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127 results on '"Equations aux dérivées partielles"'

1. Stability for quantitative photoacoustic tomography revisited

Author

Eric Bonnetier, Mourad Choulli, Faouzi Triki, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), 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]), ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Université de Lorraine (UL), 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), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), and Université Grenoble Alpes (UGA)

Subjects
Mathematics subject classification : 35R30, Applied Mathematics, Physics::Optics, Elliptic equations, multiwave imaging, Theoretical Computer Science, Computational Mathematics, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), stability estimates, absorption coefficient, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], diffusion coefficient, Analysis of PDEs (math.AP)
Abstract

This paper is concerned with the stability issue in determining absorption and diffusion coefficients in quantitative photoacoustic imaging. Assuming that the optical wave is generated by point sources in a region where the optical coefficients are known, we derive pointwise H{\"o}lder stability estimate of the inversion. This result shows that the reconstruction of the optical coefficients is stable in the region close to the optical illumination sources and deteriorate exponentially far away. Our stability estimate is therefore in accordance with known experimental observations. Mathematics subject classification : 35R30.

Published
2022
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2. The plasmonic resonances of a bowtie antenna

Author

Faouzi Triki, Charles Dapogny, Eric Bonnetier, Hai Zhang, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Calcul des Variations, Géométrie, Image (CVGI ), 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]), Equations aux Dérivées Partielles (EDP ), Hong Kong University of Science and Technology (HKUST), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Calcul des Variations, Géométrie, Image (CVGI), 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), and Equations aux Dérivées Partielles (EDP)

Subjects
business.industry, Applied Mathematics, 01 natural sciences, 010305 fluids & plasmas, 010101 applied mathematics, Mathematics - Analysis of PDEs, Optics, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Antenna (radio), business, Analysis, Plasmon, Mathematics, Analysis of PDEs (math.AP)
Abstract

International audience; Metallic bowtie-shaped nanostructures are very interesting objects in optics, due to their capability of localizing and enhancing electromagnetic fields in the vicinity of their central neck. In this article, we investigate the electrostatic plasmonic resonances of two-dimensional bowtie-shaped domains by looking at the spectrum of their Poincar\'e variational operator. In particular, we show that the latter only consists of essential spectrum and fills the whole interval $[0,1]$. This behavior is very different from what occurs in the counterpart situation of a bowtie domain with only close-to-touching wings, a case where the essential spectrum of the Poincar\'e variational operator is reduced to an interval strictly contained in $[0,1]$. We provide an explanation for this difference by showing that the spectrum of the Poincaré variational operator of bowtie-shaped domains with close-to-touching wings has eigenvalues which densify and eventually fill the remaining intervals as the distance between the two wings tends to zero.

Published
2018
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3. An inverse problem for an electroseismic model describing the coupling phenomenon of electromagnetic and seismic waves

Author

Qi Xue, Eric Bonnetier, Faouzi Triki, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), 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]), ANR-11-LABX-0025,PERSYVAL-lab,Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011), ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), 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), and ANR-11-LABX-0025-01,PERSYVAL-lab,Systèmes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011)

Subjects
Inverse problems, Boundary (topology), 010103 numerical & computational mathematics, Carleman estimate, 01 natural sciences, Stability (probability), Electromagnetic radiation, AMS: 35R30, Seismic wave, Physics::Geophysics, Theoretical Computer Science, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematical Physics, Mathematics, Coupling, Electro-seismic model, Applied Mathematics, Biot's system, Mechanics, Inverse problem, Hölder stability, Computer Science Applications, 010101 applied mathematics, Wavelength, 35R30, Signal Processing, Coupling coefficient of resonators, Analysis of PDEs (math.AP)
Abstract

International audience; The electroseismic model describes the coupling phenomenon of the electromagnetic waves and seismic waves in fluid immersed porous rock. Electric parameters have better contrast than elastic parameters while seismic waves provide better resolution because of the short wavelength. The combination of theses two different waves is prominent in oil exploration. Under some assumptions on the physical parameters, we derived a Hölder stability estimate to the inverse problem of recovery of the electric parameters and the coupling coefficient from the knowledge of the fields in a small open domain near the boundary. The proof is based on a Carleman estimate of the electroseismic model.

