and analyze the errors linked to various model and simulation parameters. Since our method provides a Fourier-type representation of the diffusion MRI signal, it can potentially facilitate new physical and biological signal interpretations in the future.
Published: 2022

6. An automatic PML for acoustic finite element simulations in convex domains of general shape

Author: Beriot, Hadrien, Modave, Axel, Siemens Industry Software S.r.l., Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Propagation des Ondes : Etude Mathématique et Simulation (POEMS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)
Subjects: perfectly matched layer, finite element method, wave propagation, Helmholtz equation, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], exterior acoustics, [PHYS.MECA.ACOU]Physics [physics]/Mechanics [physics]/Acoustics [physics.class-ph]
Abstract: International audience; This article addresses the efficient finite element solution of exterior acoustic problems with truncated computational domains surrounded by perfectly matched layers (PMLs). The PML is a popular nonreflecting technique that combines accuracy, computational efficiency, and geometric flexibility. Unfortunately, the effective implementation of the PML for convex domains of general shape is tricky because of the geometric parameters that are required to define the PML medium. In this work, a comprehensive implementation strategy is proposed. This approach, which we call the automatically matched layer (AML) implementation, is versatile and fully automatic for the end‐user. With the AML approach, the mesh of the layer is extruded, the required geometric parameters are automatically obtained during the extrusion step, and the practical implementation relies on a simple modification of the Jacobian matrix in the elementwise integrals. The AML implementation is validated and compared with other implementation strategies using numerical benchmarks in two and three dimensions, considering computational domains with regular and nonregular boundaries. A three‐dimensional application with a generally shaped domain generated using a convex hull is proposed to illustrate the interest of the AML approach for realistic industrial cases.
Published: 2021

7. Solving 2D linear isotropic elastodynamics by means of scalar potentials: a new challenge for finite elements

Author: Patrick Joly, Jerónimo Rodríguez, Sébastien Imperiale, Jorge Martínez, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela [Spain] (USC ), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Universidade de Santiago de Compostela, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine ( M3DISIM ), Laboratoire de mécanique des solides ( LMS ), École polytechnique ( X ) -MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique ( CNRS ) -École polytechnique ( X ) -MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique ( CNRS ) -Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Propagation des Ondes : Étude Mathématique et Simulation ( POEMS ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -Centre National de la Recherche Scientifique ( CNRS ), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris
Subjects: Numerical Analysis, Applied Mathematics, Isotropy, Scalar (mathematics), Mathematical analysis, General Engineering, 010103 numerical & computational mathematics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Finite element method, Theoretical Computer Science, 010101 applied mathematics, Computational Mathematics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Computational Theory and Mathematics, Displacement field, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Time domain, 0101 mathematics, Scalar field, Helmholtz decomposition, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: International audience; In this work we present a method for the computation of numerical solutions of 2D homogeneous isotropic elastodynamics equations by solving scalar wave equations. These equations act on the potentials of a Helmholtz decomposition of the displacement field and are decoupled inside the propagation domain. We detail how these equations are coupled at the boundary depending on the nature of the boundary condition satisfied by the displacement field. After presenting the case of rigid boundary conditions, that presents no specific difficulty, we tackle the challenging case of free surface boundary conditions that presents severe stability issues if a straightforward approach is used. We introduce an adequate functional framework as well as a time domain mixed formulation to circumvent these issues. Numerical results confirm the stability of the proposed approach.
Published: 2018

8. On measures in sub-Riemannian geometry

Author: Roberta Ghezzi, Frédéric Jean, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), This work was supported by iCODE (Institute for Control and Decision), research project of the IDEX Paris–Saclay, and by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Programme Gaspard Monge en Optimisation et Recherche Opérationnelle.iCODE (Institute for Control and Decision), research project of the IDEX Paris–Saclay, ANR-15-CE40-0018,SRGI,Sub-Riemannian Geometry and Interactions ( 2015 ), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), and ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Pure mathematics, 021103 operations research, 010102 general mathematics, 0211 other engineering and technologies, Hausdorff space, 02 engineering and technology, Riemannian geometry, 16. Peace & justice, 01 natural sciences, Measure (mathematics), Complement (complexity), [ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG], symbols.namesake, Mathematics - Metric Geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Metric (mathematics), symbols, [ MATH.MATH-MG ] Mathematics [math]/Metric Geometry [math.MG], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics::Differential Geometry, 0101 mathematics, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics
Abstract: International audience; In \cite{gjha} we give a detailed analysis of spherical Hausdorff measures on sub-Riemannian manifolds in a general framework, that is, without the assumption of equiregularity. The present paper is devised as a complement of this analysis, with both new results and open questions. The first aim is to extend the study to other kinds of intrinsic measures on sub-Riemannian manifolds, namely Popp's measure and general (i.e., non spherical) Hausdorff measures. The second is to explore some consequences of \cite{gjha} on metric measure spaces based on sub-Riemannian manifolds.
Published: 2018

9. Minimal graphs for matching extensions

Author: Christophe Picouleau, Marie-Christine Costa, D. de Werra, Centre d'Etude et De Recherche en Informatique et Communications ( CEDRIC ), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise ( ENSIIE ) -Conservatoire National des Arts et Métiers [CNAM] : EA4629, Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Ecole Polytechnique Fédérale de Lausanne ( EPFL ), CEDRIC. Optimisation Combinatoire (CEDRIC - OC), Centre d'études et de recherche en informatique et communications (CEDRIC), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM)-Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), and Ecole Polytechnique Fédérale de Lausanne (EPFL)
Subjects: Discrete mathematics, Factor-critical graph, expandable graph, 021103 operations research, Matching (graph theory), completable graph, Applied Mathematics, matching extension, 0211 other engineering and technologies, [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], 0102 computer and information sciences, 02 engineering and technology, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], 01 natural sciences, Graph, Combinatorics, maximum matching, 010201 computation theory & mathematics, Graph power, Bipartite graph, Discrete Mathematics and Combinatorics, Bound graph, Graph factorization, Mathematics, [ INFO.INFO-RO ] Computer Science [cs]/Operations Research [cs.RO]
Abstract: Given a positive integer n we find a graph G = ( V , E ) on | V | = n vertices with a minimum number of edges such that for any pair of non adjacent vertices x , y the graph G − x − y contains a (almost) perfect matching M . Intuitively the non edge x y and M form a (almost) perfect matching of G . Similarly we determine a graph G = ( V , E ) with a minimum number of edges such that for any matching M of the complement G of G with size ⌊ n 2 ⌋ − 1 , G − V ( M ) contains an edge e . Here M and the edge e of G form a (almost) perfect matching of G . We characterize these minimal graphs for all values of n .
Published: 2018

10. Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media

Author: Rémi Cornaggia, Marc Bonnet, Bojan B. Guzina, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Civil Engineering [Minneapolis], University of Minnesota [Twin Cities] (UMN), University of Minnesota System-University of Minnesota System, ANR-11-LABX-0020-01, Henri Lebesgue Centre, ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Propagation des Ondes : Étude Mathématique et Simulation ( POEMS ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), University of Minnesota [Minneapolis], Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
Subjects: Physics, waves in periodic media, Applied Mathematics, Macroscopic model, AMS subject classifications. 35B27, 35J05, 74H10, 74Q10, Perturbation (astronomy), periodic perturbations, Polarization (waves), Topology, 01 natural sciences, 010305 fluids & plasmas, second-order homogenization, AMS: 35B27, 35J05, 74H10, 74Q10, 010101 applied mathematics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Periodic perturbation, Topological sensitivity, 0103 physical sciences, [PHYS.MECA.SOLID]Physics [physics]/Mechanics [physics]/Solid mechanics [physics.class-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Anisotropy, Dispersion (water waves), Scalar field, [ PHYS.MECA.SOLID ] Physics [physics]/Mechanics [physics]/Mechanics of the solides [physics.class-ph]
Abstract: International audience; We consider scalar waves in periodic media through the lens of a second-order effective i.e. macroscopic description, and we aim to compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. The results demonstrate that the sought sensitivities are computable in terms of (i) three unit-cell solutions used to formulate the unperturbed macroscopic model; (ii) two adoint-field solutions driven by the mass density variation inside the unperturbed unit cell; and (iii) the usual polarization tensor, appearing in the related studies of non-periodic media, that synthesizes the geometric and constitutive features of a point-like perturbation. The proposed developments may be useful toward (a) the design of periodic media to manipulate macroscopic waves via the microstructure-generated effects of dispersion and anisotropy, and (b) sub-wavelength sensing of periodic defects or perturbations.
Published: 2018

