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529 results on '"[ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA]"'

1. The Neumann numerical boundary condition for transport equations

Author: Frédéric Lagoutière, Jean-François Coulombel, 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), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Camille Jordan [Villeurbanne] ( ICJ ), Centre National de la Recherche Scientifique ( CNRS ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon, ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion. ( 2013 ), ANR-17-CE40-0025,NABUCO,NumericAl BoUndaries and COupling, 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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Discretization, Type (model theory), 01 natural sciences, Stability (probability), 010305 fluids & plasmas, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Normal mode, numerical schemes, 0103 physical sciences, Convergence (routing), Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, convergence, Mathematical analysis, transport equations, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], stability, 010101 applied mathematics, Modeling and Simulation, Outflow boundary, AMS classification: 65M12, 65M06, 65M20, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in \begin{document}$ \ell^\infty $\end{document} for transport equations. We show in particular that the Neumann numerical boundary condition is a stable, local, and absorbing numerical boundary condition for discretized transport equations. Our main result is proved for explicit two time level numerical approximations of transport operators with arbitrarily wide stencils. The proof is based on the energy method and bypasses any normal mode analysis.
Published: 2020

2. Inference of the bottom topography in anisothermal mildly-sheared shallow ice flows

Author: Jiamin Zhu, Jerome Monnier, 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), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), 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: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [ SDU.STU.GL ] Sciences of the Universe [physics]/Earth Sciences/Glaciology, Computational Mechanics, General Physics and Astronomy, Geometry, 010103 numerical & computational mathematics, 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Data assimilation, Thermal, Interferometric synthetic aperture radar, Newtonian fluid, [SDU.STU.GL]Sciences of the Universe [physics]/Earth Sciences/Glaciology, Gravimetry, 0101 mathematics, Mechanical Engineering, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Inverse problem, Computer Science Applications, 010101 applied mathematics, Boundary layer, Mechanics of Materials, [ SPI.MECA.MEFL ] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Geology, Dimensionless quantity
Abstract: This study proposes a new inverse method to estimate the bed topography elevation beneath glaciers flows from surface observations (altimetry elevations and InSAR velocities) and (very) sparse depth measurements (e.g. acquired during airborne campaigns). To do so an original form of depth-integrated flow equations (long-wave assumption) is derived. The latter are valid for highly to mildly-sheared regimes hence including moderately fast flows; varying internal thermal profiles are taken into account. The inverse problem is particularly challenging since the surface signatures integrate the bottom features (bed elevation and friction–slip amount) plus the internal deformation. The first key ingredient of the inverse method is the derivation of this non-isothermal Reduced Uncertainty (RU) version of the classical SIA equation (lubrication type model for generalised Newtonian fluids) by intrinsically integrating the surface measurements in the formulation. This resulting multi-physics RU-SIA equation contains a unique uncertain dimensionless parameter only (the parameter γ ). The next key ingredient is an advanced Variational Data Assimilation (VDA) formulation combined with a purely data-driven extension of γ based on the trend observed in the (sparse) depth measurements (e.g. along the flight tracks). The resulting inverse method provides the first physical-based depth estimations in mildly-sheared mildly-slippery shallow flows. In poorly monitored ice-sheet areas (e.g. in Antarctica), the resulting estimations are noticeably less uncertain than the current ones (in particular compared to those obtained by gravimetry inversions). The present numerical experiments and experimental sensitivity analyses demonstrate the reliability of this new RU-SIA equation and the robustness of the inverse method. In other respect, the RU-SIA equation may provide a-posteriori estimations of the thermal boundary layer at bottom.
Published: 2019

3. Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field

Author: Florian Méhats, Xiaofei Zhao, Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, 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 ), Multi-scale numerical geometric schemes ( MINGUS ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), 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), Multi-scale numerical geometric schemes (MINGUS), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, 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), ANR-14-CE23-0007-01, Agence Nationale de la Recherche, 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), 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 (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques(2014), and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: [ INFO.INFO-MO ] Computer Science [cs]/Modeling and Simulation, 65L20, FOS: Physical sciences, 010103 numerical & computational mathematics, 01 natural sciences, [PHYS.PHYS.PHYS-COMP-PH]Physics [physics]/Physics [physics]/Computational Physics [physics.comp-ph], two-scale methods, [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], [ PHYS.PHYS.PHYS-COMP-PH ] Physics [physics]/Physics [physics]/Computational Physics [physics.comp-ph], FOS: Mathematics, medicine, Mathematics - Numerical Analysis, 0101 mathematics, 65L70, Mathematics, Algebra and Number Theory, Oscillation, Applied Mathematics, Mathematical analysis, Vlasov and Vlasov-Poisson equations, Novelty, Stiffness, high, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [ PHYS.PHYS.PHYS-PLASM-PH ] Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], Computational Physics (physics.comp-ph), [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Physics - Plasma Physics, AMS Subject Classification: 65L05, Magnetic field, Plasma Physics (physics.plasm-ph), non-homogeneous strong magnetic field, 010101 applied mathematics, Computational Mathematics, Non homogeneous, oscillations, uniform accuracy, medicine.symptom, Physics - Computational Physics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this paper, we consider the numerical solution of highly-oscillatory Vlasov and Vlasov-Poisson equations with non-homogeneous magnetic field. Designed in the spirit of recent uniformly accurate methods, our schemes remain insensitive to the stiffness of the problem, in terms of both accuracy and computational cost. The specific difficulty (and the resulting novelty of our approach) stems from the presence of a non-periodic oscillation, which necessitates a careful ad-hoc reformulation of the equations. Our results are illustrated numerically on several examples., Comment: 31 pages, 11 figures
Published: 2019

4. Stability of stationary solutions of singular systems of balance laws

Author: Nicolas Seguin, 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), 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 ), 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: Kullback–Leibler divergence, 010103 numerical & computational mathematics, 01 natural sciences, Hyperbolic systems, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Exponential stability, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, stationary state, Entropy (arrow of time), Mathematical Physics, Mathematics, Finite volume method, non-conservative systems, Applied Mathematics, Numerical analysis, relative entropy, Fluid mechanics, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], stability, Dissipation, 010101 applied mathematics, well-balanced schemes, finite volume schemes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Stationary state, Analysis of PDEs (math.AP)
Abstract: International audience; The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics, we assume that these systems are endowed with a partially convex entropy. We first construct an associated relative entropy which allows to compare two states which share the same geometric data. This way, we are able to prove the stability of some stationary states within entropy weak solutions. This result applies for instance to the shallow-water equations with bathymetry. Besides, this relative entropy can be used to study finite volume schemes which are entropy-stable and well-balanced, and due to the numerical dissipation inherent to these methods, asymptotic stability of discrete stationary solutions is obtained. This analysis does not make us of any specific definition of the non-conservative products, applies to non-strictly hyperbolic systems, and is fully multidimensional with unstructured meshes for the numerical methods.
Published: 2019

5. A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations

Author: Philippe Chartier, Florian Méhats, Mohammed Lemou, Gilles Vilmart, 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), Multi-scale numerical geometric schemes (MINGUS), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Section de mathématiques [Genève], Université de Genève (UNIGE), Invariant Preserving SOlvers ( IPSO ), 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 ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes ( UNIV-RENNES ), Université de Genève ( UNIGE ), 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), 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 (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, Université de Genève = University of Geneva (UNIGE), and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: 74Q10, micro-macro method, 010103 numerical & computational mathematics, 01 natural sciences, standard averaging, Highly-oscillatory problems, uniform accuracy AMS subject classification (2010): 65L20, Pullback method, highly-oscillatory problems, Convergence (routing), medicine, FOS: Mathematics, Applied mathematics, stroboscopic averaging, Mathematics - Numerical Analysis, ddc:510, 0101 mathematics, Mathematics, Uniform accuracy, Applied Mathematics, Numerical analysis, Spectrum (functional analysis), Stiffness, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], pullback method, 010101 applied mathematics, Computational Mathematics, Standard averaging, Transformation (function), 35K15, Stroboscopic averaging, Computational Theory and Mathematics, Micro-macro method, uniform accuracy, medicine.symptom, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; We introduce a new methodology to design uniformly accurate methods for oscil-latory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.
Published: 2020

6. Numerical quadrature in the Brillouin zone for periodic Schrödinger operators

Author: David Gontier, Antoine Levitt, Eric Cancès, Damiano Lombardi, Virginie Ehrlacher, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris sciences et lettres (PSL), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Numerical simulation of biological flows (REO), Sorbonne Université (SU)-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 de Paris (UP)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), 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)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ), MATHematics for MatERIALS ( MATHERIALS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Numerical simulation of biological flows ( REO ), 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 ), PSL Research University (PSL), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)
Subjects: Computation, [ PHYS.COND.CM-MS ] Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], FOS: Physical sciences, 010103 numerical & computational mathematics, Classification of discontinuities, 01 natural sciences, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Mathematics, Condensed Matter - Materials Science, Applied Mathematics, Numerical analysis, Mathematical analysis, Materials Science (cond-mat.mtrl-sci), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Numerical Analysis (math.NA), Numerical integration, Quadrature (mathematics), 010101 applied mathematics, Brillouin zone, Computational Mathematics, symbols, [PHYS.COND.CM-MS]Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], Ground state, Schrödinger's cat, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; As a consequence of Bloch's theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrodinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.
Published: 2020
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7. Model Consistency of Partly Smooth Regularizers

Author: Samuel Vaiter, Jalal M. Fadili, Gabriel Peyré, Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen ( GREYC ), Université de Caen Normandie ( UNICAEN ), Normandie Université ( NU ) -Normandie Université ( NU ) -Ecole Nationale Supérieure d'Ingénieurs de Caen ( ENSICAEN ), Normandie Université ( NU ) -Centre National de la Recherche Scientifique ( CNRS ), Centre National de la Recherche Scientifique (CNRS), 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), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Equipe Image - Laboratoire GREYC - UMR6072, Groupe de Recherche en Informatique, Image et Instrumentation de Caen (GREYC), Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN), Normandie Université (NU)-Normandie Université (NU)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN), Normandie Université (NU), European Research Council through the ERC Project SIGMA-Vision, European Project: 279593,EC:FP7:ERC,ERC-2011-StG_20101014,SIGMA-VISION(2011), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Université Paris Dauphine-PSL, École normale supérieure - Paris (ENS-PSL), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN), Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: FOS: Computer and information sciences, [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Partial smoothness, Design matrix, [ MATH.MATH-FA ] Mathematics [math]/Functional Analysis [math.FA], 010103 numerical & computational mathematics, 02 engineering and technology, 01 natural sciences, Regularization (mathematics), [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, Lasso (statistics), Statistics - Machine Learning, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Model consistency, Regularization, 0202 electrical engineering, electronic engineering, information engineering, Penalty method, Mathematics - Optimization and Control, [ STAT.ML ] Statistics [stat]/Machine Learning [stat.ML], Mathematics, Estimator, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], Regression, Computer Science Applications, Low-rank, [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Sensitivity analysis, Sparsity, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Information Systems, Inverse problems, [ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing, Computer Science - Information Theory, Machine Learning (stat.ML), Library and Information Sciences, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Matrix decomposition, [ INFO.INFO-IT ] Computer Science [cs]/Information Theory [cs.IT], FOS: Mathematics, Applied mathematics, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, Information Theory (cs.IT), Model selection, 020206 networking & telecommunications, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Optimization and Control (math.OC), Convex function
Abstract: International audience; This paper studies least-square regression penalized with partly smooth convex regularizers. This class of penalty functions is very large and versatile, and allows to promote solutions conforming to some notion of low complexity. Indeed, such penalties/regularizers force the corresponding solutions to belong to a low-dimensional manifold (the so-called model), which remains stable when the argument of the penalty function undergoes small perturbations. Such a good sensitivity property is crucial to make the underlying low-complexity (manifold) model robust to small noise. In a deterministic setting, we show that a generalized “irrepresentable condition” implies stable model selection under small noise perturbations in the observations and the design matrix, when the regularization parameter is tuned proportionally to the noise level. As an algorithmic implication, we also prove that this condition is almost necessary for stable model recovery. We then turn to the random setting, where the design matrix and the noise are random, and the number of observations grows large. We show that under our generalized “irrepresentable condition,” and a proper scaling of the regularization parameter, the regularized estimator is model consistent. In plain words, with a probability tending to one as the number of measurements tends to infinity, the regularized estimator belongs to the correct low-dimensional model manifold. This paper unifies and generalizes a large body of literature, where model consistency was known to hold, for instance for the Lasso, group Lasso, total variation (fused Lasso), and nuclear/trace norm regularizers. As an algorithmic implication, we show that under the deterministic model selection conditions, the forward-backward proximal splitting algorithm used to solve the penalized least-square regression problem is guaranteed to identify the model manifold after a finite number of iterations. Finally, we detail how our results extend from the quadratic loss to an arbitrary smooth and strictly convex loss function. We illustrate the usefulness of our results on the problem of low-rank matrix recovery from random measurements using nuclear norm minimization.
Published: 2018

8. Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

Author: Pierre Argoul, M. El Rhabi, A. Elmoussaoui, Abdelilah Hakim, Laboratoire de Mathématiques Appliquées, Faculté des Sciences et Techniques, Université Cadi Ayyad ( LAMAI ), Sécurité et Durabilité des Ouvrages d'Art ( IFSTTAR/MAST/SDOA ), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux ( IFSTTAR ) -Communauté Université Paris-Est, Département Ingénierie Mathématique et Informatique, Ecole des Ponts ParisTech ( IMI ), Université Cadi Ayyad [Marrakech] (UCA), Département Matériaux et Structures (IFSTTAR/MAST), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-PRES Université Paris-Est, Laboratoire de Mathématiques Appliquées, Faculté des Sciences et Techniques, Université Cadi Ayyad (LAMAI), Sécurité et Durabilité des Ouvrages d'Art (IFSTTAR/MAST/SDOA), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-Communauté Université Paris-Est, Département Ingénierie Mathématique et Informatique, École des Ponts ParisTech (IMI), and École des Ponts ParisTech (ENPC)
Subjects: [ MATH ] Mathematics [math], Crowd dynamics, [ INFO.INFO-MO ] Computer Science [cs]/Modeling and Simulation, Mathematical optimization, [ INFO ] Computer Science [cs], Complex system, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], SYSTEME DISCRET, 010103 numerical & computational mathematics, Topology, 01 natural sciences, DYNAMIQUE, Domain (mathematical analysis), PIETON, FOULE, Position (vector), SYSTEME COMPLEXE, [INFO]Computer Science [cs], Uniqueness, [MATH]Mathematics [math], 0101 mathematics, Mathematics, MODELE MATHEMATIQUE, Discrete kinetic theory, Stochastic game, Perspective (graphical), Probabilistic logic, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, Splitting scheme, Computational Mathematics, Computational Theory and Mathematics, Modeling and Simulation, Obstacle, Bounded function, Evacuation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], MOUVEMENT
Abstract: International audience; This paper concerns the mathematical modeling of the motion of a crowd in a non connected bounded domain, based on kinetic and stochastic game theories. The proposed model is a mesoscopic probabilistic approach that retains features obtained from both micro-and macro-scale representations; pedestrian interactions with various obstacles being managed from a probabilistic perspective. A proof of the existence and uniqueness of the proposed mathematical model's solution is given for large times. A numerical resolution scheme based on the splitting method is implemented and then applied to crowd evacuation in a non connected bounded domain with one rectangular obstacle. The evacuation time of the room is then calculated by our technique, according to the dimensions and position of a square-shaped obstacle, and finally compared to the time obtained by a deterministic approach by means of randomly varying some of its parameters.
Published: 2018

9. The Shannon Total Variation

Author: Lionel Moisan, Rémy Abergel, Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), and Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, Deblurring, Mathematical optimization, aliasing, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, Extrapolation, [ SPI.SIGNAL ] Engineering Sciences [physics]/Signal and Image processing, 010103 numerical & computational mathematics, 02 engineering and technology, Information theory, image restoration, 01 natural sciences, Aliasing, [ INFO.INFO-TI ] Computer Science [cs]/Image Processing, 0202 electrical engineering, electronic engineering, information engineering, Image scaling, Legendre-Fenchel duality, 0101 mathematics, Mathematics, Applied Mathematics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Shannon theory, Total variation denoising, Condensed Matter Physics, Subpixel rendering, total variation, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Modeling and Simulation, 020201 artificial intelligence & image processing, Geometry and Topology, Computer Vision and Pattern Recognition, image interpolation, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, Algorithm, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Interpolation
Abstract: Discretization schemes commonly used for total variation regularization lead to images that are difficult to interpolate, which is a real issue for applications requiring subpixel accuracy and aliasing control. In the present work, we reconciliate total variation with Shannon interpolation and study a Fourier-based estimate that behaves much better in terms of grid invariance, isotropy, artifact removal, and sub-pixel accuracy. We show that this new variant (called Shannon total variation) can be easily handled with classical primal-dual formulations, and illustrate its efficiency on several image processing tasks, including deblurring, spectrum extrapolation, and a new aliasing reduction algorithm.
Published: 2017

10. Uniformly accurate multiscale time integrators for second order oscillatory differential equations with large initial data

Author: Xiaofei Zhao, Invariant Preserving SOlvers ( IPSO ), 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 ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Invariant Preserving SOlvers (IPSO), 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)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), 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), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)
Subjects: Polynomial, Computer Networks and Communications, Differential equation, Multiscale time integrator, 010103 numerical & computational mathematics, Exponential integrator, 01 natural sciences, oscillatory equations, 65M12, 65M15, 65M70, 0101 mathematics, Fourier series, Mathematics, Oscillation, Applied Mathematics, Mathematical analysis, Ode, eror estimate, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 010101 applied mathematics, Computational Mathematics, Nonlinear system, Large data, unbounded energy, Bounded function, uniform accuracy, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Software
Abstract: International audience; We apply the modulated Fourier expansion to a class of second order differential equations which consists of an oscillatory linear part and a nonoscillatory nonlinear part, with the total energy of the system possibly unbounded when the oscillation frequency grows. We comment on the difference between this model problem and the classical energy bounded oscillatory equations. Based on the expansion, we propose the multiscale time integrators to solve the ODEs under two cases: the nonlinearity is a polynomial or the frequencies in the linear part are integer multiples of a single generic frequency. The proposed schemes are explicit and efficient. The schemes have been shown from both theoretical and numerical sides to converge with a uniform second order rate for all frequencies. Comparisons with popular exponential integrators in the literature are done.
Published: 2017

11. Set-membership identifiability of nonlinear models and related parameter estimation properties

Author: Carine Jauberthie, Nathalie Verdière, Louise Travé-MassuyèEs, Équipe DIagnostic, Supervision et COnduite ( LAAS-DISCO ), Laboratoire d'analyse et d'architecture des systèmes [Toulouse] ( LAAS ), Institut National Polytechnique [Toulouse] ( INP ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut National Polytechnique [Toulouse] ( INP ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Appliquées du Havre ( LMAH ), Université Le Havre Normandie ( ULH ), Normandie Université ( NU ) -Normandie Université ( NU ), Équipe DIagnostic, Supervision et COnduite (LAAS-DISCO), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Laboratoire de Mathématiques Appliquées du Havre (LMAH), Université Le Havre Normandie (ULH), Normandie Université (NU)-Normandie Université (NU), Centre National de la Recherche Scientifique ( CNRS ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Institut National Polytechnique [Toulouse] ( INP ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Toulouse III - Paul Sabatier ( UPS ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Institut National Polytechnique [Toulouse] ( INP ), 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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], 0209 industrial biotechnology, Mathematical optimization, [ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS], 02 engineering and technology, [ SPI.SIGNAL ] Engineering Sciences [physics]/Signal and Image processing, Set (abstract data type), 020901 industrial engineering & automation, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, [ SPI.AUTO ] Engineering Sciences [physics]/Automatic, set-membership estimation, 0202 electrical engineering, electronic engineering, information engineering, Computer Science (miscellaneous), QA1-939, Differential algebra, Well-defined, Engineering (miscellaneous), ComputingMilieux_MISCELLANEOUS, Mathematics, Soundness, Estimation theory, bounded uncertainty, Applied Mathematics, [ SDV.SPEE ] Life Sciences [q-bio]/Santé publique et épidémiologie, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], QA75.5-76.95, identifiability, Nonlinear system, Electronic computers. Computer science, Identifiability, 020201 artificial intelligence & image processing, nonlinear dynamic models
Abstract: Identifiability guarantees that the mathematical model of a dynamic system is well defined in the sense that it maps unambiguously its parameters to the output trajectories. This paper casts identifiability in a set-membership (SM) framework and relates recently introduced properties, namely, SM-identifiability, μ-SM-identifiability, and ε-SM-identifiability, to the properties of parameter estimation problems. Soundness and ε-consistency are proposed to characterize these problems and the solution returned by the algorithm used to solve them. This paper also contributes by carefully motivating and comparing SM-identifiability, μ-SM-identifiability and ε-SM-identifiability with related properties found in the literature, and by providing a method based on differential algebra to check these properties.
Published: 2016

12. Fast and accurate evaluation of a generalized incomplete gamma function

Author: Abergel , Rémy, Moisan , Lionel, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), and Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS )
Subjects: floating-point arithmetic, ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.2: Approximation/G.1.2.9: Special function approximations, [ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM], incomplete gamma function, incomplete gamma integral, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], continued fraction, ACM : G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.2: Approximation/G.1.2.9: Special function approximations, [ INFO.INFO-MS ] Computer Science [cs]/Mathematical Software [cs.MS], numerical cancellation, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Romberg's method, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS]
Abstract: We present a computational procedure to evaluate the integral ∫xy sp-1 e-μs ds, for 0 ≤ x < y ≤+∞, μ = ±1, p > 0, which generalizes the lower (x=0) and upper (y=+∞) incomplete gamma functions. To allow for large values of x, y, and p while avoiding under/overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, integrations by parts, or, when x ≈ y, with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and is essentially optimal considering the limitations imposed by the floating point arithmetic. Moreover, the admissible parameter range of our algorithm (0 ≤ p,x,y ≤ 1015) is much larger than competing algorithms and its robustness is assessed through massive usage in an image processing application.
Published: 2019