Published
2018
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4. A unified approach to solving some inverse problems for evolution equations by using observability inequalities

Author

Ammari, Kaïs, Choulli, Mourad, Triki, Faouzi, Département de Mathématiques [Monastir], Faculté des Sciences de Monastir (FSM), Université de Monastir - University of Monastir (UM)-Université de Monastir - University of Monastir (UM), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), 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]), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, 35R30, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

International audience; We survey some of our recent results on inverse problems for evolution equations. The goal is to provide an unified approach to solve various type of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities, and a special choice of initial conditions we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their original versions.

Published
2017
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5. Small perturbations in the type of boundary conditions for an elliptic operator

Author

Bonnetier, Eric, Dapogny, Charles, Vogelius, Michael, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Department of Mathematics - Rutgers School of Arts and Sciences, Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Applied Mathematics, General Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $\omega_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $\omega_\varepsilon $ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $\omega_\varepsilon $ differs in the two cases: while it is the harmonic capacity of $\omega_\varepsilon $ in the former case, we introduce a notion of ``Neumann capacity'' to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general $\omega_\varepsilon $, under the sole assumption that it is ``small'' in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset $\omega_\varepsilon $ is a vanishing surfacic ball.

Published
2022
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6. Exponential decay for the damped wave equation in unbounded domains

Author

Romain Joly, Nicolas Burq, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), ANR-13-BS01-0010-03,ANAE,Analyse asymptotique des équations aux dérivées partielles d’évolution, NOSEVOL (ANR 2011 BS01019 01), ANR blanc DYFICOLTI,NOSEVOL (ANR 2011 BS01019 01), ANR blanc DYFICOLTI, Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013), ANR-11-BS01-0019,NOSEVOL,Opérateurs non-autoadjoints, analyse semiclassique et problèmes d'évolution(2011), Joly, Romain, Blanc 2013 - Analyse asymptotique des Equations aux dérivées partielles d'évolution - - ANAÉ2013 - ANR-13-BS01-0010 - Blanc 2013 - VALID, and BLANC - Opérateurs non-autoadjoints, analyse semiclassique et problèmes d'évolution - - NOSEVOL2011 - ANR-11-BS01-0019 - BLANC - VALID

Subjects
0209 industrial biotechnology, Logarithm, General Mathematics, 02 engineering and technology, exponential decay, 01 natural sciences, Domain (mathematical analysis), symbols.namesake, 020901 industrial engineering & automation, Mathematics - Analysis of PDEs, Exponential growth, FOS: Mathematics, Geometric control, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], damped wave equation, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Exponential decay, unbounded domains, Klein–Gordon equation, Mathematics, Semigroup, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Damped wave, 35Q99, 93D15, 93B05, 35B41, Carleman estimates, uniform stabilisation, symbols, variable damping, 35Q99, 93D15, 93B05, 35B41, Analysis of PDEs (math.AP)
Abstract

We study the decay of the semigroup generated by the damped waveequation in an unbounded domain. We first prove under the natural {\em geometriccontrol condition} the exponential decay of the semigroup. Then we prove undera weaker condition the logarithmic decay of the solutions (assuming that theinitial data are smoother). As corollaries, we obtain several extensionsof previous results of stabilisation and control., On étudie la décroissance du semi-groupe des ondes amorties dans un domainenon borné. Notre premier résultat est que, sous une hypothèse naturelle decontrôle géométrique, le semi-groupe décroît exponentiellementvite. On démontre ensuite sous une hypothèse plus faible la décroissancelogarithmique des solutions associées à des données initiales plusrégulières. On obtient en corollaire plusieurs généralisations derésultats de stabilisation et de contrôle.