11. Influence of the neck shape for Helmholtz resonators

Author: Jean-François Mercier, Agnès Maurel, Jean-Jacques Marigo, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Centre National de la Recherche Scientifique (CNRS)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS), Institut Langevin ondes et images, Centre National de la Recherche Scientifique (CNRS)-ESPCI ParisTech-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut Langevin - Ondes et Images (UMR7587) (IL), Ecole Superieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Acoustics and Ultrasonics, Quantitative Biology::Tissues and Organs, Physics::Medical Physics, 02 engineering and technology, 01 natural sciences, Homogenization (chemistry), law.invention, Resonator, symbols.namesake, Arts and Humanities (miscellaneous), law, 0103 physical sciences, [PHYS.MECA.SOLID]Physics [physics]/Mechanics [physics]/Solid mechanics [physics.class-ph], [SPI.NANO]Engineering Sciences [physics]/Micro and nanotechnologies/Microelectronics, 010301 acoustics, Helmholtz resonator, ComputingMilieux_MISCELLANEOUS, Physics, [SPI.ACOU]Engineering Sciences [physics]/Acoustics [physics.class-ph], Mathematical analysis, Resonance, 021001 nanoscience & nanotechnology, Helmholtz free energy, symbols, 0210 nano-technology, Acoustic resonance
Abstract: International audience; The resonance of a Helmholtz resonator is studied with a focus on the influence of the neck shape. This is done using a homogenization approach developed for an array of resonators, and the resonance of an array is discussed when compared to that of a single resonator. The homogenization makes a parameter ℬ appear which determines unambiguously the resonance frequency of any neck. As expected, this parameter depends on the length and on the minimum opening of the neck, and it is shown to depend also on the surface of air inside the neck. Once these three geometrical parameters are known, ℬ has an additional but weak dependence on the neck shape, with explicit bounds.
Published: 2017

12. Shape Optimization of Peristaltic Pumps Transporting Rigid Particles in Stokes Flow

Author: Marc Bonnet, Ruowen Liu, Shravan Veerapaneni, Hai Zhu, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics - University of Michigan, University of Michigan [Ann Arbor], and University of Michigan System-University of Michigan System
Subjects: Physics::Fluid Dynamics, integral equations, Computational Mathematics, fast algorithms, Optimization and Control (math.OC), Applied Mathematics, FOS: Mathematics, Shape sensitivity analysis, particulate flows, Mathematics - Optimization and Control, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation
Abstract: This paper presents a computational approach for finding the optimal shapes of peristaltic pumps transporting rigid particles in Stokes flow. In particular, we consider shapes that minimize the rate of energy dissipation while pumping a prescribed volume of fluid, number of particles and/or distance traversed by the particles over a set time period. Our approach relies on a recently developed fast and accurate boundary integral solver for simulating multiphase flows through periodic geometries of arbitrary shapes. In order to fully capitalize on the dimensionality reduction feature of the boundary integral methods, shape sensitivities must ideally involve evaluating the physical variables on the particle or pump boundaries only. We show that this can indeed be accomplished owing to the linearity of Stokes flow. The forward problem solves for the particle motion in a slip-driven pipe flow while the adjoint problems in our construction solve quasi-static Dirichlet boundary value problems backwards in time, retracing the particle evolution. The shape sensitivities simply depend on the solution of one forward and one adjoint (for each shape functional) problems. We validate these analytic shape derivative formulas by comparing against finite-difference based gradients and present several examples showcasing optimal pump shapes under various constraints., Comment: 24 pages, 8 figures
Published: 2023

13. Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

Author: Thomas Strauszer‐Caussade, Luiz M. Faria, Agustín Fernandez‐Lado, Carlos Pérez‐Arancibia, Pontificia Universidad Católica de Chile (UC), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Intel Corporation [USA], University of Twente, Mathematics of Computational Science, and MESA+ Institute
Subjects: Applied Mathematics, FOS: Mathematics, UT-Hybrid-D, FOS: Physical sciences, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Computational Physics (physics.comp-ph), [MATH]Mathematics [math], Physics - Computational Physics, Physics - Optics, Optics (physics.optics)
Abstract: International audience; 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.
Published: 2023

14. Variational methods for solving numerically magnetostatic systems

Author: Ciarlet, Patrick, Jamelot, E., Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), CEA- Saclay (CEA), and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)
Subjects: variational formulations, edge finite elements, T-coercivity, Magnetostatic systems, [MATH]Mathematics [math], Lagrange finite elements
Abstract: In this paper, we study some techniques for solving numerically magnetostatic systems. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements, or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of T-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on on an augmented variational formulation obtained with the help of the Weighted Regularization Method.
Published: 2023

15. Study of a degenerate non-elliptic equation to model plasma heating

Author: Ciarlet, Patrick, Kachanovska, Maryna, PEILLON, Etienne, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH]Mathematics [math]
Abstract: In this manuscript, we study solutions to resonant Maxwell's equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω 2 u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [13] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series a numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence.
Published: 2023

16. On the convergence analysis of one-shot inversion methods

Author: Bonazzoli, Marcella, Haddar, Houssem, Vu, Tuan Anh, Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)
Subjects: parameter identification, one-shot methods, inverse problems, [MATH]Mathematics [math], convergence analysis
Abstract: When an inverse problem is solved by a gradient-based optimization algorithm, the corresponding forward and adjoint problems, which are introduced to compute the gradient, can be also solved iteratively. The idea of iterating at the same time on the inverse problem unknown and on the forward and adjoint problem solutions yields to the concept of one-shot inversion methods. We are especially interested in the case where the inner iterations for the direct and adjoint problems are incomplete, that is, stopped before achieving a high accuracy on their solutions. Here, we focus on general linear inverse problems and generic fixed-point iterations for the associated forward problem. We analyze variants of the so-called multi-step one-shot methods, in particular semi-implicit schemes with a regularization parameter. We establish sufficient conditions on the descent step for 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 gradient descent, where the forward and adjoint problems are solved exactly by a direct solver instead. We observe that very few inner iterations are enough to guarantee good convergence of the inversion algorithm, even in the presence of noisy data.
Published: 2023

17. Global solution of non-convex quadratically constrained quadratic programs

Author: Amélie Lambert, Sourour Elloumi, Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Centre d'Etude et De Recherche en Informatique du Cnam ( CEDRIC ), Conservatoire National des Arts et Métiers [CNAM] ( CNAM ), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), CEDRIC. Optimisation Combinatoire (CEDRIC - OC), Centre d'études et de recherche en informatique et communications (CEDRIC), and Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM)-Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM)
Subjects: Global optimiza- tion, Mathematical optimization, Control and Optimization, 0211 other engineering and technologies, 010103 numerical & computational mathematics, 02 engineering and technology, 01 natural sciences, Quadratic residue, Quadratic programming, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, [ INFO.INFO-RO ] Computer Science [cs]/Operations Research [cs.RO], Quadratic growth, Quadratically constrained quadratic program, 021103 operations research, Applied Mathematics, General mixed-integer quadratic programming, [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], Quadratic function, Isotropic quadratic form, Spatial branch-and-bound, Second-order cone programming, Binary quadratic form, Quadratic convex relaxation, Experiments, Software
Abstract: International audience; The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QC-QPs with the following restriction : all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P *). This equivalent problem (P *) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y = xx T , where x are the initial variables of problem (P). We then propose to solve (P *) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO.
Published: 2017

18. Solving chance constrained optimal control problems in aerospace via Kernel Density Estimation

Author: Emmanuel Trélat, Achille Sassi, Max Cerf, Jean-Baptiste Caillau, Hasnaa Zidani, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Mathematics for Control, Transport and Applications ( McTAO ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Airbus Safran Launchers, Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Control And GEometry ( CaGE ), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria de Paris, Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), ANR : IDEX PARIS SACLAY,ANR-11-IDEX-0003-02 IDEX Paris-Saclay, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), ANR-11-IDEX-0003,IPS,Idex Paris-Saclay(2011), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), ANR: IDEX PARIS SACLAY,ANR-11-IDEX-0003-02 IDEX Paris-Saclay, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Mathematical optimization, Control and Optimization, chance constrained optimization, Kernel density estimation, 0211 other engineering and technologies, Probability density function, 02 engineering and technology, 01 natural sciences, Kernel Density Estimation, 010104 statistics & probability, 0101 mathematics, Mathematics, 021103 operations research, Applied Mathematics, Constrained optimization, Trajectory optimization, stochastic optimization, Optimal control, Distribution (mathematics), Aerospace engineering, Control and Systems Engineering, Stochastic optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Random variable, Software
Abstract: International audience; The goal of this paper is to show how non-parametric statistics can be used to solve some chance constrained optimization and optimal control problems. We use the Kernel Density Estimation method to approximate the probability density function of a random variable with unknown distribution , from a relatively small sample. We then show how this technique can be applied and implemented for a class of problems including the God-dard problem and the trajectory optimization of an Ariane 5-like launcher.
Published: 2017