13. Microwave Tomographic Imaging of Cerebrovascular Accidents by Using High-Performance Computing

Author: Marcella Bonazzoli, Ibtissam El Kanfoud, Christian Pichot, Pierre Jolivet, Ioannis Aliferis, Pierre-Henri Tournier, Claire Migliaccio, Maya de Buhan, Serguei Semenov, Frédéric Hecht, Victorita Dolean, Marion Darbas, Frédéric Nataf, 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)), 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 d'Electronique, Antennes et Télécommunications (LEAT), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Shape reconstruction and identification (DeFI ), 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), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA), Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Univ Strathclyde], University of Strathclyde [Glasgow], Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), 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)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Centre National de la Recherche Scientifique (CNRS), EMTensor GmbH, This work was granted access to the HPC resources of TGCC@CEA under the allocations 2016-067519 and 2016-067730 made by GENCI. This work has been supported in part by ANR through project MEDIMAX, ANR-13-MONU-0012, ANR-13-MONU-0012,MEDIMAX,Résolution des Equations de Maxwell pour un système d'Imagerie Médicale complet(2013), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA), Department of Mathematics and Statistics, University of Strathclyde, Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), 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 ), Algorithms and parallel tools for integrated numerical simulations ( ALPINES ), 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 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Laboratoire d'Electronique, Antennes et Télécommunications ( LEAT ), Université Nice Sophia Antipolis ( UNS ), Université Côte d'Azur ( UCA ) -Université Côte d'Azur ( UCA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Jean Alexandre Dieudonné ( JAD ), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Amiénois de Mathématique Fondamentale et Appliquée ( LAMFA ), Université de Picardie Jules Verne ( UPJV ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de recherche en informatique de Toulouse ( IRIT ), Institut National Polytechnique [Toulouse] ( INP ) -Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique ( CNRS ), ANR-13-MONU-0012,MEDIMAX,Résolution des Equations de Maxwell pour un système d'Imagerie Médicale complet ( 2013 ), Université Nice Sophia Antipolis (1965 - 2019) (UNS), 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), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)
Subjects: QA75, FOS: Computer and information sciences, Computer Networks and Communications, Computer science, Parallel algorithm, FOS: Physical sciences, 010103 numerical & computational mathematics, 01 natural sciences, Theoretical Computer Science, Reduction (complexity), Computational Engineering, Finance, and Science (cs.CE), symbols.namesake, Artificial Intelligence, Microwave imaging, 0101 mathematics, Computer Science - Computational Engineering, Finance, and Science, Tomographic reconstruction, Domain decomposition methods, Reconstruction algorithm, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Inverse problem, Computational Physics (physics.comp-ph), Supercomputer, Computer Graphics and Computer-Aided Design, Scalable preconditioners, 010101 applied mathematics, [SPI.ELEC]Engineering Sciences [physics]/Electromagnetism, Maxwell's equations, Computer Science - Distributed, Parallel, and Cluster Computing, Hardware and Architecture, symbols, Distributed, Parallel, and Cluster Computing (cs.DC), Algorithm, Physics - Computational Physics, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; The motivation of this work is the detection of cerebrovascular accidents by microwave tomographic imaging. This requires the solution of an inverse problem relying on a minimization algorithm (for example, gradient-based), where successive iterations consist in repeated solutions of a direct problem. The reconstruction algorithm is extremely computationally intensive and makes use of efficient parallel algorithms and high-performance computing. The feasibility of this type of imaging is conditioned on one hand by an accurate reconstruction of the material properties of the propagation medium and on the other hand by a considerable reduction in simulation time. Fulfilling these two requirements will enable a very rapid and accurate diagnosis. From the mathematical and numerical point of view, this means solving Maxwell's equations in time-harmonic regime by appropriate domain decomposition methods, which are naturally adapted to parallel architectures.
Published: 2019

14. Optimality and resonances in a class of compact finite difference schemes of high order

Author: Joackim Bernier, Multi-scale numerical geometric schemes (MINGUS), 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)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, 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 ), Invariant Preserving SOlvers ( IPSO ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, and Institut National de Recherche en Informatique et en Automatique ( Inria )
Subjects: Dirichlet problem, Class (set theory), Algebra and Number Theory, Algebraic structure, Compact finite difference, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 010103 numerical & computational mathematics, Diophantine approximation, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Theory of computation, Convergence (routing), FOS: Mathematics, Padé approximant, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: International audience; In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an algebraic structure. We give some general criteria of convergence and we apply them to obtain two new results. On the one hand, we use Padé approximant theory to construct, for each given order of consistency, the most efficient schemes and we prove their convergence. On the other hand, we use diophantine approximation theory to prove that almost all of these schemes are convergent at the same rate as the consistency order, up to some logarithmic correction.
Published: 2019

15. A loosely coupled scheme for fictitious domain approximations of fluid-structure interaction problems with immersed thin-walled structures

Author: Miguel Angel Fernández, Ludovic Boilevin-Kayl, Jean-Frédéric Gerbeau, COmputational Mathematics for bio-MEDIcal Applications (COMMEDIA), 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), Numerical simulation of biological flows (REO), Sorbonne Université (SU)-Inria de Paris, Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), This work was supported by the project MIVANA, a collaborative project for the development of new technologies for mitral valve repair, which was led by the start-up company Kephalios, with the participation of the start-up company Epygon, who received funds from theFrench government, in the context of the program 'Investissement d’Avenir., Numerical simulation of biological flows ( REO ), 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, and Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria )
Subjects: Scheme (programming language), Discretization, Fictitious domain method, Coupling schemes, Applied Mathematics, Unfitted meshes, Stability (learning theory), 010103 numerical & computational mathematics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Domain (software engineering), Computational Mathematics, Coupling (physics), Fluid–structure interaction, Fluid-structure interaction, Compressibility, Applied mathematics, Incompressible fluid, 0101 mathematics, Immersed thin-walled structure, computer, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], computer.programming_language, Mathematics
Abstract: International audience; Fictitious domain approximations of fluid-structure interaction problems are generally discretized in time using strongly coupled schemes. This guarantees unconditional stability but at the price of solving a computationally demanding coupled system at each time-step. The design of loosely coupled schemes (i.e., methods that invoke the fluid and solid solvers only once per time-step) is of fundamental interest, especially for three-dimensional simulations, but the existing approaches are known to suffer from severe stability and/or time accuracy issues. We propose a new approach that overcomes these difficulties in the case of the coupling with immersed thin-walled structures.
Published: 2019

16. Vortex patterns in the almost-bosonic anyon gas

Author: Douglas Lundholm, Romain Duboscq, Michele Correggi, Nicolas Rougerie, Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] (Sapienza University of Rome), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA), Department of Mathematics [Sweden] (KTH), Stockholm University, Laboratoire de physique et modélisation des milieux condensés (LPM2C), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), European Project: 758620,CORFRONMAT, Correggi, Michele, Duboscq, Romain, Lundholm, Dougla, Rougerie, Nicolas, Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] ( Sapienza University of Rome ), Università degli Studi di Roma 'La Sapienza' [Rome], Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ), Department of Mathematics [Sweden] ( KTH ), Laboratoire de physique et modélisation des milieux condensés ( LPM2C ), Université Joseph Fourier - Grenoble 1 ( UJF ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), European Project : 758620,CORFRONMAT, Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT), and European Project: 758620,H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC),CORFRONMAT(2018)
Subjects: [ PHYS.QPHY ] Physics [physics]/Quantum Physics [quant-ph], [PHYS.COND.GAS]Physics [physics]/Condensed Matter [cond-mat]/Quantum Gases [cond-mat.quant-gas], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Anyon, General Physics and Astronomy, FOS: Physical sciences, Trapping, 01 natural sciences, Topological quantum computer, [ PHYS.COND.GAS ] Physics [physics]/Condensed Matter [cond-mat]/Quantum Gases [cond-mat.quant-gas], 010305 fluids & plasmas, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Quantum mechanics, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Abelian group, 010306 general physics, Settore MAT/07 - Fisica Matematica, Mathematical Physics, Energy functional, Physics, Condensed Matter::Quantum Gases, Quantum Physics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Mathematical Physics (math-ph), Numerical Analysis (math.NA), Gauge (firearms), Vortex, Quantum Gases (cond-mat.quant-gas), [ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph], Ground state, Quantum Physics (quant-ph), Condensed Matter - Quantum Gases, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: International audience; We study theoretically and numerically the ground state of a gas of 2D abelian anyons in an external trapping potential. We treat anyon statistics in the magnetic gauge picture, perturbatively around the bosonic end. This leads to a mean-field energy functional, whose ground state displays vortex lattices similar to those found in rotating Bose-Einstein condensates. A crucial difference is however that the vortex density is proportional to the underlying matter density of the gas.
Published: 2019

17. Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field

Author: Bruno Dubroca, Stéphane Brull, Xavier Lhébrard, lhebrard, xavier, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), 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 Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire d'Analyse et de Mathématiques Appliquées ( LAMA ), Université Paris-Est Marne-la-Vallée ( UPEM ) -Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 ( UPEC UP12 ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bordeaux ( IMB ), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux ( UB ) -Institut Polytechnique de Bordeaux ( Bordeaux INP ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: General Computer Science, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, Entropy (classical thermodynamics), hydrodynamic limit, Relaxation system, 0103 physical sciences, discrete entropy minimum principle, 0101 mathematics, BGK models, Physics, Finite volume method, relaxation method, Mathematical analysis, General Engineering, Relaxation (iterative method), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Euler system, Solver, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], non-conservative hyperbolic system, discrete entropy inequalities, 010101 applied mathematics, Maxwell's equations, Transverse magnetic field, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: The present paper concerns the study of the nonconservative bitem-perature Euler system with transverse magnetic field. We firstly introduce an underlying conservative kinetic model coupled to Maxwell equations. The nonconservative bitemperature Euler system with transverse magnetic field is then established from this kinetic model by hydrodynamic limit. Next we present the derivation of a finite volume method to approximate weak solutions. It is obtained by solving a relaxation system of Suliciu type, and is similar to HLLC type solvers. The solver is shown in particular to preserve positivity of density and internal energies. Moreover we use a local minimum entropy principle to prove discrete entropy inequalities, ensuring the robustness of the scheme.
Published: 2019

18. Linear stability of a vectorial kinetic relaxation scheme with a central velocity

Author: Courtès, Clémentine, Franck, Emmanuel, 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), TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, 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)-Université de Strasbourg (UNISTRA), 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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), TOkamaks and NUmerical Simulations ( TONUS ), Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria )
Subjects: [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], kinetic relaxation, von Neumann analysis, semi-Lagrangian method, eigenvalues, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], linear stability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], implicit splitting scheme
Abstract: This article deals with the linear stability of an implicit vectorial kinetic relaxation scheme with a central velocity used to solve numerically some multi-scale hyperbolic systems.Keywords : linear stability, von Neumann analysis, kinetic relaxation, semi-Lagrangian method, implicit splitting scheme, eigenvalues.
Published: 2019