Published
2014

7. An interface formulation for the Poisson equation in the presence of a semiconducting single-layer material

Author

Jourdana, Clément, Pietra, Paola, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI), and National Research Council of Italy | Consiglio Nazionale delle Ricerche (CNR)

Subjects
domain decomposition, saddle-point problem, finite element method, FOS: Mathematics, single-layer material, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), interface model, Poisson equation, graphene Field-Effect Transistor, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 35J20, 65N30, 65N55, 65Z05
Abstract

In this paper, we consider a semiconducting device with an active zone made of a single-layer material. The associated Poisson equation for the electrostatic potential (to be solved in order to perform self-consistent computations) is characterized by a surface particle density and an out-of-plane dielectric permittivity in the region surrounding the single-layer. To avoid mesh refinements in such a region, we propose an interface problem based on the natural domain decomposition suggested by the physical device. Two different interface continuity conditions are discussed. Then, we write the corresponding variational formulations adapting the so called three-fields formulation for domain decomposition and we approximate them using a proper finite element method. Finally, numerical experiments are performed to illustrate some specific features of this interface approach.

Published
2023

8. On the numerical approximation of Blaschke-Santaló diagrams using Centroidal Voronoi Tessellations

Author

Bogosel, Beniamin, Buttazzo, Giuseppe, Oudet, Edouard, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica [Pisa], University of Pisa - Università di Pisa, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), and Université Grenoble Alpes (UGA)

Subjects
Blaschke-Santaló diagrams, optimal transport, Lloyd's algorithm, FOS: Mathematics, Voronoi tessellations, Monte Carlo methods, Numerical Analysis (math.NA), 49Q10, 49M20, 65C05, 65C20, 52B35, [MATH]Mathematics [math]
Abstract

Identifying Blaschke-Santaló diagrams is an important topic that essentially consists in determining the image $Y=F(X)$ of a map $F:X\to{\mathbb{R}}^d$, where the dimension of the source space $X$ is much larger than the one of the target space. In some cases, that occur for instance in shape optimization problems, $X$ can even be a subset of an infinite-dimensional space. The usual Monte Carlo method, consisting in randomly choosing a number $N$ of points $x_1,\dots,x_N$ in $X$ and plotting them in the target space ${\mathbb{R}}^d$, produces in many cases areas in $Y$ of very high and very low concentration leading to a rather rough numerical identification of the image set. On the contrary, our goal is to choose the points $x_i$ in an appropriate way that produces a uniform distribution in the target space. In this way we may obtain a good representation of the image set $Y$ by a relatively small number $N$ of samples which is very useful when the dimension of the source space $X$ is large (or even infinite) and the evaluation of $F(x_i)$ is costly. Our method consists in a suitable use of {\it Centroidal Voronoi Tessellations} which provides efficient numerical results. Simulations for two and three dimensional examples are shown in the paper., 26 pages, 16 figures

Published
2023
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9. Entropy-regularized Wasserstein distributionally robust shape and topology optimization

Author

Charles Dapogny, Franck Iutzeler, Andrea Meda, Boris Thibert, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and Données, Apprentissage et Optimisation (DAO)

Subjects
Control and Optimization, Control and Systems Engineering, Optimization and Control (math.OC), FOS: Mathematics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Computer Graphics and Computer-Aided Design, Software, Computer Science Applications
Abstract

This brief note aims to introduce the recent paradigm of distributional robustness in the field of shape and topology optimization. Acknowledging that the probability law of uncertain physical data is rarely known beyond a rough approximation constructed from observed samples, we optimize the worst-case value of the expected cost of a design when the probability law of the uncertainty is ``close'' to the estimated one up to a prescribed threshold. The ``proximity'' between probability laws is quantified by the Wasserstein distance, a notion pertaining to optimal transport theory. The combination of the classical entropic regularization technique in this field with recent results from convex duality theory allows to reformulate the distributionally robust optimization problem in a way which is tractable for computations. Two numerical examples are presented, in the different settings of density-based topology optimization and geometric shape optimization. They exemplify the relevance and applicability of the proposed formulation regardless of the selected optimal design framework.