19. Optimization of wireless sensor networks deployment with coverage and connectivity constraints

Author: Olivier Hudry, Sourour Elloumi, Agnès Plateau, Stephane Rovedakis, Estel Marie, Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Centre d'Etude et De Recherche en Informatique du Cnam ( CEDRIC ), Conservatoire National des Arts et Métiers [CNAM] ( CNAM ), Institut Télécom, Télécom ParisTech, Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Centre d'études et de recherche en informatique et communications (CEDRIC), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM), CEDRIC. Réseaux et Objets Connectés (CEDRIC - ROC), and Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM)-Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM)
Subjects: Engineering, 021103 operations research, business.industry, 0211 other engineering and technologies, 0102 computer and information sciences, 02 engineering and technology, [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], 01 natural sciences, Key distribution in wireless sensor networks, Tree (data structure), Base station, 010201 computation theory & mathematics, Video tracking, Sensor node, Mobile wireless sensor network, Central processing unit, business, Wireless sensor network, ComputingMilieux_MISCELLANEOUS, Computer network, [ INFO.INFO-RO ] Computer Science [cs]/Operations Research [cs.RO]
Abstract: Wireless sensor networks have been widely deployed in the last decades to provide various services, like environmental monitoring or object tracking. Such a network is composed of a set of sensor nodes which are used to sense and transmit collected information to a base station. To achieve this goal, two properties have to be guaranteed: (i) the sensor nodes must be placed such that all the environment of interest is covered, and (ii) every sensor node can transmit its data to the base station (through other sensor nodes). In this paper, we consider the Minimum Connected Coverage (MCC) problem. We propose two mathematical programming formulations for the MCC problem on square grid graphs. We compare them to a recent model proposed by Rebai et al [1]. Our mathematical programming formulations yield a better LP-bound at the root of the branch-and-cut process than the model of Rebai et al. Moreover, the presented formulations outperform the proportion of solved instances in their work as well as the CPU computation time and the number of nodes explored in the tree search.
Published: 2017

20. Active Design of Diffuse Acoustic Fields in Enclosures

Author: Aquino, Wilkins, Rouse, Jerry, Bonnet, Marc, Duke University [Durham], Sandia National Laboratories [Albuquerque] (SNL), Sandia National Laboratories - Corporation, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), and This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly-owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-NA-0003525.
Subjects: [SPI.ACOU]Engineering Sciences [physics]/Acoustics [physics.class-ph]
Abstract: This paper presents a numerical framework for designing diffuse fields in rooms of any shape and size, driven at arbitrary frequencies. That is, we aim at overcoming the Schroeder frequency limit for generating diffuse fields in an enclosed space. We formulate the problem as a Tikhonov regularized inverse problem and propose a lowrank approximation of the spatial correlation that results in significant computational gains. Our approximation is applicable to arbitrary sets of target points and allows us to produce an optimal design at a computational cost that grows only linearly with the (potentially large) number of target points. We demonstrate the feasibility of our approach through numerical examples where we approximate diffuse fields at frequencies well below the Schroeder limit.
Published: 2023

21. Solvability results for the transient acoustic scattering by an elastic obstacle

Author: Bonnet, Marc, Chaillat, Stéphanie, Nassor, Alice, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Fluid-solid interaction, [SPI]Engineering Sciences [physics], Transient acoustic-elastic scattering, well-posedness, Transient acoustic-elastic scattering Fluid-solid interaction well-posedness, [MATH]Mathematics [math]
Abstract: The well-posedness of the linear evolution problem governing the transient scattering of acoustic waves by an elastic obstacle is investigated. After using linear superposition in the acoustic domain, the analysis focuses on an equivalent causal transmission problem. The proposed analysis provides existence and uniqueness results, as well as continuous data-to-solution maps. Solvability results are established for three cases, which differ by the assumed regularity in space on the transmission data on the acoustic-elastic interface Γ. The first two results consider data with "standard" H −1/2 (Γ) and improved H 1/2 (Γ) regularity in space, respectively, and are established using the Hille-Yosida theorem and energy identities. The third result assumes data with L 2 (Γ) regularity in space and follows by Sobolev interpolation. Obtaining the latter result was motivated by the key role it plays (in a separate study) in the justification of an iterative numerical solution method based on domain decomposition. A numerical example is presented to emphasize the latter point.
Published: 2023

22. A new class of uniformly stable time-domain Foldy-Lax models for scattering by small particles. Acoustic sound-soft scattering by circles. Extended version

Author: Kachanovska, Maryna, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Wave equation, asymptotic model, time-domain boundary integral equations, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this work we study time-domain sound-soft scattering by small circles. Our goal is to derive an asymptotic model for this problem valid when the size of the particles tends to zero. We present a systematic approach to constructing such models, based on a well-chosen Galerkin discretization of a boundary integral equation. The convergence of the method is achieved by decreasing the asymptotic parameter rather than increasing the number of basis functions. For the case of circular obstacles, we prove the second-order convergence of the field error with respect to the particle size. Our findings are illustrated with numerical experiments.
Published: 2023

23. REDUCED DISSIPATION EFFECT IN STOCHASTIC TRANSPORT BY GAUSSIAN NOISE WITH REGULARITY GREATER THAN 1/2

Author: Flandoli, Franco, Russo, Francesco, Scuola Normale Superiore di Pisa (SNS), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), ANR-22-CE40-0015,SDAIM,Analyse stochastique et déterministe pour des modèles irréguliers(2022), and European Project: ERC,Noisy Fluid
Subjects: Transport equation, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Stochastic turbulence modeling, Probability (math.PR), Stationary Gaussian noise, 76M35, 60H07, 60H15, 60G22, 76F55, FOS: Mathematics, Malliavin Calculus, Fractional Brownian motion, Mathematics - Probability, Stochastic PDEs
Abstract: Diffusion with stochastic transport is investigated here when the random driving process is a very general Gaussian process including Fractional Brownian motion. The purpose is the comparison with a deterministic PDE which in certain cases represents the equation for the mean value. From this equation we observe a reduced dissipation property with respect to delta correlated noise when regularity is higher than the one of Brownian motion, a fact interpreted qualitatively here as a signature of the reduced dissipation observed for 2D turbulent fluids due to the inverse cascade. We give results also for the variance of the solution and for a scaling limit of a two-component noise input.
Published: 2023

24. A second order asymptotic model for diffusion MRI in permeable media

Author: Kchaou, Marwa, Li, Jing-Rebecca, Université de Tunis El Manar (UTM), Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)
Subjects: diffusion MRI, Homogenization, Bloch–Torrey equation, higher-order diffusion tensor, [MATH]Mathematics [math]
Abstract: International audience; Starting from a reference partial differential equation model of the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses, one can use periodic homogenization theory to establish macroscopic models. A previous work introduced an asymptotic model that accounted for permeable interfaces in the imaging medium. In this paper we formulate a higher order asymptotic model to treat higher values of permeability. We explicitly solved this new asymptotic model to obtain a system of ordinary differential equations that can model the diffusion MRI signal and we present numerical results showing the improved accuracy of the new model in the regime of higher permeability.
Published: 2023

25. Long-time behaviour of the solution of Maxwell’s equations in dissipative generalized Lorentz materials (I): a frequency-dependent Lyapunov function approach

Author: Cassier, Maxence, Joly, Patrick, Rosas Martínez, Luis Alejandro, EPSILON (EPSILON), Institut FRESNEL (FRESNEL), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)
Subjects: passive electromagnetic media, Applied Mathematics, General Mathematics, FOS: Physical sciences, General Physics and Astronomy, dissipative generalized Lorentz models, Mathematical Physics (math-ph), [SPI.ELEC]Engineering Sciences [physics]/Electromagnetism, frequency-dependent Lyapunov estimates, Mathematics - Analysis of PDEs, Maxwell's equations, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 35Q60, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematical Physics, Analysis of PDEs (math.AP), long time electromagnetic energy decay rate
Abstract: It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances., 33 pages, 1 figure
Published: 2023

26. Maxwell's equations with hypersingularities at a negative index material conical tip

Author: Dhia, Anne-Sophie Bonnet-Ben, Chesnel, Lucas, Rihani, Mahran, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Inversion of Differential Equations For Imaging and physiX (IDEFIX), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École polytechnique (X)-EDF (EDF), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), and École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)
Subjects: black hole singularities, MSC: 78A25, 35Q61, 78A48, 35B65, 78M30, Mathematics - Analysis of PDEs, Maxwell's equations, Mandelstam radiation principle, limiting radiation principle, FOS: Mathematics, negative index materials, [MATH]Mathematics [math], Kondratiev theory, Analysis of PDEs (math.AP)
Abstract: We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity $\varepsilon$ and the permeability $\mu$ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in $\varepsilon$ and in $\mu$, the corresponding scalar operators are not of Fredholm type in the usual $H^1$ spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical $L^2$ framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the $\mathrm{T}$-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.
Published: 2023