19. Analysis of an asymptotic preserving scheme for stochastic linear kinetic equations in the diffusion limit

Author: Erwan Faou, Nathalie Ayi, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Multi-scale numerical geometric schemes (MINGUS), 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)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), 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)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-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)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Inria Rennes – Bretagne Atlantique, 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 ), Multi-scale numerical geometric schemes ( MINGUS ), 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 ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, and Institut National de Recherche en Informatique et en Automatique ( Inria )
Subjects: Statistics and Probability, 010103 numerical & computational mathematics, diffusion limit, stiff terms, Kinetic energy, stability analysis, asymptotic preserving scheme, 01 natural sciences, 010104 statistics & probability, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Decomposition (computer science), FOS: Mathematics, Discrete Mathematics and Combinatorics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Mathematics, stochastic transport equations, Applied Mathematics, Probability (math.PR), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Numerical Analysis (math.NA), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Kinetic equations, Modeling and Simulation, Scheme (mathematics), Statistics, Probability and Uncertainty, Diffusion limit, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: International audience; We present an asymptotic preserving scheme based on a micro-macro decomposition for stochastic linear transport equations in kinetic and diffusive regimes. We perform a mathematical analysis and prove that the scheme is uniformly stable with respect to the mean free path of the particles in the simple telegraph model and in the general case. We present several numerical tests which validate our scheme.
Published: 2019

20. Analysis of 3D non-linear Stokes problem coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications to digital rock physics and mucociliary clearance

Author: Robin Chatelin, David Sanchez, Philippe Poncet, Laurène Hume, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UPS), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), 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-Centre National de la Recherche Scientifique (CNRS), 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), 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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), 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), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et de leurs Applications [Pau] ( LMAP ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Tribologie et Dynamique des Systèmes ( LTDS ), École Centrale de Lyon ( ECL ), and Université de Lyon-Université de Lyon-École Nationale des Travaux Publics de l'État ( ENTPE ) -Ecole Nationale d'Ingénieurs de Saint Etienne-Centre National de la Recherche Scientifique ( CNRS )
Subjects: Generalized Stokes equations, 010103 numerical & computational mathematics, 01 natural sciences, 010305 fluids & plasmas, Physics::Fluid Dynamics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], mucociliary clearance, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Diffusion (business), MSC: 35Q30, 76D03, 76D07, 65M25, 68U20, 76Z05, 92B05, Microscale chemistry, porous media flows, Physics, Numerical Analysis, Shear thinning, digital rock physics, Applied Mathematics, Numerical analysis, shear-thinning, variable viscosity, Mechanics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], bio-mathematics, Action (physics), Computational Mathematics, Nonlinear system, Flow (mathematics), Modeling and Simulation, Porous medium, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 3D flows
Abstract: This study provides the analysis of the generalized 3D Stokes problem in a time dependent domain, modeling a solid in motion. The fluid viscosity is a non-linear function of the shear-rate and depends on a transported and diffused quantity. This is a natural model of flow at very low Reynolds numbers, typically at the microscale, involving a miscible, heterogeneous and shear-thinning incompressible fluid filling a complex geometry in motion. This one-way coupling is meaningful when the action produced by a solid in motion has a dominant effect on the fluid. Several mathematical aspects are developed. The penalized version of this problem is introduced, involving the penalization of the solid in a deformable motion but defined in a simple geometry (a periodic domain and/or between planes), which is of crucial interest for many numerical methods. All the equations of this partial differential system are analyzed separately, and then the coupled model is shown to be well-posed and to converge toward the solution of the initial problem. In order to illustrate the pertinence of such models, two meaningful micrometer scale real-life problems are presented: on the one hand, the dynamics of a polymer percolating the pores of a real rock and miscible in water; on the other hand, the dynamics of the strongly heterogeneous mucus bio-film, covering the human lungs surface, propelled by the vibrating ciliated cells. For both these examples the mathematical hypothesis are satisfied.
Published: 2019

21. Asymptotic Preserving numerical schemes for multiscale parabolic problems

Author: Mohammed Lemou, Nicolas Crouseilles, Gilles Vilmart, 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 ), Invariant Preserving SOlvers ( IPSO ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), Section de mathématiques [Genève], Université de Genève ( UNIGE ), 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), Invariant Preserving SOlvers (IPSO), 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 (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Genève = University of Geneva (UNIGE), ANR-14-CE23-0007-01, Agence Nationale de la Recherche, ERC-279389, European Research Council, 200020_144313/1, Swiss National Foundation, ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques(2014), 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), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, Université de Genève (UNIGE), and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: Mathematical optimization, Discretization, 65M12, 74Q10, 35K15, 65M12, 74Q10, 35K5, 010102 general mathematics, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 010103 numerical & computational mathematics, General Medicine, 01 natural sciences, Homogenization (chemistry), FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, ddc:510, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale $\varepsilon$. Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behaviour as $\varepsilon \rightarrow 0$, without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, known such homogenization schemes are in general not accurate for both the highly oscillatory regime $\varepsilon \rightarrow 0$ and the non oscillatory regime $\varepsilon \sim 1$. In this paper, we introduce an Asymptotic Preserving method based on an exact micro-macro decomposition of the solution which remains consistent for both regimes., 7 pages, to appear in C. R. Acad. Sci. Paris; Ser. I
Published: 2016

22. Semialgebraic Geometry of Nonnegative Tensor Rank

Author: Yang Qi, Lek-Heng Lim, Pierre Comon, CICS ( GIPSA-CICS ), Département Images et Signal ( GIPSA-DIS ), Grenoble Images Parole Signal Automatique ( GIPSA-lab ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Grenoble Images Parole Signal Automatique ( GIPSA-lab ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Department of Mathematics, University of California [Berkeley], European Project : 320594,EC:FP7:ERC,ERC-2012-ADG_20120216,DECODA ( 2013 ), GIPSA - Communication Information and Complex Systems (GIPSA-CICS), Département Images et Signal (GIPSA-DIS), Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Department of Mathematics [Berkeley], University of California-University of California, European Project: 320594,EC:FP7:ERC,ERC-2012-ADG_20120216,DECODA(2013), University of California [Berkeley] (UC Berkeley), and University of California (UC)-University of California (UC)
Subjects: [ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing, Rank (linear algebra), Boundary (topology), 010103 numerical & computational mathematics, Nonnegative rank, 01 natural sciences, Combinatorics, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, FOS: Mathematics, Tensor, Nonnegative matrix, 0101 mathematics, Mathematics, Conjecture, Direct sum, 010102 general mathematics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Mathematics - Rings and Algebras, Metzler matrix, tensor, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], rank, Rings and Algebras (math.RA), nonnegative, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract: We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. We show that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank. In addition, such nonnegative tensors always have unique nonnegative rank-$r$ decompositions if the real tensor space is $r$-identifiable. We determine conditions under which a best nonnegative rank-$r$ approximation has a unique nonnegative rank-$r$ decomposition: for $r \le 3$, this is always the case; for general $r$, this is the case when the best nonnegative rank-$r$ approximation does not lie on the boundary of $D_r$. Many of our general identifiability results also apply to real tensors and real symmetric tensors., 25 pages, to appear in SIMAX
Published: 2016

23. A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes

Author: Rémi Abgrall, University of Zurich, Abgrall, Rémi, University of Zürich [Zürich] ( UZH ), and Universität Zürich [Zürich] = University of Zurich (UZH)
Subjects: Discretization, Physics and Astronomy (miscellaneous), 010103 numerical & computational mathematics, 01 natural sciences, 510 Mathematics, 2604 Applied Mathematics, Discontinuous Galerkin method, FOS: Mathematics, 1706 Computer Science Applications, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Entropy (energy dispersal), 3101 Physics and Astronomy (miscellaneous), 2612 Numerical Analysis, Mathematics, Numerical Analysis, Applied Mathematics, Mathematical analysis, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Finite element method, 3100 General Physics and Astronomy, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Nonlinear system, 10123 Institute of Mathematics, Modeling and Simulation, Dissipative system, A priori and a posteriori, 2605 Computational Mathematics, Joint quantum entropy, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 2611 Modeling and Simulation
Abstract: We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting from the discretization of the original PDE, how to construct a scheme that is consistent with the original PDE and the additional conservation relation. Since one interesting example is given by the systems endowed by an entropy, we provide one explicit solution, and show that the accuracy of the new scheme is at most degraded by one order. In the case of a discontinuous Galerkin scheme and a Residual distribution scheme, we show how not to degrade the accuracy. This improves the recent results obtained in [1] , [2] , [3] , [4] in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved. We study the behavior of the method on a non linear scalar problem. However, the method is not restricted to scalar problems.
Published: 2018

24. Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

Author: Victor Michel-Dansac, Raphaël Loubère, Marie-Hélène Vignal, Giacomo Dimarco, 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 Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Ferrara], Università degli Studi di Ferrara (UniFE), Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), 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), ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques(2014), Università degli Studi di Ferrara ( UniFE ), Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques ( 2014 ), Università degli Studi di Ferrara = University of Ferrara (UniFE), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), 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: Physics and Astronomy (miscellaneous), Discretization, High-order schemes, Monotonic function, 010103 numerical & computational mathematics, 01 natural sciences, NO, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Stability theory, Total variation diminishing, FOS: Mathematics, Low Mach, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, Mathematics, Hyperbolic, Numerical Analysis, Asymptotic preserving, High-order, IMEX schemes, SSP-TVD, Applied Mathematics, Courant–Friedrichs–Lewy condition, Mathematical analysis, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Euler system, SSP-TVD property, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Mach number, Modeling and Simulation, Hyperbolic conservation laws, Euler's formula, symbols, Asymptotic Preserving, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Low Mach number limit
Abstract: In this work, we consider the development of implicit-explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The proposed scheme is asymptotically stable with a CFL condition independent of the Mach number. In addition, it degenerates, in the low Mach number regime, to a consistent discretization of the incompressible system. Since it has been proved that implicit schemes of order higher than one cannot be TVD (SSP) [30] , we construct a new paradigm of implicit time integrators by coupling first-order in time schemes with second-order ones in the same spirit as highly accurate shock-capturing TVD methods in space. For this particular class of schemes, the TVD property is first proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first- to the second-order both in space and time. It preserves the monotonicity of the solution, and is highly accurate for all choices of the Mach number. Moreover, the time step is only restricted by the non-stiff part of the system. We finally show, thanks to one- and two-dimensional test cases, that the method indeed possesses the claimed properties.
Published: 2018

25. Dispersion analysis of compatible Galerkin schemes for the 1D shallow water model

Author: Daniel Y. Le Roux, Christopher Eldred, Mathematics and computing applied to oceanic and atmospheric flows (AIRSEA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), ANR-14-CE23-0010,HEAT,Modélisation atmosphérique hautement efficace(2014), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Mathematics and computing applied to oceanic and atmospheric flows ( AIRSEA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Institut National Polytechnique de Grenoble ( INPG ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), and Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: [ MATH ] Mathematics [math], Work (thermodynamics), dispersion relationship, 010504 meteorology & atmospheric sciences, Physics and Astronomy (miscellaneous), mimetic Galerkin differences, 010103 numerical & computational mathematics, fluid dynamics, 01 natural sciences, geophysical, geophysical fluid dynamics, Fluid dynamics, dynamical core, mixed finite elements, 0101 mathematics, [MATH]Mathematics [math], Dispersion (water waves), Galerkin method, Shallow water equations, 0105 earth and related environmental sciences, Physics, Numerical Analysis, shallow water equations, Applied Mathematics, Mathematical analysis, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], exterior calculus, Mass matrix, Finite element method, Computer Science Applications, Quadrature (mathematics), Computational Mathematics, finite element, Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; In this work, we study the dispersion properties of two compatible Galerkin schemes for the 1D linearized shallow water equations: the P C n −P DG n−1 and the GD n −DGD n element pairs. Compatible Galerkin methods have many desirable properties, including energy conservation, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, this does not guarantee good wave dispersion properties. Previous work on the P C 2 − P DG 1 pair has indeed indicated the presence of spectral gaps, and it is extended in this paper to the study of the P C n − P DG n−1 pair for arbitrary n. Additionally, an alternative element pair is introduced, the GD n − DGD n pair, that is free of spectral gaps while benefiting from the desirable properties of compatible elements. Asymptotic convergence rates are established for both element pairs, including the use of inexact quadrature (which di-agonalizes the velocity mass matrix) for the P C n − P DG n−1 pair and reduced quadrature for the GD n − DGD n pair. Plots of the dispersion relationship and group velocities for a wide range of n and Rossby radii are shown. A brief investigation into the utility of mass lumping to remove the spectral gaps for the P C 3 − P DG 2 pair is performed. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two elements.
Published: 2018