Published
2022

10. Coefficient identification in parabolic equations with final data

Author

Faouzi Triki, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Inverse problems, General Mathematics, 35R30, 35K20, 01 natural sciences, Lipschitz stability estimate, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, MSC: 35R30, 35K20, 35K15, Uniqueness, 0101 mathematics, Eigenvalues and eigenvectors, Mathematics, Parabolic equation, Final data, Applied Mathematics, 010102 general mathematics, Eigenfunction, Lipschitz continuity, Parabolic partial differential equation, 010101 applied mathematics, Elliptic operator, Nonlinear system, Convection–diffusion equation, Analysis of PDEs (math.AP)
Abstract

International audience; In this work we determine the second-order coefficient in a parabolic equation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate for the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of the stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest.

Published
2021
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11. Strong c-concavity and stability in optimal transport

Author

Gallouët, Anatole, Mérigot, Quentin, Thibert, Boris, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), and ANR-16-CE40-0014,MAGA,Monge-Ampère et Géométrie Algorithmique(2016)

Subjects
Optimization and Control (math.OC), FOS: Mathematics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +$\infty$ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.

Published
2022

12. Three-species drift-diffusion models for memristors

Author

Jourdana, Clément, Jüngel, Ansgar, Zamponi, Nicola, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Institute for Analysis and Scientific Computing [Wien], and Vienna University of Technology (TU Wien)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35B25, 35K51, 35Q81, Analysis of PDEs (math.AP)
Abstract

A system of drift-diffusion equations for the electron, hole, and oxygene vacancy densities in a semiconductor, coupled to the Poisson equation for the electric potential, is analyzed in a bounded domain with mixed Dirichlet-Neumann boundary conditions. This system describes the dynamics of charge carriers in a memristor device. Memristors can be seen as nonlinear resistors with memory, mimicking the conductance response of biological synapses. In the fast-relaxation limit, the system reduces to a drift-diffusion system for the oxygene vacancy density and electric potential, which is often used in neuromorphic applications. The following results are proved: the global existence of weak solutions to the full system in any space dimension; the uniform-in-time boundedness of the solutions to the full system and the fast-relaxation limit in two space dimensions; the global existence and weak-strong uniqueness analysis of the reduced system. Numerical experiments in one space dimension illustrate the behavior of the solutions and reproduce hysteresis effects in the current-voltage characteristics.

Published
2022

13. A posteriori error estimates for Darcy-Forchheimer's problem

Author

Semaan, Georges, Sayah, Toni, Triki, Faouzi, Université Saint-Joseph de Beyrouth (USJ), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Numerical Analysis (math.NA), Mathematics - Numerical Analysis, Analysis of PDEs (math.AP)
Abstract

This work deals with the a posteriori error estimates for the Darcy-Forchheimer problem. We introduce the corresponding variational formulation and discretize it by using the finite-element method. A posteriori error estimate with two types of computable error indicators is showed. The first one is linked to the linearization and the second one to the discretization. Finally, numerical computations are performed to show the effectiveness of the error indicators., arXiv admin note: text overlap with arXiv:2104.04864

Published
2022

14. Localization of moving sources: uniqueness, stability, and Bayesian inference

Author

Wang, Sára, Karamehmedović, Mirza, Triki, Faouzi, Danmarks Tekniske Universitet = Technical University of Denmark (DTU), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, 35L05, 35R30, 62F15, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

We consider the subsonic moving point source problem for the scalar wave equation in $\pmb{R}^3$, proving a regularity result for the direct problem, and uniqueness and stability results for the inverse problem. We then present and investigate numerically a Bayesian framework for the inference of the source trajectory and intensity from wave field measurements. The framework employs Gaussian process priors, the pre-conditioned Crank-Nicholson scheme with Markov Chain Monte Carlo sampling, and conditioning on functionals to include prior information on the source trajectory.