27. On the development and use of higher-order asymptotics for solving inverse scattering problems

Author: Cornaggia, Rémi, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay, Marc Bonnet, Bojan Guzina, STAR, ABES, Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Université Paris Saclay (COmUE), University of Minnesota, and Cornaggia, Rémi
Subjects: [PHYS.MECA.SOLID] Physics [physics]/Mechanics [physics]/Solid mechanics [physics.class-ph], méthodes asymptotiques, homogénéisation, homogenization, asymptotic methods, inverse scattering, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], élastodynamique, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, elastodynamics, transmission eigenvalues, topological derivatives, Derivée topologique, diffraction inverse, [PHYS.MECA.SOLID]Physics [physics]/Mechanics [physics]/Solid mechanics [physics.class-ph], Topological derivative, dérivées topologiques, [INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation, Valeurs propres de tranmission, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], valeurs propres de transmission
Abstract: The purpose of this work was to develop new methods to address inverse problems in elasticity, taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptotic expansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity B in a 3D elastic domain. In this goal, we focus on the study of functionals J(Ba) quantifying the misfit between B and a trial homogeneity Ba . Such functionals are to be minimized w.r.t. some or allthe characteristics of the trial inclusion Ba (location, size, mechanical properties ...) to find the best agreement with B. To this end, we produce an expansion of J with respect to the size of Ba , providing a polynomial approximation easier to minimize. This expansion, established up to the sixth order in a volume integral equations framework, is justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space.The main purpose of this second part is to characterize a microstructured two-phases layered 1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigenvalues of the so-called interior transmission problem (ITP). To providea convenient invertible model, while accounting for the microstructure effects, we rely on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method to recover the macroscopic parameters (length and material contrast) of such inclusion. To access the key features of the microstructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions., L'objectif de ce travail est le développement de nouvelles méthodes pour aborder certains problèmes inverses en élasticité, en tirant parti de la présence d'un petit paramètre dans ces problèmes pour construire des approximations asymptotiques d'ordre élevé.La première partie est consacrée à l’identification de la taille et la position d’une inhomogénéité B enfouie dans un domaine élastique tridimensionnel. Nous nous concentrons sur l’étude de fonctions-coûts J(Ba) quantifiant l’écart entre B et une hétérogénéité “test” Ba . Une telle fonction-coût peut en effet être minimisée par rapport à tout ou partie des caractéristiques de l’inclusion “test” Ba (position, taille,propriétés mécaniques ...) pour établir la meilleure correspondance possible entre Ba et B. A cet effet, nous produisons un développement asymptotique de J en la taille de Ba , qui en constitue une approximation polynomiale plus aisée à minimiser. Ce développement, établi jusqu’à l’ordre six, est justifié par une estimation du résidu. Une méthode d’identification adaptée est ensuite présentée et illustrée par des exemples numériques portant sur des obstacles de formes simples dans l’espace libre.L’objet de la seconde partie est de caractériser une inclusion microstructurée de longueur L, modélisée en une dimension, composée de couches de deux matériaux alternés périodiquement, en supposant que les plus basses de ses fréquences propres de transmission (TEs) sont connues. Ces fréquences sont les valeurs propres d’un problème dit de transmission intérieur (ITP). Afin de disposer d’un modèle propice à l’inversion, tout en prenant en compte les effets de la microstructure, nous nous reposons sur des approximations de l’ITP exact obtenues par homogénéisation. A partir du modèle homogénéisé d’ordre 0, nous établissons tout d’abord une méthode simple pour déterminer les paramètres macroscopiques(longueur et contrastes matériaux) d’une telle inclusion. Pour avoir accès à la période de la microstructure, nous nous intéressons ensuite à des modèles homogénéisés d’ordre élevé, pour lesquels nous soulignons le besoin de conditions aux limites adaptées.
Published: 2016

28. Développement et utilisation de méthodes asymptotiques d'ordre élevé pour la résolution de problèmes de diffraction inverse

Author: Cornaggia, Rémi, Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay (COmUE), University of Minnesota, and Marc Bonnet
Subjects: Homogenization, Derivée topologique, Homogénéisation, Asymptotic methods, Inverse scattering, Élastodynamique, Elastodynamics, Transmission eigenvalues, Topological derivative, Diffraction inverse, Méthodes asymptotiques, Valeurs propres de tranmission, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: The purpose of this work was to develop new methods to address inverse problems in elasticity,taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptoticexpansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity Bᵗʳᵘᵉ in a 3Delastic domain. In this goal, we focused on the study of functionals ��(Bₐ) quantifying the misfit between Bᵗʳᵘᵉ and a trial homogeneity Bₐ. Such functionals are to be minimized w.r.t. some or all the characteristics of the trial inclusion Bₐ (location, size, mechanical properties ...) to find the best agreement with Bᵗʳᵘᵉ. To this end, we produced an expansion of �� with respect to the size a of Bₐ, providing a polynomial approximation easier to minimize. This expansion, established up to O(a⁶) in a volume integral equations framework, is justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space ℝ³.The main purpose of this second part is to characterize a microstructured two-phases layered1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigen values of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we then relied on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method tore cover the macroscopic parameters (L and material contrast) of such inclusion. To access to the period of themicrostructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.; L'objectif de ce travail fut le développement de nouvelles méthodes pour aborder certains problèmes inverses en élasticité, en tirant parti de la présence d'un petit paramètre dans ces problèmes pour construire des approximation asymptotiques d'ordre élevé. La première partie est consacrée à l'identification de la taille et la position d'une inhomogénéité Bᵗʳᵘᵉ enfouie dans un domaine élastique tridimensionnel. Nous nous concentrons sur l'étude de fonctions-coûts ��(Bₐ) quantifiant l'écart entre Bᵗʳᵘᵉ et une hétérogénéité ``test'' Bₐ. Une telle fonction-coût peut en effet être minimisée par rapport à tout ou partie des caractéristiques de l'inclusion ``test'' Bₐ (position, taille, propriétés mécaniques ...) pour établir la meilleure correspondance possible entre Bₐ et Bᵗʳᵘᵉ. A cet effet, nous produisons un développement asymptotique de �� en la taille a de Bₐ, qui en constitue une approximation polynomiale plus aisée à minimiser. Ce développement, établi jusqu'à l'ordre O(a⁶), est justifié par une estimation du résidu. Une méthode d'identification adaptée est ensuite présentée et illustrée par des exemples numériques portant sur des obstacles de formes simples dans l'espace libre ℝ³.L'objet de la seconde partie est de caractériser une inclusion microstructurée de longueur L, modélisée en une dimension, composée de couches de deux matériaux alternés périodiquement, en supposant que les plus basses de ses fréquences propres de transmission (TEs) sont connues. Ces fréquences sont les valeurs propres d'un problème dit de transmission intérieur (ITP). Afin de disposer d'un modèle propice à l'inversion, tout en prenant en compte les effets de la microstructure, nous nous reposons sur des approximations de l'ITP exact obtenues par homogénéisation. A partir du modèle homogénéisé d'ordre 0, nous établissons tout d'abord une méthode simple pour déterminer les paramètres macroscopiques (L et contrastes matériaux)d'une telle inclusion. Pour avoir accès à la période de la microstructure, nous nous intéressons ensuite à des modèles homogénéisés d'ordre élevé, pour lesquels nous soulignons le besoin de conditions aux limites adaptées.
Published: 2016

29. Using a conic bundle method to accelerate both phases of a quadratic convex reformulation

Author: Amélie Lambert, Sourour Elloumi, Alain Billionnet, Angelika Wiegele, Centre d'études et de recherche en informatique et communications (CEDRIC), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), CEDRIC. Optimisation Combinatoire (CEDRIC - OC), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM)-Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM), Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Austria, Centre d'Etude et De Recherche en Informatique du Cnam ( CEDRIC ), Conservatoire National des Arts et Métiers [CNAM] ( CNAM ), Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), and École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech )
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Mathematical optimization, [ INFO ] Computer Science [cs], 0211 other engineering and technologies, MathematicsofComputing_NUMERICALANALYSIS, 0102 computer and information sciences, 02 engineering and technology, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], 01 natural sciences, Quadratic equation, FOS: Mathematics, [INFO]Computer Science [cs], Subgradient method, Mathematics - Optimization and Control, ComputingMilieux_MISCELLANEOUS, [ INFO.INFO-RO ] Computer Science [cs]/Operations Research [cs.RO], Mathematics, Semidefinite programming, Quadratically constrained quadratic program, 021103 operations research, General Engineering, Regular polygon, [ INFO.INFO-DM ] Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], Solver, Optimization and Control (math.OC), 010201 computation theory & mathematics, Conic section, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Conic optimization
Abstract: We present algorithm MIQCR-CB that is an advancement of MIQCR. MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem (SDP); in the second phase, the equivalent formulation is solved by a standard solver. Because the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers even for medium-sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving (SDP) that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method, which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm. First, we apply MIQCR-CB to the k-cluster problem that can be formulated by a binary quadratic program. As an illustration of the efficiency of our new algorithm, for instances of size 80 and of density 25%, MIQCR-CB is on average 78 times faster for phase 1 and 24 times faster for phase 2 than the original MIQCR. We also compare MIQCR-CB with QCR and with BiqCrunch, two methods devoted to binary quadratic programming. We show that MIQCR-CB is able to solve most of the 225 considered instances within three hours of CPU time. We also present experiments on two classes of general integer instances where we compare MIQCR-CB with MIQCR, Couenne, and Cplex12.6. We demonstrate the significant improvement over the original MIQCR approach. Finally, we show that MIQCR-CB is able to solve almost all of the considered instances, whereas Couenne and Cplex12.6 are not able to solve half of them.
Published: 2016