26. The Local Average Contact (LAC) method

Author: Guillaume Drouet, Mickaël Abbas, Patrick Hild, Matériaux et Mécanique des Composants (EDF R&D MMC), EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), EDF (EDF), Institut de Mathématiques de Toulouse UMR5219 (IMT), 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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), 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), MMC et MAI ( EDF R&D ), EDF R&D ( EDF R&D ), EDF ( EDF ) -EDF ( EDF ), EDF ( EDF ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), ElectRotechnique et MEcanique des Structures (EDF R&D ERMES), Institut des Sciences de la mécanique et Applications industrielles (IMSIA - UMR 9219), and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-EDF R&D (EDF R&D)
Subjects: Unilateral contact, Discretization, Computational Mechanics, General Physics and Astronomy, Boundary (topology), 010103 numerical & computational mathematics, [SPI.MECA.SOLID]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Solid mechanics [physics.class-ph], 01 natural sciences, Quadratic equation, Convergence (routing), [ SPI.MECA.SOLID ] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanics of the solides [physics.class-ph], Polygon mesh, 0101 mathematics, [SPI.MECA.SOLID]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanics of the solides [physics.class-ph], ComputingMilieux_MISCELLANEOUS, Local average, Mathematics, nonmatching meshes, Mechanical Engineering, Mathematical analysis, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Finite element method, Computer Science Applications, 010101 applied mathematics, Mechanics of Materials, geometrical nonconformity, Local Average Contact (LAC) condition, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: The Local Average Contact (LAC) method allows the handling of nonmatching meshes in an easy way by averaging locally the interpenetration between the contacting bodies. In this paper we consider the frictionless unilateral contact problem and several numerical experiments involving two and three-dimensional bodies discretized with various linear and quadratic finite elements. We also present a convergence analysis of the method in the geometrical nonconforming case in which the boundary points of the candidate contact areas do not coincide.
Published: 2018

27. Efficient algorithms to perform linear algebra operations on 3D arrays in vector languages

Author: Cuvelier, Francois, Laboratoire Analyse, Géométrie et Applications ( LAGA ), Université Paris 8 Vincennes-Saint-Denis ( UP8 ) -Université Paris 13 ( UP13 ) -Université Sorbonne Paris Cité ( USPC ) -Institut Galilée-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and Cuvelier, Francois
Subjects: [ INFO.INFO-NA ] Computer Science [cs]/Numerical Analysis [cs.NA], [INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In a few number of applications, a need arises to do some usual linear algebra operations on a very large number of very small matrices of the same size, refered in this report by 3D-array. These operations could be as simple as sum or products, or more complex like computation of determinants, factorizing, solving, ... The aim of this report is to describe some vectorized algorithms for each one of these operations and to give eciency measures. For example, computing the LU decomposition with partial pivoting of one million of 8-by-8 matrices on our reference computer is performed in 3.1 seconds with Matlab, 5.6 seconds with Octave and 9.7 seconds with Python.
Published: 2018

28. Stabilization of walls in notched ferromagnetic nanowires

Author: Carbou, Gilles, Sanchez, David, Université de Pau et des Pays de l'Adour (UPPA), 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), Université de Pau et des Pays de l'Adour ( UPPA ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Centre National de la Recherche Scientifique ( CNRS ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -PRES Université de Toulouse-Université Paul Sabatier - Toulouse 3 ( UPS ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse 1 Capitole ( UT1 ), 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: Physics::Fluid Dynamics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Landau-Lifshitz equation, nanowire, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], stability, ferromagnetism, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this paper we study a one-dimensional model of ferromagnetic nanowire presenting notches. We prove the existence of stable wall profiles even under a small applied magnetic field with the walls localized in notches. Moreover, in order to illustrate wall depinning by applied magnetic field, we prove the non-existence of stationary wall profiles in the presence of a large applied magnetic field.
Published: 2018

29. Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation

Author: Mark Kärcher, Sébastien Boyaval, Karen Veroy, Martin A. Grepl, Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (LHSV), École des Ponts ParisTech (ENPC)-Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Rheinisch-Westfälische Technische Hochschule Aachen ( RWTH ), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics ( Saint-Venant ), École des Ponts ParisTech ( ENPC ) -Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement ( Cerema ) -EDF R&D ( EDF R&D ), EDF ( EDF ) -EDF ( EDF ), MATHematics for MatERIALS ( MATHERIALS ), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Rheinisch-Westfälische Technische Hochschule Aachen (RWTH), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (Saint-Venant), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)
Subjects: Reduced-order models, Control and Optimization, Optimization problem, 010504 meteorology & atmospheric sciences, Aerospace Engineering, Weak-constraint, 010103 numerical & computational mathematics, 01 natural sciences, Strong-constraint, Data assimilation, FOS: Mathematics, Parameter estimation, Initial value problem, Applied mathematics, Variational data assimilation, 0101 mathematics, Electrical and Electronic Engineering, Mathematics - Optimization and Control, 0105 earth and related environmental sciences, Civil and Structural Engineering, Mathematics, 65K10, Basis (linear algebra), Estimation theory, Mechanical Engineering, Function (mathematics), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], A posteriori error estimation, Optimization and Control (math.OC), 4D-Var, A priori and a posteriori, Errors-in-variables models, PDE-constrained optimization, Reduced basis method, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach.
Published: 2018

30. Direct simulation of rigid particles in a viscoelastic fluid

Author: Astrid Decoene, Sébastien Martin, Bertrand Maury, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), Martin, Sébastien, Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), MAP5 - Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Centre National de la Recherche Scientifique ( CNRS ) -Institut National des Sciences Mathématiques et de leurs Interactions-Université Paris Descartes - Paris 5 ( UPD5 ), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Paris (ENS Paris)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Inertial frame of reference, General Chemical Engineering, dense suspensions, fluid-structure interaction, [SPI.MECA.MEFL] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 01 natural sciences, Viscoelasticity, characteris, 010305 fluids & plasmas, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Physics::Fluid Dynamics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 0103 physical sciences, Fluid–structure interaction, Newtonian fluid, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], General Materials Science, Penalty method, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Physics, 010304 chemical physics, penalty method, Applied Mathematics, Mechanical Engineering, Numerical analysis, Mechanics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Condensed Matter Physics, Ellipsoid, Oldroyd model, Finite element method, fictitious domain, characteristics method, tics method, [ SPI.MECA.MEFL ] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], viscoelastic fluid, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this paper, we present a numerical method which performs the direct simulation of 2D viscoelastic suspensions. Objectives. Interactions between viscoelastic fluids of Oldroyd type, including inertial effects, and a large number of rigid spheres or ellipsoids are addressed. Method. The numerical method is built upon a fictitious domain approach which consists in defining the fluid-structure problem over the whole domain (fluid+solid). In the Newtonian case, the resulting variational formulation, although classical, uses non-standard functional spaces which include the rigidity constraints. From the numerical point of view, these constraints can be addressed by penalty methods, leading to the possible use of standard finite elements solvers with fixed structured meshes. We show how this approach can be adapted to viscoelastic fluids of Oldroyd type. Results. Two types of configurations are investigated in dilute and dense regimes: sedimentation of spherical particles and shear flows with ellipsoidal particles.
Published: 2018

31. Models and methods for complex flows: application to atmospheric reentry and particle / fluid interactions

Author: Mathiaud, Julien, Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA-CESTA), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), University of Bordeaux, Pauline Lafitte-Godillon, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Commissariat à l'Energie Atomique et aux Energies Alternatives ( CEA-CESTA ), Commissariat à l'énergie atomique et aux énergies alternatives ( CEA ), Institut de Mathématiques de Bordeaux ( IMB ), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux ( UB ) -Institut Polytechnique de Bordeaux ( Bordeaux INP ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: ACM : G.: Mathematics of Computing, [ INFO.INFO-NA ] Computer Science [cs]/Numerical Analysis [cs.NA], Théorie cinétique des gaz, Fokker Planck equation, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Equation de Fokker-Planck, ACM : G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Modèle BGK, BGK model, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], KInetic theory of gases, Fluid particle model, ACM: G.: Mathematics of Computing, interaction particule fluide, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: This manuscript is devoted to the presentation of the different studies that I have done since 2003 and the beginning of my Phd at the French Atomic Agency (CEA). I will try to put in perspective all the things done since then and give guidelines of what I should explore later. The works presented here essentially deal with the physics of atmospheric reentry and fluid-particles interactions. For these two mains applications we focus on models, mathematical properties, numerical results and experimental results.
Published: 2018

32. Hybrid discretization methods with adaptive yield surface detection for Bingham pipe flows

Author: Alexandre Ern, Xavier Chateau, Karol L. Cascavita, Jeremy Bleyer, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Navier (NAVIER UMR 8205), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, This work has benefited from a French government grant managed by ANR within the framework of the National Program Investments for the Future ANR-11-LABX-0022-01, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ), Simulation for the Environment: Reliable and Efficient Numerical Algorithms ( SERENA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Laboratoire Navier ( NAVIER UMR 8205 ), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux ( IFSTTAR ) -Centre National de la Recherche Scientifique ( CNRS ) -École des Ponts ParisTech ( ENPC ), and ANR-11-LABX-0022,MMCD,Modélisation et Expérimentation Multi-Echelles des Matériaux pour la Cons-truction Durable(2011)
Subjects: [ INFO.INFO-MO ] Computer Science [cs]/Modeling and Simulation, Discretization, Yield surface, Context (language use), 010103 numerical & computational mathematics, 01 natural sciences, Theoretical Computer Science, symbols.namesake, Applied mathematics, Polygon mesh, 0101 mathematics, Mathematics, ComputingMethodologies_COMPUTERGRAPHICS, Numerical Analysis, Adaptive mesh refinement, Augmented Lagrangian method, Applied Mathematics, [INFO.INFO-CE]Computer Science [cs]/Computational Engineering, Finance, and Science [cs.CE], General Engineering, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, Computational Mathematics, Computational Theory and Mathematics, Lagrange multiplier, Piecewise, symbols, [ INFO.INFO-CE ] Computer Science [cs]/Computational Engineering, Finance, and Science [cs.CE], Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; We devise a hybrid low-order method for Bingham pipe flows, where the velocity is discretized by means of one unknown per mesh face and one unknown per mesh cell which can be eliminated locally by static condensation. The main advantages are local conservativity and the possibility to use polygonal/polyhedral meshes. We exploit this feature in the context of adaptive mesh refinement to capture the yield surface by means of local mesh refinement and possible coarsening. We consider the augmented Lagrangian method to solve iteratively the variational inequalities resulting from the discrete Bingham problem, using piecewise constant fields for the auxiliary variable and the associated Lagrange multiplier. Numerical results are presented in pipes with circular and eccentric annulus cross-section for different Bingham numbers.
Published: 2018