Published
2022
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15. A note on maximal Fourier restriction for spheres in all dimensions

Author

Vitturi, Marco, Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), European Project: 637510,H2020,ERC-2014-STG,FAnFArE(2015), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Classical Analysis and ODEs, MSC 42B10, 42B25, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Mathematics::Classical Analysis and ODEs, 42B10, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Fourier restriction, maximal operators, Fourier restriction, maximal operators
Abstract

We prove a maximal Fourier restriction theorem for the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple observation. When $d=3$ the range corresponds exactly to the full Stein-Tomas one, but is otherwise a proper subset when $d>3$. We also present an application regarding the Lebesgue points of functions in $\mathcal{F}(L^p)$ when $p$ is sufficiently close to 1., 5 pages, no figures

Published
2022

16. Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations

Author

Kian, Yavar, Soccorsi, Éric, Triki, Faouzi, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017), Equations aux Dérivées Partielles (EDP), and Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

In this paper we study the inverse problem of identifying a source or an initial state in a time-fractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both inversions. These results show that the ill-posedness increases exponentially when the fractional derivative order tends to zero, while it exponentially decreases when the regularity of the source or the initial state becomes larger. The stability estimate concerning the problem of recovering the initial state can be considered as a weak observability inequality in control theory. The analysis is mainly based on Laplace inversion techniques and a precise quantification of the unique continuation property for the resolvent of the time-fractional diffusion operator as a function of the frequency in the complex plane. We also determine a global time regularity for the time-fractional diffusion equation which is of interest itself.

Published
2021

17. Recovering simultaneously a potential and a point source from Cauchy data

Author

Bao, Gang, Liu, Yuantong, Triki, Faouzi, Zhejiang University, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, point sources, stability estimate, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Schrödinger equation, MSC: 35R30, 35C20, inverse potential, Dirichlet to Neumann map, Analysis of PDEs (math.AP)
Abstract

International audience; This paper is devoted to the inverse problem of recovering simultaneously a potential and a point source in a Schrödinger equation from the associated nonlinear Dirichlet to Neumann map. The uniqueness of the inversion is proved and logarithmic stability estimates are derived. It is well known that the inverse problem of determining only the potential while knowing the source, is ill-posed. In contrast the problem of identifying a point source when the potential is given is well posed. The obtained results show that the nonlinear Dirichlet to Neumann map contains enough information to determine simultaneously the potential and the point source. However recovering a point source imbedded in an unknown background medium becomes an ill-posed inversion.

Published
2021

18. Inverse moving point source problem for the wave equation

Author

Hanin Al Jebawy, Abdellatif Elbadia, Faouzi Triki, Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, Moving source, Applied Mathematics, Signal Processing, FOS: Mathematics, Inverse source problem, Wave equation, [MATH]Mathematics [math], Mathematical Physics, Analysis of PDEs (math.AP), Computer Science Applications, Theoretical Computer Science
Abstract

In this paper, we consider the problem of identifying a single moving point source for a three-dimensional wave equation from boundary measurements. Precisely, we show that the knowledge of the field generated by the source at six different points of the boundary over a finite time interval is sufficient to determine uniquely its trajectory. We also derive a Lipschitz stability estimate for the inversion.

Published
2022
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19. A mollifier approach to regularize a Cauchy problem for the inhomogeneous Helmholtz equation

Author

Marechal, Pierre, Lee, Walter Simo Tao, Triki, Faouzi, 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), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, Applied Mathematics, FOS: Mathematics, Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Numerical Analysis (math.NA), Mathematics - Numerical Analysis, 47A52, 35J05, Analysis of PDEs (math.AP)
Abstract

The Cauchy problem for the inhomogeneous Helmholtz equation with non-uniform refraction index is considered. The ill-posedness of this problem is tackled by means of the variational form of mollification. This approach is proved to be consistent, and the proposed numerical simulations are quite promising.

Published
2021

20. Longest minimal length partitions

Author

Bogosel, Beniamin, Oudet, Edouard, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)

Subjects
Optimization and Control (math.OC), Applied Mathematics, TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY, FOS: Mathematics, Mathematics::Metric Geometry, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Computer Science::Computational Geometry, Mathematics - Optimization and Control, MathematicsofComputing_DISCRETEMATHEMATICS
Abstract

International audience; This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimizations steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points.