30. Characterization and Genetic Analyses of New Genes Coding for NOD2 Interacting Proteins

Author: Jean-Pierre Hugot, Stephane Parnis, Patrick Lécine, Jean-Paul Borg, Leigh Pascoe, Raphaële Thiébaut, Sophie Esmiol, Cendrine Nicoletti, Aurélie Hermant, Julie Perroy, Vincent Ollendorff, Batoul Mahfouz, Centre de Recherche en Cancérologie de Marseille (CRCM), Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut Paoli-Calmettes, Fédération nationale des Centres de lutte contre le Cancer (FNCLCC)-Fédération nationale des Centres de lutte contre le Cancer (FNCLCC)-Aix Marseille Université (AMU), Dynamique Musculaire et Métabolisme (DMEM), Institut National de la Recherche Agronomique (INRA)-Université de Montpellier (UM), Centre de recherche sur l'Inflammation (CRI (UMR_S_1149 / ERL_8252 / U1149)), Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7), Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems (Commands), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Institut des Sciences Moléculaires de Marseille (ISM2), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Génomique Fonctionnelle (IGF), Université de Montpellier (UM)-Université Montpellier 1 (UM1)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Montpellier 2 - Sciences et Techniques (UM2)-Centre National de la Recherche Scientifique (CNRS), Fondation Jean Dausset CEPH, Assistance publique - Hôpitaux de Paris (AP-HP) (APHP), Aix Marseille Université (AMU)-Institut Paoli-Calmettes, Fédération nationale des Centres de lutte contre le Cancer (FNCLCC)-Fédération nationale des Centres de lutte contre le Cancer (FNCLCC)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Ollendorff, Vincent, Université Paris Diderot - Paris 7 (UPD7)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS), Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), AP-HP Hôpital universitaire Robert-Debré [Paris], Assistance publique - Hôpitaux de Paris (AP-HP) (AP-HP), ANR-11-IDEX-0005,USPC,Université Sorbonne Paris Cité(2011), Université de Montpellier (UM)-Institut National de la Recherche Agronomique (INRA), Centre National de la Recherche Scientifique (CNRS), Institut National de la Santé et de la Recherche Médicale (INSERM), Université de Montpellier (UM), Fondation Jean Dausset - Centre d’Étude du Polymorphisme Humain, ANR-11-IDEX-0005-02/10-LABX-0017,INFLAMEX,INSTITUTE OF INFLAMMATORY DISEASES(2011), Aix Marseille Université (AMU)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Institut de Chimie du CNRS (INC), ProdInra, Archive Ouverte, and Université Sorbonne Paris Cité - - USPC2011 - ANR-11-IDEX-0005 - IDEX - VALID
Subjects: 0301 basic medicine, Co-immunoprecipitation, [SDV]Life Sciences [q-bio], Nod2 Signaling Adaptor Protein, lcsh:Medicine, Biochemistry, Interactome, Crohn Disease, Intestinal mucosa, NOD2, Protein Interaction Mapping, Yellow fluorescent protein, Small interfering RNAs, lcsh:Science, Genetics, Multidisciplinary, NF-kappa B, Complementary DNA, Precipitation Techniques, Enzymes, [SDV] Life Sciences [q-bio], Nucleic acids, CHMP5, Oxidoreductases, Luciferase, Research Article, Forms of DNA, Colon, Two-hybrid screening, Biology, Research and Analysis Methods, Transfection, Polymorphism, Single Nucleotide, ESCRT, Cell Line, 03 medical and health sciences, Transcription factors, Humans, Immunoprecipitation, Non-coding RNA, Molecular Biology Techniques, Molecular Biology, Gene, Transcription factor, Evolutionary Biology, Population Biology, Macrophages, lcsh:R, Biology and Life Sciences, Proteins, DNA, digestive system diseases, Gene regulation, Luminescent Proteins, 030104 developmental biology, Haplotypes, Mutation, Enzymology, RNA, lcsh:Q, Gene expression, Population Genetics, Cloning
Abstract: International audience; NOD2 contributes to the innate immune response and to the homeostasis of the intestinal mucosa. In response to its bacterial ligand, NOD2 interacts with RICK and activates the NF-κB and MAPK pathways, inducing gene transcription and synthesis of proteins required to initiate a balanced immune response. Mutations in NOD2 have been associated with an increased risk of Crohn's Disease (CD), a disabling inflammatory bowel disease (IBD). Because NOD2 signaling plays a key role in CD, it is important to further characterize the network of protein interacting with NOD2. Using yeast two hybrid (Y2H) screens, we identified new NOD2 interacting proteins (NIP). The primary interaction was confirmed by coim-munoprecipitation and/or bioluminescence resonance energy transfer (BRET) experiments for 11 of these proteins (ANKHD1, CHMP5, SDCCAG3, TRIM41, LDOC1, PPP1R12C, DOCK7, VIM, KRT15, PPP2R3B, and C10Orf67). These proteins are involved in diverse functions, including endosomal sorting complexes required for transport (ESCRT), cyto-skeletal architecture and signaling regulation. Additionally, we show that the interaction of 8 NIPs is compromised with the 3 main CD associated NOD2 mutants (R702W, G908R and 1007fs). Furthermore, to determine whether these NOD2 protein partners could be encoded by IBD susceptibility genes, a transmission disequilibrium test (TDT) was performed on 101 single nucleotide polymorphisms (SNPs) and the main corresponding haplo-types in genes coding for 15 NIPs using a set of 343 IBD families with 556 patients. Overall this work did not increase the number of IBD susceptibility genes but extends the NOD2 protein interaction network and suggests that NOD2 interactome and signaling dependupon the NOD2 mutation profile in CD.
Published: 2016

31. Fiabilité et optimisation des calculs obtenus par des formulations intégrales en propagation d'ondes

Author: Bakry, Marc, STAR, ABES, Propagation des Ondes : Étude Mathématique et Simulation ( POEMS ), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -Centre National de la Recherche Scientifique ( CNRS ), ONERA - The French Aerospace Lab ( Toulouse ), ONERA, Université Paris-Saclay, Patrick Ciarlet, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Saclay (COmUE)
Subjects: Optimization, Formulation intégrale, Équation d'onde, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Wave equation, Optimisation, Integral formulation, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]
Abstract: The aim of this work is to participate to the popularization of methods for the resolution of wave propagation problems based on integral equations formulations by developping a posteriori error estimates in the context of autoadaptive mesh refinement strategies. The development of such estimates is difficult because of the non-locality of the norms associated to the Sobolev spaces and of the involved integral operators. Estimates from the literature are extended in the case of the propagation of an acoustic wave. The proofs of quasi-optimal convergence of the autoadaptive algorithms are established. We then introduce a new approach with respect to the literature which is based on a new norm-localization technique based on the use of a well-chosen Λ operator and not on inverse inequalities as it was the case previously.We then establish new a posteriori error estimates which are reliable, efficient, local and asymptotically exact with respect to the Galerkin norm of the error. We give a method for the construction of such estimates. Numerical applications on 2D and 3D geometries confirm the asymptotic exactness and the optimality of the autoadaptive algorithm.These estimates are extended in the case of the propagation of an electromagnetic wave. More precisely, we are interested in the EFIE. We suggest generalization of the estimates of the literature. A proof for quasi-optimal convergence is given for an estimate based on a localization of the norm of the residual. The principle of Λ is used to construct the first reliable, efficient, local error estimate for this equation. We give a second forme which is eventually theoretically asymptotically exact., Dans cette thèse, on se propose de participer à la popularisation des méthodes de résolution de problèmes de propagation d'onde basées sur des formulations intégrales en fournissant des indicateurs d'erreur a posteriori utilisable dans le cadre d'algorithmes de raffinement autoadaptatif. Le développement de tels indicateurs est complexe du fait de la non-localité des normes associées aux espaces de Sobolev et des opérateurs entrant en jeu. Des indicateurs de la littérature sont étendus au cas de la propagation d'une onde acoustique. On étend les preuves de convergence quasi-optimale (de la littérature) des algorithmes autoadaptatifs associés dans ce cas. On propose alors une nouvelle approche par rapport à la littérature qui consiste à utiliser une technique de localisation des normes, non pas basée sur des inégalités inverses, mais sur l'utilisation d'un opérateur Λ de localisation bien choisi.On peut alors construire des indicateurs d'erreur a posteriori fiables, efficaces, locaux et asymptotiquement exacts par rapport à la norme de Galerkin de l'erreur. On donne ensuite une méthode pour la construction de tels indicateurs. Les applications numériques sur des géométries 2D et 3D confirment l'exactitude asymptotique ainsi que l'optimalité du guidage de l'algorithme autoadaptatif.On étend ensuite ces indicateurs au cas de la propagation d'une onde électromagnétique. Plus précisément, on s'intéresse au cas de l'EFIE. On propose des généralisations des indicateurs de la littérature. On effectue la preuve de convergence quasi-optimale dans le cas d'un indicateur basé sur une localisation de la norme du résidu. On utilise le principe du Λ pour obtenir le premier indicateur d'erreur fiable, efficace et local pour cette équation. On en propose une seconde forme qui est également, théoriquement asymptotiquement exacte.
Published: 2016