33. Optimal input design for parameter estimation in a bounded-error context for nonlinear dynamical systems

Author: Carine Jauberthie, Qiaochu Li, Zohra Cherfi-Boulanger, Lilianne Denis-Vidal, Roberval, Université de Technologie de Compiègne ( UTC ), Équipe DIagnostic, Supervision et COnduite ( LAAS-DISCO ), Laboratoire d'analyse et d'architecture des systèmes [Toulouse] ( LAAS ), Centre National de la Recherche Scientifique ( CNRS ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Institut National Polytechnique [Toulouse] ( INP ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Toulouse III - Paul Sabatier ( UPS ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Institut National Polytechnique [Toulouse] ( INP ), Laboratoire de Mathématiques Appliquées de Compiègne ( LMAC ), Équipe DIagnostic, Supervision et COnduite (LAAS-DISCO), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), Roberval (Roberval), Roberval ( LRURM ), Laboratoire d'analyse et d'architecture des systèmes [Toulouse] (LAAS), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UPS), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique [Toulouse] (INP)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UPS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique [Toulouse] (INP), 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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], 0209 industrial biotechnology, Computer science, Gaussian, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS], [ SPI.SIGNAL ] Engineering Sciences [physics]/Signal and Image processing, 02 engineering and technology, Interval arithmetic, symbols.namesake, 020901 industrial engineering & automation, [ SPI.AUTO ] Engineering Sciences [physics]/Automatic, 0202 electrical engineering, electronic engineering, information engineering, Nonlinear systems, Parameter estimation, Applied mathematics, Interval analysis, Electrical and Electronic Engineering, Finite set, Aerospace models, Gramian matrix, Estimation theory, Sensitivity functions, Design of experiments, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Nonlinear system, Bounded noise, Optimal input, Continuous-time systems, Control and Systems Engineering, symbols, A priori and a posteriori, 020201 artificial intelligence & image processing, State estimation, Experiment design
Abstract: International audience; This paper deals with optimal input design for parameter estimation in a bounded-error context. Uncertain controlled nonlinear dynamical models, when the input can be parametrized by a finite number of parameters, are considered. The main contribution of this paper concerns criteria for obtaining optimal inputs in this context. Two input design criteria are proposed and analysed. They involve sensitivity functions. The first criterion requires the inversion of the Gram matrix of sensitivity functions. The second one does not require this inversion and is then applied for parameter estimation of a model taken from the aeronautical domain. The estimation results obtained using an optimal input are compared with those obtained with an input optimized in a more classical context (Gaussian measurement noise and parameters a priori known to belong to some boxes). These results highlight the potential of optimal input design in a bounded-error context.
Published: 2018

34. Nonsmooth contact dynamics for the numerical simulation of collisions in musical string instruments

Author: Jean-Loïc Le Carrou, Clara Issanchou, Vincent Acary, Cyril Touzé, Franck Pérignon, Institut des Sciences de la mécanique et Applications industrielles (IMSIA - UMR 9219), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Lutheries - Acoustique - Musique (IJLRDA-LAM), Institut Jean Le Rond d'Alembert (DALEMBERT), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Modélisation, simulation et commande des systèmes dynamiques non lisses (TRIPOP ), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-EDF R&D (EDF R&D), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut des Sciences de la mécanique et Applications industrielles ( IMSIA - UMR 9219 ), Commissariat à l'énergie atomique et aux énergies alternatives ( CEA ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Paris-Saclay-EDF R&D ( EDF R&D ), EDF ( EDF ) -EDF ( EDF ), Modélisation, simulation et commande des systèmes dynamiques non lisses ( TRIPOP ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Institut Jean Le Rond d'Alembert ( DALEMBERT ), and Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: [SPI.ACOU]Engineering Sciences [physics]/Acoustics [physics.class-ph], Bass guitar, Acoustics and Ultrasonics, Computer simulation, [ SPI.ACOU ] Engineering Sciences [physics]/Acoustics [physics.class-ph], Computer science, [SPI.MECA.VIBR]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 010103 numerical & computational mathematics, Musical, 01 natural sciences, [ SPI.MECA.VIBR ] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph], Contact force, Musical acoustics, Vibration, [SPI]Engineering Sciences [physics], Arts and Humanities (miscellaneous), Control theory, 0103 physical sciences, [ SPI ] Engineering Sciences [physics], C++ string handling, Contact dynamics, 0101 mathematics, 010301 acoustics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; Collisions in musical string instruments play a fundamental role in explaining the sound production in various instruments such as sitars, tanpuras and electric basses. Contacts occuring during the vibration provide a nonlinear effect which shapes a specific tone due to energy transfers and enriches the hearing experience. As such, they must be carefully simulated for the purpose of physically-based sound synthesis. Most of the numerical methods presented in the literature rely on a compliant modeling of the contact force between the string and the obstacle. In this contribution, numerical methods from nonsmooth contact dynamics are used to integrate the problem in time. A Moreau-Jean time-stepping scheme is combined with an exact scheme for phases with no contact, thus controlling the numerical dispersion. Results for a two-point bridge mimicking a tanpura and an electric bass are presented, showing the ability of the method to deal efficiently with such problems while invoking, as compared to a compliant approach, less modelling parameters and a reduced computational burden.
Published: 2018

35. Infinite Impulse Response Filters for Nonuniform Data

Author: Bidégaray-Fesquet, Brigitte, Fesquet, Laurent, Equations aux Dérivées Partielles ( EDP ), Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Techniques of Informatics and Microelectronics for integrated systems Architecture ( TIMA ), Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut polytechnique de Grenoble - Grenoble Institute of Technology ( Grenoble INP ) -Institut National Polytechnique de Grenoble ( INPG ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), ANR-11-LABX-0025-01,PERSYVAL-lab,Systèmes et Algorithmes Pervasifs au confluent des mondes physique et numérique ( 2011 ), Equations aux Dérivées Partielles (EDP ), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Techniques de l'Informatique et de la Microélectronique pour l'Architecture des systèmes intégrés (TIMA), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), ANR-11-LABX-0025,PERSYVAL-lab,Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011), Bidégaray-Fesquet, Brigitte, Laboratoires d'excellence - Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique - - PERSYVAL-lab2011 - ANR-11-LABX-0025 - LABX - VALID, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Techniques of Informatics and Microelectronics for integrated systems Architecture (TIMA), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), and ANR-11-LABX-0025-01,PERSYVAL-lab,Systèmes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011)
Subjects: Infinite Impulse Response Filters, [SPI.NANO] Engineering Sciences [physics]/Micro and nanotechnologies/Microelectronics, [ SPI.NANO ] Engineering Sciences [physics]/Micro and nanotechnologies/Microelectronics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [ SPI.SIGNAL ] Engineering Sciences [physics]/Signal and Image processing, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [SPI.NANO]Engineering Sciences [physics]/Micro and nanotechnologies/Microelectronics, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, Electro-cardiogramm, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], [SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processing, Asynchronous and Event-Driven Systems
Abstract: International audience; Capturing and storing samples only when needed is a way to ensure a drastic reduction of the data to be processed, which is a crucial issue in many applications, such as autonomous and communicating smart devices (Internet of Things). This leads to nonuniform data for which specific processing chains have to be designed. In this paper, we discuss the issues to be address to generalize Infinite Impulse Response filters to the nonuniform case. We illustrate the performance of the constructed filters on an electrocardiogram signal, for two ways to obtain the nonuniform samples, and select linear decimation and the bilinear scheme as a good combination for this application.
Published: 2018

36. Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates

Author: Tony Lelièvre, Wei Zhang, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ), MATHematics for MatERIALS ( MATHERIALS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Zuse Institute Berlin ( ZIB ), European Project : 614492,EC:FP7:ERC,ERC-2013-CoG,MSMATH ( 2014 ), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Zuse Institute Berlin (ZIB), The first author's research was supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC grant agreement 614492. The second author's research was supported by the Einstein Foundation Berlin of the Einstein Center for Mathematics (ECMath) through project CH21, European Project: 614492,EC:FP7:ERC,ERC-2013-CoG,MSMATH(2014), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)
Subjects: Work (thermodynamics), Reaction coordinate, General Physics and Astronomy, Conditional expectation, 01 natural sciences, Continuation, Approximation error, AMS Subject Classifications: 60J60, 60H10, 82C31, Pathwise estimates, effective dynamics, FOS: Mathematics, Diffusion process, Applied mathematics, 0101 mathematics, Mathematics, 60J60, 60H10, 82C31, Stochastic process, Ecological Modeling, 010102 general mathematics, Probability (math.PR), reaction coordinate, General Chemistry, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Computer Science Applications, 010101 applied mathematics, Nonlinear system, Modeling and Simulation, Effective dynamics, 60J60, 60H10, 82C31 [AMS Subject Classifications], pathwise estimates, time scale separation, Time scale separation, diffusion process, Mathematics - Probability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Lelièvre, Nonlinearity, 23 (2010), pp. 2131--2163] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Lelièvre, and S. Olla, Stochastic Process. Appl., 127 (2017), pp. 2841--2863], in this paper we obtain pathwise estimates of effective dynamics for both nonlinear and vector-valued reaction coordinate functions.
Published: 2018

37. Schémas numériques pour les EDPs anisotropes sur grille cartésienne

Author: Mirebeau, Jean-Marie, Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Sud XI, J. Frédéric Bonnans, Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques d'Orsay ( LMO ), and Université Paris-Saclay-Centre National de la Recherche Scientifique ( CNRS )
Subjects: Voronoi reduction, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Image processing, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Reduction de Voronoi, Anisotropic eikonal equation, Anisotropic diffusion, Equation eikonale anisotrope, Traitement de l'image, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Diffusion anisotrope
Published: 2018

38. Logarithmic Mathematical Morphology: a new framework robust to illumination changes

Author: Noyel , Guillaume, International Prevention Research Institute ( IPRI ), and University of Strathclyde Institute of Global Public Health ( SIGPH@iPRI )
Subjects: Contrast variations, pattern recognition, ACM : I.: Computing Methodologies/I.5: PATTERN RECOGNITION/I.5.1: Models, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, Logarithmic Image Processing, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [ SPI.SIGNAL ] Engineering Sciences [physics]/Signal and Image processing, [ INFO.INFO-CV ] Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], [ MATH.MATH-GN ] Mathematics [math]/General Topology [math.GN], ACM : I.: Computing Methodologies/I.4: IMAGE PROCESSING AND COMPUTER VISION, illumination changes, Computer Science::Computer Vision and Pattern Recognition, ACM : I.: Computing Methodologies/I.5: PATTERN RECOGNITION, Mathematical Morphology
Abstract: A new set of mathematical morphology (MM) operators robust to illumination changes in images is defined thanks to the Logarithmic Image Processing (LIP) model which is based on physical laws of acquisition and which is consistent with human vision. The fundamental operators of logarithmic-MM, the logarithmic-dilation and the logarithmic-erosion, are defined with the LIP-addition of a structuring function. The combination of these two adjunct operators gives morphological filters, namely the logarithmic-opening and closing, useful for pattern recognition. The mathematical relation existing between " classical " dilation and erosion-with an additive structuring function-and their logarithmic-versions is established facilitating their implementation. Results on simulated and real images show that logarithmic-MM is more efficient on low-contrasted information than " classical " MM.
Published: 2018

39. A moment-matching method to study the variability of phenomena described by partial differential equations

Author: Jean-Frédéric Gerbeau, Damiano Lombardi, Eliott Tixier, Numerical simulation of biological flows (REO), Sorbonne Université (SU)-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)-Centre National de la Recherche Scientifique (CNRS), Numerical simulation of biological flows ( REO ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -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 ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), and Tixier, Eliott
Subjects: Matching (statistics), Partial differential equation, Differential equation, Applied Mathematics, Principle of maximum entropy, Backward uncertainty quantification, 010103 numerical & computational mathematics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, 010101 applied mathematics, Moment (mathematics), Computational Mathematics, Moment matching, Maximum entropy, Stochastic inverse problem, Calculus, Mathematics::Metric Geometry, Applied mathematics, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: International audience; Many phenomena are modeled by deterministic differential equations , whereas the observation of these phenomena, in particular in life sciences, exhibits an important variability. This paper addresses the following question: how can the model be adapted to reflect the observed variability? Given an adequate model, it is possible to account for this variability by allowing some parameters to adopt a stochastic behavior. Finding the parameters probability density function that explains the observed variability is a difficult stochastic inverse problem, especially when the computational cost of the forward problem is high. In this paper, a non-parametric and non-intrusive procedure based on offline computations of the forward model is proposed. It infers the probability density function of the uncertain parameters from the matching of the statistical moments of observable degrees of freedom (DOFs) of the model. This inverse procedure is improved by incorporating an algorithm that selects a subset of the model DOFs that both reduces its computational cost and increases its robustness. This algorithm uses the pre-computed model outputs to build an approximation of the local sensitivities. The DOFs are selected so that the maximum information on the sensitivities is conserved. The proposed approach is illustrated with elliptic and parabolic PDEs. In the Appendix, an nonlinear ODE is considered and the strategy is compared with two existing ones.
Published: 2018

40. Existence and stability of traveling waves for discrete nonlinear Schroedinger equations over long times

Author: Joackim Bernier, Erwan Faou, Multi-scale numerical geometric schemes ( MINGUS ), 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 ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), Multi-scale numerical geometric schemes (MINGUS), 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)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), 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)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: 01 natural sciences, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, Traveling wave, Discrete nonlinear schroedinger equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Nonlinear Schrödinger equation, Mathematics, Continuum (measurement), Applied Mathematics, Cubic nonlinearity, 010102 general mathematics, Mathematical analysis, Finite difference, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 010101 applied mathematics, Computational Mathematics, Nonlinear system, solitary waves, symbols, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: International audience; We consider the problem of existence and stability of solitary traveling waves for the one dimensional discrete non linear Schroedinger equation (DNLS) with cubic nonlinearity, near the continuous limit.We construct a family of solutions close to the continuous traveling waves and prove their stability over long times. Applying a modulation method, we also show that we can describe the dynamics near these discrete traveling waves over long times.
Published: 2018

41. BOUNDS ON THE GROWTH OF HIGH DISCRETE SOBOLEV NORMS FOR THE CUBIC DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS ON hZ

Author: Bernier , Joackim, Multi-scale numerical geometric schemes ( MINGUS ), 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 ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, and Institut National de Recherche en Informatique et en Automatique ( Inria )
Subjects: [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA]
Abstract: We consider the discrete nonlinear Schrödinger equations on a one dimensional lattice of mesh h, with a cubic focusing or defocusing nonlinearity. We prove a polynomial bound on the growth of the discrete Sobolev norms, uniformly with respect to the stepsize of the grid. This bound is based on a construction of higher modified energies.
Published: 2018

42. Well-suited and adaptive post-processing for the visualization of hp simulation results

Author: Christophe Besse, Vincent Mouysset, Matthieu Maunoury, Pol-André Haas, Sébastien Pernet, ONERA - The French Aerospace Lab ( Toulouse ), ONERA, Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Centre National de la Recherche Scientifique ( CNRS ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -PRES Université de Toulouse-Université Paul Sabatier - Toulouse 3 ( UPS ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse 1 Capitole ( UT1 ), ONERA / DTIS, Université de Toulouse [Toulouse], ONERA-PRES Université de Toulouse, 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), 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: Physics and Astronomy (miscellaneous), Computer science, 010103 numerical & computational mathematics, 01 natural sciences, high-order finite element, Discontinuous Galerkin method, 0101 mathematics, Representation (mathematics), Visualization, Pointwise, Software visualization, Numerical Analysis, Applied Mathematics, Numerical analysis, post-processing, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Finite element method, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, A priori and a posteriori, Algorithm, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], discontinuous Galerkin
Abstract: International audience; While high order methods became very popular as they allow to perform very accurate solutions with low computational time and memory cost, there is a lack of tools to visualize and post-treat the solutions given by these methods. Originally, visualization softwares were developed to post-process results from methods such that finite differences or usual finite elements and therefore process linear primitives. In this paper, we present a methodology to visualize results of high order methods. Our approach is based on the construction of an optimized affine approximation of the high order solution which can therefore be handled by any visualization software. A representation mesh is constructed and the process is guided by an a posteriori estimate which control the error between the numerical solution and its representation pointwise. This point by point control is crucial as under their picture form, data correspond to values mapped on elements where anyone can pick up a pointwise information. A strategy is established to ensure that discontinuities are well represented. These discontinuities come either from the physical problem (material change) or the numerical method (discontinuous Galerkin method) and are pictured accurately. Several numerical examples are presented to demonstrate the potential of the method.
Published: 2018

43. Accurate, Automatic and Compressed Visualization of Radiated Helmholtz Fields from Boundary Element Solutions

Author: Maunoury , Matthieu, Besse , Christophe, Mouysset , Vincent, Pernet , Sébastien, ONERA - The French Aerospace Lab ( Toulouse ), ONERA, Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Centre National de la Recherche Scientifique ( CNRS ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -PRES Université de Toulouse-Université Paul Sabatier - Toulouse 3 ( UPS ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse 1 Capitole ( UT1 ), ONERA / DTIS, Université de Toulouse [Toulouse], ONERA-PRES Université de Toulouse, Institut de Mathématiques de Toulouse UMR5219 (IMT), 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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), 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), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
Subjects: visualisation, integral representation formulas, high order reconstruction, méthode des éléments de frontière, reconstruction d'ordre élevé, directional interpolation, Boundary Equation Methods, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], interpolation directionnelle, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], visualization, formule de représentation intégrale
Abstract: International audience; We propose a methodology to generate an accurate and efficient reconstruction of radiated fields based on high order interpolation. As the solution is obtained with the convolution by a smooth but potentially high frequency oscillatory kernel, our basis functions therefore incorporate plane waves. Directional interpolation is shown to be efficient for smart directions. An adaptive subdivision of the domain is established to limit the oscillations of the kernel in each element. The new basis functions , combining high order polynomials and plane waves, provide much better accuracy than low order ones. Finally, as standard visualization softwares are generally unable to represent such fields, a method to have a well-suited visualization of high order functions is used. Several numerical results confirm the potential of the method.; Nous proposons une méthodologie pour générer la reconstruction précise et efficace des champs rayonnés à partir d'interpolations d'ordre élevé. Comme la solution est obtenue par convolution avec un noyau régulier mais potentiellement oscillant à hautes fréquences, nos fonctions de base incorporent donc des ondes planes. Nous montrons que l'interpolation directionnelle est efficace pour les bonnes directions. Une subdivision adaptative du domaine est créée afin de limiter les oscillations du noyau dans chaque élément. Les nouvelles fonctions de base, combinant des polynômes d'ordre élevé et des ondes planes, donnent une bien meilleure précision que celles de bas ordre. Enfin, les logiciels de visualisation étant généralement incapables de représenter de telles solutions, une méthode permettant une visualisation correcte des fonctions d'ordre élevé est utilisée. Plusieurs exemples numériques confirment le potentiel de cette méthode.
Published: 2018

44. Analyzing the Effect of Local Rounding Error Propagation on the Maximal Attainable Accuracy of the Pipelined Conjugate Gradient Method

Author: Siegfried Cools, Wim Vanroose, Emrullah Fatih Yetkin, Emmanuel Agullo, Luc Giraud, Universiteit Antwerpen [Antwerpen], High-End Parallel Algorithms for Challenging Numerical Simulations ( HiePACS ), Laboratoire Bordelais de Recherche en Informatique ( LaBRI ), Centre National de la Recherche Scientifique ( CNRS ) -École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Université Sciences et Technologies - Bordeaux 1-Université Bordeaux Segalen - Bordeaux 2-Centre National de la Recherche Scientifique ( CNRS ) -École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Université Sciences et Technologies - Bordeaux 1-Université Bordeaux Segalen - Bordeaux 2-Inria Bordeaux - Sud-Ouest, 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 ) -École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Université Sciences et Technologies - Bordeaux 1-Université Bordeaux Segalen - Bordeaux 2, Universiteit Antwerpen = University of Antwerpen [Antwerpen], High-End Parallel Algorithms for Challenging Numerical Simulations (HiePACS), Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Inria Bordeaux - Sud-Ouest, Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), and Yetkin, Emrullah Fatih
Subjects: latency hiding, Global Communication, MathematicsofComputing_NUMERICALANALYSIS, 010103 numerical & computational mathematics, 02 engineering and technology, 01 natural sciences, conjugate gradients, Mathematics::Numerical Analysis, Computer Science::Hardware Architecture, [ INFO.INFO-DC ] Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC], Conjugate gradient method, parallelization, 0202 electrical engineering, electronic engineering, information engineering, Conjugate Gradients, FOS: Mathematics, rounding errors, 65F10, 65F50, 65G50, 65Y05, 65Y20, maximal attainable accuracy, Mathematics - Numerical Analysis, 0101 mathematics, Scaling, Mathematics, Rounding Errors, AMS subject classifications 65F10 65Y20 65Y05 65G50 65F50, 020203 distributed computing, Linear system, Astrophysics::Instrumentation and Methods for Astrophysics, Parallelization, Pipelining, Krylov subspace, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Computer Science::Numerical Analysis, Maximal Attainable Accuracy, pipelin-global communication, Computer Science::Mathematical Software, Round-off error, [INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC], Algorithm, Latency Hiding, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations. However, to achieve this communication hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the influence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare to the traditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG, while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems., Comment: 26 pages, 6 figures, 2 tables, 4 algorithms
Published: 2018

45. Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes and Darcy problems

Author: Joaquín Mura, Franz Chouly, Laura Blank, Alfonso Caiazzo, Alexei Lozinski, Weierstrass Institute for Applied Analysis and Stochastics ( WIAS ), Forschungsverbund Berlin e.V. (FVB), Laboratoire de Mathématiques de Besançon (UMR 6623) ( LMB ), Centre National de la Recherche Scientifique ( CNRS ) -Université de Franche-Comté ( UFC ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin] (WIAS), Forschungsverbund Berlin e.V. (FVB) (FVB), 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), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), and Biomedical Imaging Center, Pontificia Universidad Católica de Chile, Santiago, Chile.
Subjects: Weak boundary conditions, 010103 numerical & computational mathematics, Classification of discontinuities, stabilized finite elements, 01 natural sciences, Physics::Fluid Dynamics, Brinkman problem, Applied mathematics, Boundary value problem, 0101 mathematics, Normal velocity, Mathematics, Brinkman, Nitsche, Numerical Analysis, 65N30, MSC: 65N30, 65N12, 65N15, Applied Mathematics, Finite element approach, 65N12, Penalty-free Nitsche method, 65N15, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Finite element method, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Norm (mathematics), Compressibility, A priori and a posteriori, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (Boiveau and Burman, IMA J. Numer. Anal. 36 (2016) 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy ows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.
Published: 2018