32. Chance constrained optimization of a three-stage launcher

Author: Caillau, Jean-Baptiste, Cerf, Max, Sassi, Achille, Trélat, Emmanuel, Zidani, Hasnaa, Sassi, Achille, Mathematics for Control, Transport and Applications ( McTAO ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), ASTRIUM, EADS - European Aeronautic Defense and Space, Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Sorbonne Universités, Université Pierre et Marie Curie - Paris 6 ( UPMC ), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Sorbonne Université (SU), Université Pierre et Marie Curie - Paris 6 (UPMC), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [ INFO.INFO-NA ] Computer Science [cs]/Numerical Analysis [cs.NA], [INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA], [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA]
Abstract: Journées SMAI-MODE 2016 (Toulouse)
Published: 2016

33. Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion

Author: Bayen, Térence, Mairet, Francis, Martinon, Pierre, Sebbah, Matthieu, Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme (MODEMIC), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Biological control of artificial ecosystems (BIOCORE), Laboratoire d'océanographie de Villefranche (LOV), Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA), Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems (Commands), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Universidad Tecnica Federico Santa Maria [Valparaiso] (UTFSM), Programa de Financiamiento Basal from Center of Mathematical Modeling, Universidad de Chile, INRIA Associated team with Chile: DYMECOS, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), Institut National de la Recherche Agronomique (INRA)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire océanologique de Villefranche-sur-mer (OOVM), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme ( MODEMIC ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie ( MISTEA ), Institut National de la Recherche Agronomique ( INRA ) -Institut national d’études supérieures agronomiques de Montpellier ( Montpellier SupAgro ) -Institut National de la Recherche Agronomique ( INRA ) -Institut national d’études supérieures agronomiques de Montpellier ( Montpellier SupAgro ), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie ( MISTEA ), Institut National de la Recherche Agronomique ( INRA ) -Institut national d’études supérieures agronomiques de Montpellier ( Montpellier SupAgro ), Biological control of artificial ecosystems ( BIOCORE ), Laboratoire d'océanographie de Villefranche ( LOV ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Institut national des sciences de l'Univers ( INSU - CNRS ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Institut national des sciences de l'Univers ( INSU - CNRS ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de la Recherche Agronomique ( INRA ), Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems ( Commands ), Centre de Mathématiques Appliquées - Ecole Polytechnique ( CMAP ), École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ) -École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ), Universidad Tecnica Federico Santa Maria [Valparaiso] ( UTFSM ), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National de la Recherche Agronomique (INRA)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [ MATH ] Mathematics [math], [ SDV ] Life Sciences [q-bio], Optimal Control, [SDV]Life Sciences [q-bio], Micro-algae, Periodic Differential Equations, [SPI]Engineering Sciences [physics], Chemostat, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, [ INFO.INFO-AU ] Computer Science [cs]/Automatic Control Engineering, [ SPI ] Engineering Sciences [physics], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math]
Abstract: International audience; In this work, we study the coupling of a culture of microalgae limited by light and an anaerobic digester in a two-tank bioreactor. The model for the reactor combines a periodic day-night light for the culture of microalgae and a classical chemostat model for the digester. We first prove the existence and attraction of periodic solutions of this problem for a 1 day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain timeframe, the control on the system being the dilution rate (the input flow of microalgae in the digester). We apply Pontryagin's Maximum Principle in order to characterize optimal controls, including the computation of singular controls. We present numerical simulations by direct and indirect methods for different light models and compare the optimal 1-day periodic solution to the optimal strategy over larger timeframes. Finally, we also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks.
Published: 2015

34. Hausdorff volume in non equiregular sub-Riemannian manifolds

Author: Roberta Ghezzi, Frédéric Jean, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), iCODE (Institute for Control and Decision), research project of the IDEX Paris- Saclay, European Project: 246923,EC:FP7:ERC,ERC-2009-AdG,GEMETHNES(2010), European Project: 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO(2011), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), European Project : 246923,EC:FP7:ERC,ERC-2009-AdG,GEMETHNES ( 2010 ), European Project : 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO ( 2011 ), and Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Pure mathematics, Applied Mathematics, Hausdorff space, Characterization (mathematics), 16. Peace & justice, Lebesgue integration, Manifold, [ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG], Geometric measure theory, symbols.namesake, Mathematics - Metric Geometry, 53C17, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Radon measure, symbols, [ MATH.MATH-MG ] Mathematics [math]/Metric Geometry [math.MG], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Analysis, Mathematics, Volume (compression)
Abstract: Version preprint; International audience; In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).
Published: 2015

35. Modélisation des neurones, équation de Bloch-Torrey et leur application à l'estimation de la microstructure du cerveau par IRM de diffusion

Author: Fang, Chengran, Modèles et inférence pour les données de Neuroimagerie (MIND), IFR49 - Neurospin - CEA, Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Université Paris-Saclay, Jing-Rebecca Li, and Demian Wassermann
Subjects: Bloch-Torrey equation, Modélisation des neurones, Numerical simulation, IRM de diffusion, Apprentissage automatique, Diffusion MRI, [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], Équation de Bloch-Torrey, Imagerie de la microstructure du cerveau, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, Neuron modeling, Simulation numérique, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Brain microstructure imaging, Machine learning, [INFO.INFO-IM]Computer Science [cs]/Medical Imaging, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: Non-invasively estimating brain microstructure that consists of a very large number of neurites, somas, and glial cells is essential for future neuroimaging. Diffusion MRI (dMRI) is a promising technique to probe brain microstructural properties below the spatial resolution of MRI scanners. Due to the structural complexity of brain tissue and the intricate diffusion MRI mechanism, in vivo microstructure estimation is challenging.Existing methods typically use simplified geometries, particularly spheres, and sticks, to model neuronal structures and to obtain analytical expressions of intracellular signals. The validity of the assumptions made by these methods remains undetermined. This thesis aims to facilitate simulationdriven brain microstructure estimation by replacing simplified geometries with realistic neuron geometry models and the analytical intracellular signal expressions with diffusion MRI simulations. Combined with accurate neuron geometry models, numerical dMRI simulations can give accurate intracellular signals and seamlessly incorporate effects arising from, for instance, neurite undulation or water exchange between soma and neurites.Despite these advantages, dMRI simulations have not been widely adopted due to the difficulties in constructing realistic numerical phantoms, the high computational cost of dMRI simulations, and the difficulty in approximating the implicit mappings between dMRI signals and microstructure properties. This thesis addresses the above problems by making four contributions. First, we develop a high-performance opensource neuron mesh generator and make publicly available over a thousand realistic cellular meshes.The neuron mesh generator, swc2mesh, can automatically and robustly convert valuable neuron tracing data into realistic neuron meshes. We have carefully designed the generator to maintain a good balance between mesh quality and size. A neuron mesh database, NeuronSet, which contains 1213 simulation-ready cell meshes and their neuroanatomical measurements, was built using the mesh generator. These meshes served as the basis for further research. Second, we increased the computational efficiency of the numerical matrix formalism method by accelerating the eigendecomposition algorithm and exploiting GPU computing. The speed was increased tenfold. With similar accuracy, the optimized numerical matrix formalism is 20 times faster than the FEM method and 65 times faster than a GPU-based Monte Carlo method. By performing simulations on realistic neuron meshes, we investigated the effect of water exchange between somas and neurites, and the relationship between soma size and signals. We then implemented a new simulation method that provides a Fourier-like representation of the dMRI signals. This method was derived theoretically and implemented numerically. We validated the convergence of the method and showed that the error behavior is consistent with our error analysis. Finally, we propose a simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI. By exploiting the powerful modeling and computational capabilities that are mentioned above, we have built a synthetic database containing the dMRI signals and microstructure parameters of 1.4 million artificial brain voxels. We have shown that this database can help approximate the underlying mappings of the dMRI signals to volume and surface fractions using artificial neural networks.; L'estimation non invasive de la microstructure du cerveau, qui se compose de nombreux neurites, de somas et de cellules gliales, est essentielle pour l'imagerie cérébrale. L'IRM de diffusion (IRMd) est une technique prometteuse pour sonder les propriétés microstructurelles du cerveau en dessous de la résolution spatiale des scanners IRM. En raison de la complexité structurelle du tissu cérébral et du mécanisme complexe de l'IRM de diffusion, l'estimation de la microstructure in vivo est un défi. Les méthodes existantes utilisent généralement des géométries simplifiées, notamment des sphères et des bâtons, pour modéliser les structures neuronales et obtenir des expressions analytiques des signaux intracellulaires. La validité des hypothèses faites par ces méthodes reste indéterminée. Cette thèse vise à faciliter l'estimation de la microstructure du cerveau par simulation en remplaçant les géométries simplifiées par des modèles réalistes de la géométrie des neurones et les expressions analytiques des signaux intracellulaires par des simulations d'IRM de diffusion. Combinées à des modèles précis de la géométrie des neurones, les simulations numériques d'IRMd peuvent donner des signaux intracellulaires précis et incorporer les effets dus, par exemple, à l'ondulation des neurites ou à l'échange d'eau entre le soma et les neurites.Malgré ces avantages, les simulations d'IRMd n'ont pas été largement adoptées en raison de l'inaccessibilité des fantômes numériques, de la faible efficacité de calcul des simulateurs d'IRMd et de la difficulté d'approximer les mappings implicites entre les signaux d'IRMd et les propriétés de la microstructure. Cette thèse contribue à la résolution des problèmes susmentionnés de la manière suivante : (1) en développant un générateur de maillage de neurones open-source et en rendant accessibles au public plus d'un millier de maillages cellulaires réalistes ; (2) en augmentant d'un facteur dix l'efficacité de calcul de la méthode du formalisme matriciel numérique ; (3) en mettant en œuvre une nouvelle méthode de simulation qui fournit une représentation de type Fourier des signaux IRMd ; (4) en proposant un cadre d'apprentissage supervisé basé sur la simulation pour estimer la microstructure du cerveau par IRM de diffusion.
Published: 2023