46. Numerical study of the Kadomtsev–Petviashvili equation and dispersive shock waves

Author: Tamara Grava, Christian Klein, Giuseppe Pitton, Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies ( SISSA / ISAS ), School of Mathematics [Bristol], University of Bristol [Bristol], Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Leverhulme Trust. Grant number: RF-2015-442, RISE network IPaDEGAN, Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS), Institut de Mathématiques de Bourgogne [Dijon] (IMB), 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: Shock wave, Breather, General Mathematics, General Physics and Astronomy, Semiclassical physics, FOS: Physical sciences, Pattern Formation and Solitons (nlin.PS), Kadomtsev–Petviashvili equation, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 0103 physical sciences, Modulation (music), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Settore MAT/07 - Fisica Matematica, Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, Line (formation), Physics, Kadomtsev-Petviashvili equation, Kadomtsev Petviashvili equatuon, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Dispersive Shock waves, 010102 general mathematics, General Engineering, Numerical Analysis (math.NA), Dispersive shock waves, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Whitham equations, Nonlinear Sciences - Pattern Formation and Solitons, Lumps, Kadomtsev Petviashvili equation, dispersive shock waves, Classical mechanics, Nonlinear Sciences::Exactly Solvable and Integrable Systems, symbols, Soliton, Exactly Solvable and Integrable Systems (nlin.SI), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Kadomtsev Petviashvili equation, Analysis of PDEs (math.AP)
Abstract: A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrodinger equation in the semiclassical limit., Comment: Improved resolution on several figures. The derivation of the Whitham modulation equations has been revised. Long version w.r.t. the published one
Published: 2018

47. A multiscale medium approximation method for the propagation P-waves in highly heterogeneous geophysical media

Author: Chaumont-Frelet , Théophile, Barucq , Hélène, Calandra , H., Gout , C., Basque Center for Applied Mathematics ( BCAM ), Basque Center for Applied Mathematics, Advanced 3D Numerical Modeling in Geophysics ( Magique 3D ), Laboratoire de Mathématiques et de leurs Applications [Pau] ( LMAP ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Laboratoire de Mathématiques de l'INSA de Rouen Normandie ( LMI ), Institut national des sciences appliquées Rouen Normandie ( INSA Rouen Normandie ), Normandie Université ( NU ) -Normandie Université ( NU ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ), Total E&P, Basque Center for Applied Mathematics (BCAM), Advanced 3D Numerical Modeling in Geophysics (Magique 3D), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), 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), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [ MATH ] Mathematics [math], [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH]Mathematics [math], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: Approximating the solution of Helmholtz problems is a tricky task, especially at high frequency. Indeed, because of the numerical dispersion, it is mandatory to increase the number of discretization points per wavelength for higher frequencies. Using high degree polynomial shape functions on a coarse mesh is an interesting approach because it contributes to reduce such numerical dispersion as compared to linear functions set on a fine mesh. While this solution is very efficient for wave propagation in homogeneous media, it is not obvious how the variations of the medium inside a cell have to be taken into account. Recently the Multiscale Medium Approximation method (MMAm) has been proposed to overcome this difficulty. High-order shape functions are used with a coarse mesh to reduce dispersion and hence, the pollution effect. Then, the key idea is to introduce a multiscale approximation of the medium to take into account fine scale heterogeneities on the coarse mesh. This approach has proven to be efficient on academic test cases and a convergence theory has been developed. In this work, we further validate the MMAm by demonstrating its efficiency on geophysical benchmarks. Highly heterogeneous media, Seismic wave propagation, Time-harmonic modelling, High-order methods, Mulsticale methods, Finite element methods
Published: 2018

48. Asymptotic Numerical Method for the resolution of Navier-Stokes equations

Author: Guevel , Yann, Cadou , Jean-Marc, Girault , Gregory, Institut de Recherche Dupuy de Lôme ( IRDL ), Université de Bretagne Sud ( UBS ) -Université de Brest ( UBO ) -ENSTA Bretagne-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire d'Ingénierie des Matériaux de Bretagne ( LIMATB ), Université de Bretagne Sud ( UBS ) -Université de Brest ( UBO ) -Institut Brestois du Numérique et des Mathématiques ( IBNM ), Université de Brest ( UBO ) -Université de Brest ( UBO ), IRDL - Institut de Recherche Dupuy de Lôme ( IRDL ), Laboratoire Energétique Explosions Structures ( LEES ), Université d'Orléans ( UO ), Institut de Recherche Dupuy de Lôme (IRDL), Université de Bretagne Sud (UBS)-Université de Brest (UBO)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Ingénierie des Matériaux de Bretagne (LIMATB), Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Université de Brest (UBO), Laboratoire Énergétique Explosions Structures [1998-2007] (LEES), Université d'Orléans (UO), and Guevel, Yann
Subjects: [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], [SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS], [ PHYS.MECA.MEFL ] Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], [SPI.FLUID] Engineering Sciences [physics]/Reactive fluid environment, [PHYS.PHYS.PHYS-FLU-DYN] Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [ SPI.FLUID ] Engineering Sciences [physics]/Reactive fluid environment, [PHYS.MECA.MEFL] Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], [ PHYS.PHYS.PHYS-FLU-DYN ] Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], ComputingMilieux_MISCELLANEOUS
Abstract: International audience
Published: 2018

49. Méthode de Nitsche pour le contact et l'auto contact avec frottement: étude mathématique et numérique

Author: MLIKA , Rabii, Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne] ( LaMCoS ), Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut national des sciences appliquées de Lyon, Yves Renard, Franz Chouly, Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne] (LaMCoS), Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and MLIKA, Rabii
Subjects: Nitsche's method, [PHYS.MECA.STRU] Physics [physics]/Mechanics [physics]/Structural mechanics [physics.class-ph], méthode de Nitsche, formulation non biaisée, [PHYS.MECA]Physics [physics]/Mechanics [physics], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], unbiased methods, numerical quadrature for contact, [ PHYS.MECA.STRU ] Physics [physics]/Mechanics [physics]/Mechanics of the structures [physics.class-ph], contact frottant en petites et grandes déformations, intégration numérique, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.STRU]Physics [physics]/Mechanics [physics]/Mechanics of the structures [physics.class-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], frictional contact for small and large strain, [ PHYS.MECA ] Physics [physics]/Mechanics [physics], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA] Physics [physics]/Mechanics [physics], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this thesis, we present and study a new formulation of frictional contact between two elastic bodiesbased on Nitsche's method. This method aims to treat the interface conditions in a weak sense, thanksto a consistent additional term stabilized with the parameter gamma. At first, we introduce the study carriedout in the small strain framwork for an unbiased version of the method. The non-distinction between amaster surface and a slave one will allow the method to be more generic and directly applicable to theself-contact problem. The restrictive framework of small strain allowed us to obtain theoretical results onthe consistency and convergence of the method. Then, we present the extension of the Nitsche methodto the large strain case more relevant for industrial applications and situations of self-contact. ThisNitsche's method is formulated for an hyper-elastic material and declines in the two versions: biased andunbiased. We describe a class of methods through a generalisation parameter theta . Particular variantshave different properties from a numerical point of view, in terms of accuracy and robustness. To provethe accuracy of the method for large deformations, we provide several academic and industrial tests. Wealso study the inuence of numerical quadrature on the accuarcy and convergence of the method. This study covers a comparison of several integration rules proposed in the literature for other integral methods., Dans cette thèse, nous présentons et étudions une nouvelle formulation du problème de contact frottant entre deux corps élastiques se basant sur la méthode de Nitsche. Dans cette méthode les conditions de contact sont imposées faiblement, grâce à un terme de pénalité consistant et stabilisé par un paramètre gamma.Nous introduisons, en premier lieu, l'étude effectuée en petites déformations pour une version non biaisée de la méthode. La non-distinction entre une surface maitre et une surface esclave permettrera à la méthode d’être plus générique et applicable directement au problème d'auto-contact. Le cadre restrictif des petites déformations nous permet d’obtenir des résultats théoriques sur la stabilité et la convergence de la méthode. Ces résultats sont complétés par une validation numérique. Ensuite, nous introduisons l'extension de la méthode de Nitsche au cadre des grandes déformations qui est d’avantage pertinent pour les applications industrielles et les situations d’auto-contact. La méthode de Nitsche est formulée pour un matériau hyperélastique avec frottement de Coulomb et se décline en deux versions : biaisée ou non. La formulation est généralisée à travers un paramètre theta pour couvrir toute une famille de méthodes et chaque variante particulière a des propriétés différentes du point de vue théorique et numérique, en termes de précision et de robustesse.La méthode est testée et validée à travers plusieurs tests académiques et industriels. Nous effectuons aussi une étude de l'influence de l'intégration numérique sur la précision et la convergence de la méthode. Cette étude couvre une comparaison entre plusieurs schémas d’intégration proposés dans la littérature pour d'autres méthodes intégrales.
Published: 2018

50. Recycling of Solution Spaces in Multi-Preconditioned FETI Methods Applied to Structural Dynamics

Author: Leistner, Michael, Gosselet, Pierre, Rixen, Daniel, Institute of Applied Mechanics [Garching], Technische Universität München [München] ( TUM ), Laboratoire de Mécanique et Technologie ( LMT ), École normale supérieure - Cachan ( ENS Cachan ) -Centre National de la Recherche Scientifique ( CNRS ), Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Laboratoire de Mécanique et Technologie (LMT), and École normale supérieure - Cachan (ENS Cachan)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [SPI.MECA.STRU]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph], Ritz vectors, multipreconditioned, adaptive, geneo, [ SPI.MECA.STRU ] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanics of the structures [physics.class-ph], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], recycling, Computer Science::Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Feti, MSC 65F10 * 65N22 * 65N30 * 65N55, Mathematics::Numerical Analysis
Abstract: International audience; This article presents a new method to recycle the solution space of an adaptive multi-preconditioned finite element tearing and interconnecting (AMP-FETI) algorithm in the case where the same operator is solved for multiple right-hand sides like in linear structural dynamics. It accelerates the computation from the second time step on by applying a coarse-space that is generated from Ritz approximations of local eigenproblems, using the solution space of the first time step. These eigenproblems are known to provide very efficient coarse-spaces but must usually be solved a priori at high computational cost. Their Ritz approximations are much smaller and less expensive to solve. Recycling methods based on Ritz approximations of global eigenproblems have been published for classical FETI algorithms but their efficient application to multi-preconditioned variants is not possible. The article also presents the application of a simpler recycling procedure, which reuses plain solution spaces, to AMP-FETI. Numerical results of the application of the presented methods to four test cases are shown. The new Ritz approximation method leads to coarse-spaces which turn out to be as efficient as those obtained from solving the unreduced eigenproblems. It is the most efficient recycling method currently available for multi-preconditioned dual domain decomposition techniques.
Published: 2018

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