36. Analysis of time-harmonic Maxwell impedance problems in anisotropic media

Author: Chicaud, Damien, Ciarlet, Patrick, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), and Ciarlet, Patrick
Subjects: Maxwell's equations, edge finite elements, surface PDEs, impedance boundary condition, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], anisotropic media, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; We consider the time-harmonic Maxwell's equations in anisotropic media. The problem to be solved is an approximation of the diffraction problem, or scattering from bounded objects, that is usually set in some exterior domain in $\mathbb{R}^3$. We consider perfectly conducting objects, so the equations are supplemented with a Dirichlet boundary condition on those objects, and we truncate the exterior domain by imposing an impedance condition on an artificial boundary, to model an approximate radiation condition. The resulting problem is then posed in a bounded domain, with Dirichlet and impedance boundary conditions. In this work, we focus on the mathematical meaning of the impedance condition, precisely in which function space it holds. This relies on a careful analysis of the regularity of the traces of electromagnetic fields, which can be derived thanks to the study of the regularity of the solution to second-order surface PDEs. Then, we prove well-posedness of the model, and we determine the a priori regularity of the fields in the domain and on the boundaries, depending on the geometry, the coefficients and the data. Finally, the discretization of the formulations is presented, with an approximation based on edge finite elements. Error estimates are derived, and a benchmark is provided to discuss those estimates.
Published: 2023

37. A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

Author: Chaabane, Slim, Haddar, Houssem, Jerbi, Rahma, Université de Sfax - University of Sfax, Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), and Haddar, Houssem
Subjects: Inverse problems, Impedance tomography, Iterative methods Mathematics, Kohn-Vogelius cost function, Domain decomposition method, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simplified case and numerically show its efficiency for some two dimensional experiments.
Published: 2023

38. Spectrum of the Dirichlet Laplacian in a thin cubic lattice

Author: Chesnel, Lucas, Nazarov, Sergei, Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École polytechnique (X)-EDF (EDF), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Institute of Mechanical Engineering Problems [St. Petersburg] (IPME), Russian Academy of Sciences [Moscow] (RAS), and Chesnel, Lucas
Subjects: thin periodic lattice, Quantum waveguide, [MATH] Mathematics [math], [MATH]Mathematics [math], threshold resonance, trapped waves
Abstract: We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width ε ≪ 1) which have a square cross section. This spectrum coincides with the union of segments which all go to +∞ as ε tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length O(e^{−δ/ε}), δ > 0, while the length of the next spectral segments is O(ε). To establish these results, we need to study in detail the properties of the Dirichlet Laplacian A^Ω in the geometry Ω obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that A^Ω has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for A^Ω , that is no non trivial bounded solution at the threshold frequency for A^Ω. This implies that the correct 1D model of the lattice for the next spectral segments is a graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.
Published: 2023

39. Wave propagation in one-dimensional quasiperiodic media

Author: Amenoagbadji, Pierre, Fliss, Sonia, Joly, Patrick, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Transparent Boundary Conditions, Mathematics - Analysis of PDEs, FOS: Mathematics, Waveguide, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35J05, 78A50, 42A75, Helmholtz Equations, Quasiperiodic functions, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: This work is devoted to the resolution of the Helmholtz equation $-(\mu u')' - \rho \omega^2 u = f$ in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation., Comment: 45 pages, 15 figures
Published: 2023
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40. Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer

Author: Chesnel, Lucas, Nazarov, Sergei A., Taskinen, Jari, Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Institute for Problems of Mechanical Engineering (IPME RAS), Russian Academy of Sciences [Moscow] (RAS), Department of Mathematics and Statistics [Helsinki], Falculty of Science [Helsinki], and Helsingin yliopisto = Helsingfors universitet = University of Helsinki-Helsingin yliopisto = Helsingfors universitet = University of Helsinki
Subjects: Mathematics - Spectral Theory, discrete spectrum, Mathematics - Analysis of PDEs, chamfered quarter of layer, Fichera layer, FOS: Mathematics, Laplacian with mixed boundary conditions, [MATH]Mathematics [math], Spectral Theory (math.SP), trapped modes, Analysis of PDEs (math.AP)
Abstract: We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. The geometry depends on two parameters gathered in some vector $\kappa=(\kappa_1,\kappa_2)$ which characterizes the domain at the edges. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to $\kappa$. In particular, we show that for a given $\kappa_1>0$, there is some $h(\kappa_1)>0$ such that discrete spectrum exists for $\kappa_2\in(-\kappa_1,0)\cup(h(\kappa_1),\kappa_1)$ whereas it is empty for $\kappa_2\in[0;h(\kappa_1)]$. 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.
Published: 2023
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41. On particular solutions of linear partial differential equations with polynomial right-hand-sides

Author: Anderson, Thomas G., Bonnet, Marc, Faria, Luiz M., Pérez-Arancibia, Carlos, Computational & Applied Mathematics, Rice University [Houston], Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics [Twente], and University of Twente
Subjects: Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: This paper introduces general methodologies for constructing closed-form solutions to several important partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. The covered equations include the isotropic and anisotropic Poisson, Helmholtz, Stokes, and elastostatic equations, as well as the time-harmonic linear elastodynamic and Maxwell equations. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and have potential applications in certain kinds of Trefftz finite element methods. Our approach to all of these PDEs relates the particular solution to polynomial solutions of the Poisson and Helmholtz polynomial particular solutions, solutions that can in turn be obtained, respectively, from expansions using homogeneous polynomials and the Neumann series expansion of the operator $(k^2+\Delta)^{-1}$. No matrix inversion is required to compute the solution. The method naturally incorporates divergence constraints on the solution, such as in the case of Maxwell and Stokes flow equations. This work is accompanied by a freely available Julia library, \texttt{PolynomialSolutions.jl}, which implements the proposed methodology in a non-symbolic format and efficiently constructs and provides access to rapid evaluation of the desired solution.
Published: 2023
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42. Multi-domain FEM-BEM coupling for acoustic scattering

Author: Bonazzoli, Marcella, Claeys, Xavier, Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Algorithms and parallel tools for integrated numerical simulations (ALPINES), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and ANR-15-CE23-0017,NonLocalDD,Méthodes non-locales en décomposition de domaines pour l'électromagnétisme(2015)
Subjects: FEM-BEM coupling, multi-trace formulations, cross-points, FOS: Mathematics, boundary integral equations, acoustic scattering, Helmholtz equation, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with domain variational formulations for the heterogeneous subdomain. This is an extension of Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains abut. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects; here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a G{\aa}rding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.
Published: 2023
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43. A posteriori error estimates for mixed finite element discretizations of the Neutron Diffusion equations

Author: Patrick Ciarlet, Minh Hieu Do, François Madiot, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Service d’Études des Réacteurs et de Mathématiques Appliquées (SERMA), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Service des Réacteurs et de Mathématiques Appliquées (SERMA)
Subjects: a posteriori error estimates, Mathematics Subject Classification: 65J10 / 65N15 / 65N30 / 65N50, diffusion equation, Neutronics, low regularity solution, eigenvalue problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], mesh refinement, mixed formulation, [MATH]Mathematics [math], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; We analyse $a\ posteriori$ error estimates for the discretization of the neutron diffusion equations with mixed finite elements. We provide guaranteed and locally efficient estimators on a base block equation, the one-group neutron diffusion equation. We pay particular attention to AMR strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications. We exhibit a robust marker strategy for this specific constraint, the $direction\ marker$ strategy. The approach is further extended to a Domain Decomposition Method, the so-called DD+$L^2$ jumps method, as well as to the multigroup neutron diffusion equation.
Published: 2023

44. RELATIONSHIP BETWEEN THE MAXIMUM PRINCIPLE AND DYNAMIC PROGRAMMING FOR MINIMAX PROBLEMS

Author: Cristopher Hermosilla, Hasnaa Zidani, Departamento de Matematica, Universidad Tecnica Federico Santa Maria, Universidad Tecnica Federico Santa Maria [Valparaiso] (UTFSM), Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), This work was supported by ECOS-ANID (Grant Number ECOS200064). C. Hermosilla has received research support from FONDECYT Grant Number 11190456 and H. Zidani received research support from Région Normandie and EU through ERDF program (Chaire COPTI)., and European Project: ERDF
Subjects: Control and Optimization, Applied Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [SPI.AUTO]Engineering Sciences [physics]/Automatic
Abstract: International audience; This paper is concerned with the relationship between the maximum principle and dynamic programming for a large class of optimal control problems with maximum running cost. Inspired by a technique introduced by Vinter in the 1980s, we are able to obtain jointly a global and a partial sensitivity relation that link the coextremal with the value function of the problem at hand. One of the main contributions of this work is that these relations are derived by using a single perturbed problem, and therefore, both sensitivity relations hold, at the same time, for the same coextremal. As a by-product, and thanks to the level-set approach, we obtain a new set of sensitivity relations for Mayer problems with state constraints. One important feature of this last result is that it holds under mild assumptions, without the need of imposing strong compatibility assumptions between the dynamics and the state constraints set. Minimax optimal control problems and Maximum principle and Dynamic programming and Sensitivity analysis and State constraints
Published: 2023

45. Asymptotic Expansion of Transmission Eigenvalues for Anisotropic Thin Layers

Author: H. Boujlida, H. Haddar, M. Khenissi, Ecole Supérieure des Sciences et de Technologie de Hammam Sousse (ESSTHS), Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), and Haddar, Houssem
Subjects: transmission eigenvalues thin layer asymptotic inverse scattering, thin layer asymptotic, Applied Mathematics, [INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA], inverse scattering, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], Analysis, transmission eigenvalues
Abstract: International audience; We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [19]. We also provide formal derivation in more general cases validating the obtained theoretical result.
Published: 2023

46. Eddy-current asymptotics of the Maxwell PMCHWT formulation for multiple bodies and conductivity levels

Author: Marc Bonnet, Edouard Demaldent, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Département Imagerie et Simulation pour le Contrôle (DISC), Laboratoire d'Intégration des Systèmes et des Technologies (LIST (CEA)), Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Bonnet, Marc
Subjects: Computational Mathematics, Computational Theory and Mathematics, Modeling and Simulation, [INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation
Abstract: International audience; In eddy current (EC) testing applications, ECs σE (E : electric field, σ: conductivity) are induced in tested metal parts by a low-frequency (LF) source idealized as a closed current loop in air. In the case of highly conducting (HC) parts, a boundary integral equation (BIE) of the first kind under the magneto-quasi-static approximation-which neglects the displacement current-was shown in a previous work to coincide with the leading order of an asymptotic expansion of the Maxwell BIE in a small parameter reflecting both LF and HC assumptions. The main goal of this work is to generalize the latter approach by establishing a unified asymptotic framework that is applicable to configurations that may involve multiple moderately-conducting (σ = O(1)) and non-conducting objects in addition to (possibly multiply-connected) HC objects. Leading-order approximations of the quantities relevant to EC testing, in particular the impedance variation, are then found to be computable from a reduced set of primary unknowns (three on HC objects and two on other objects, instead of four per object for the Maxwell problem). Moreover, when applied to the Maxwell BIE, the scalings suggested by the asymptotic approach stabilize the condition number at low frequencies and remove the low-frequency breakdown effect. The established asymptotic properties are confirmed on 3D numerical examples for simple geometries as well as two EC testing configurations, namely a classical benchmark and a steam generator tube featured in pressurized water reactors of nuclear power plants.
Published: 2023

47. Modelling of the fatigue cracking resistance of grid reinforced asphalt concrete by coupling fast BEM and FEM

Author: A. Dansou, S. Mouhoubi, C. Chazallon, M. Bonnet, Laboratoire de Tribologie et Dynamique des Systèmes (LTDS), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-École Nationale des Travaux Publics de l'État (ENTPE)-Ecole Nationale d'Ingénieurs de Saint Etienne (ENISE)-Centre National de la Recherche Scientifique (CNRS), Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), This work was supported in part by the French National Research Agency (SolDuGri project ANR-14-CE22-0019) and in part by the Grand-Est region, France., ANR-14-CE22-0019,SolDuGri,Pour une solution durable du renforcement des infrastructures par grilles en fibre de verre(2014), Bonnet, Marc, Appel à projets générique - Pour une solution durable du renforcement des infrastructures par grilles en fibre de verre - - SolDuGri2014 - ANR-14-CE22-0019 - Appel à projets générique - VALID, Department of Mathematics [Ann Arbor], University of Michigan [Ann Arbor], and University of Michigan System-University of Michigan System
Subjects: Galerkin BEM, Grid reinforcement, Crack propagation, Fast Multipole Method, [INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation, Surface-breaking crack, Sciences de l'ingénieur [physics]/Génie civil, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Civil and Structural Engineering, Physics::Geophysics
Abstract: International audience; We present a computational modeling approach aimed at investigating the effect of fiber grid reinforcement on crack opening displacement and fatigue crack propagation. Grid reinforcements are modeled using elastic membrane finite elements, while the cracked concrete is treated using a symmetric boundary element method (BEM), which in particular allows easy geometrical modelling and meshing of cracks. The BEM is accelerated by the fast multipole method, allowing the handling of potentially large BEM models entailed by three-dimensional configurations hosting multiple cracks. Fatigue crack growth is modelled using the Paris law. The proposed computational approach is first verified on a reinforced cracked beam, and then applied to a three-dimensional configuration featuring a grid-reinforced asphalt pavement.
Published: 2023

48. Inversion of Eddy-Current Signals Using a Level-Set Method and Block Krylov Solvers

Author: Audibert, Lorenzo, Girardon, Hugo, Haddar, Houssem, Jolivet, Pierre, Inversion of Differential Equations For Imaging and physiX (IDEFIX), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Shape reconstruction and identification (DeFI ), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Ecole Nationale Supérieure d'Electrotechnique, d'Electronique, d'Informatique, d'Hydraulique et de Télécommunications (ENSEEIHT), Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université de Toulouse (UT), This work was granted access to the GENCI-sponsored HPC resources of TGCC@CEA under allocations A0070607519 and A0090607519., Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), ENSEEIHT, and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées
Subjects: large-scale inverse problems AMS subject classifications. 35Q61, 49Q10, block Krylov methods, 78A46, eddy currents, [MATH]Mathematics [math], 65F10, domain decomposition preconditioners, large-scale inverse problems
Abstract: International audience; The application motivating this work is related to the identification of deposits inside nuclear power plant steam generators using eddy-current probes. We consider a realistic experimental process that relies on the scan of a domain by sweeping along a tube axis a probe made out of coils, playing the role of the sources/receivers. Solving the inverse shape problem associated with these measurements using a least squares method requires solutions to the eddy-current and the adjoint problems for a large number of right-hand sides at each gradient-descent iteration. Additional cost in the forward solver comes from the use of a potential formulation of the problem that has the advantage of being independent from the topology of the conductive media (that may vary during iterations). We use a level-set approach to avoid remeshing and handle unknown topologies. The crucial ingredient in our algorithm is an optimized way of handling high numbers of right-hand sides for iterative solvers of large-scale problems. We first benchmark various block Krylov methods, block GMRES and block BGCRODR, to test their effectiveness compared to their standard counterpart, i.e., GMRES and GCRODR. Then, we propose for BGCRODR a new implementation for recycling information from previously generated Krylov bases that scales better than traditional approaches. This part is independent from the practical inverse problem at hand. The efficiency of the overall inversion procedure is finally demonstrated on realistic synthetic 3D examples.
Published: 2023

49. Special issue on New trends in optimal control

Author: Caillau, Jean-Baptiste, Grüne, Lars, De Pinho, Maria Do Rosário, Trélat, Emmanuel, Zidani, Hasnaa, Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Mathematisches Institut [Bayreuth], Universität Bayreuth, Faculdade de Engenharia da Universidade do Porto (FEUP), Universidade do Porto = University of Porto, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Mathematics for Control, Transport and Applications ( McTAO ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Faculdade de Engenharia [Porto] ( FEUP ), Universidade do Porto [Porto], Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), and Universidade do Porto
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; This special volume gathers a number of new contributions addressing various topics related to the field of optimal control theory and sensitivity analysis. The field has a rich and varied mathematical theory, with a long tradition and a vibrant body of applications. It has attracted a growing interest across the last decades, with the introduction of new ideas and techniques, and thanks to various new applications.
Published: 2015

50. Some Loci of Rational Cubic Fourfolds

Author: Giovanni Staglianò, Michele Bolognesi, Francesco Russo, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Università degli studi di Catania [Catania], Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Università degli studi di Catania = University of Catania (Unict), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Unité de Mathématiques Appliquées ( UMA ), and École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech )
Subjects: Double point, General Mathematics, 010102 general mathematics, Pfaffian, 01 natural sciences, Rational normal scroll, Moduli space, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Combinatorics, Mathematics - Algebraic Geometry, Hypersurface, Mathematics::Algebraic Geometry, Quartic function, 0103 physical sciences, FOS: Mathematics, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in $\mathbb P^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to $\mathcal C_{14}$ is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus is not open in $\mathcal C_{14}$., 24 pages. Final, rewritten and expanded version with some new results, to appear on Math. Annalen
Published: 2015

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