Published
2021
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21. Extending nonstandard finite difference schemes rules to systems of nonlinear ODEs with constant coefficients

Author

Songolo, Marc, Bidégaray-Fesquet, Brigitte, Université de Lubumbashi (UNILU), Institut d'Électronique et des Technologies du numéRique (IETR), Université de Nantes (UN)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Université de Lubumbashi, Nantes Université (NU)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-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)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), and Université de Nantes (UN)-Université de Rennes 1 (UR1)

Subjects
Ordinary Differential Equations, Nonstandard finite difference schemes, MathematicsofComputing_NUMERICALANALYSIS, FOS: Mathematics, Exact finite difference schemes, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Matrix exponential, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) scheme to adapt them to systems of ODEs. This leads to exact schemes in the linear case, and also improve the accuracy in the nonlinear case. In the Hamiltonian nonlinear case, it consists in adding correction terms to schemes derived by Mickens.

Published
2021
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22. Strang splitting schemes for N-level Bloch models

Author

Songolo, Marc, Bidégaray-Fesquet, Brigitte, Université de Lubumbashi, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and Université de Lubumbashi (UNILU)

Subjects
Nonstandard finite difference schemes, Bloch equation, FOS: Mathematics, MathematicsofComputing_NUMERICALANALYSIS, FOS: Physical sciences, Exact finite difference schemes, Splitting method, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Computational Physics (physics.comp-ph), Exponential of a matrix, Physics - Computational Physics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We extend to the N-level Bloch model the splitting scheme which use exact numerical solutions of sub-equations. These exact solutions involve matrix exponentials which we want to avoid to calculate at each time step. We use Newton interpolation to reduce the computational cost. The resulting scheme is nonstandard and preserves all qualitative properties of the Bloch equations. We show numerical simulations to compare this approach with a few other schemes.

Published
2021

23. Diffusion-redistanciation schemes for 2D and 3D constrained Willmore flow: application to the equilibrium shapes of vesicles

Author

Emmanuel Maitre, Mourad E. H. Ismail, Arnaud Sengers, Thibaut Metivet, ModELisation de l'apparence des phénomènes Non-linéaires (ELAN), 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 National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), DYnamique des Fluides COmplexes et Morphogénèse [Grenoble] (DYFCOM-LIPhy ), Laboratoire Interdisciplinaire de Physique [Saint Martin d’Hères] (LIPhy ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Most of the computations presented in this paper were performed using the Froggy platform of the GRICAD infrastructure (https://gricad.univ-grenoble-alpes.fr), which is supported by the Rhône-Alpes region (GRANT CPER07_13 CIRA) and the Equip@Meso project (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche, and the Atlas cluster from the Research Institute in Advanced Mathematics (IRMA - UMR7501)., ANR-10-EQPX-0029,EQUIP@MESO,Equipement d'excellence de calcul intensif de Mesocentres coordonnés - Tremplin vers le calcul petaflopique et l'exascale(2010), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), DYnamique des Fluides COmplexes et Morphogénèse [Grenoble] [2020-….] (DYFCOM-LIPhy [2020-….]), Laboratoire Interdisciplinaire de Physique [Saint Martin d’Hères] [2020-….] (LIPhy [2020-….]), and Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)

Subjects
Volume and area preserving, Vol- ume and area preserving, Diffusion generated motion, Physics and Astronomy (miscellaneous), Computer science, Interface (Java), and phrases: Diffusion generated motion, Signed distance function, Clifford torus, 010103 numerical & computational mathematics, Willmore flow, 01 natural sciences, Level set, Dimension (vector space), FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Diffusion (business), [MATH]Mathematics [math], High-order geometrical flow, Numerical Analysis, Level set (data structures), Applied Mathematics, Numerical Analysis (math.NA), Thresholding, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Flow (mathematics), Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper we present a novel algorithm for simulating geometrical flows, and in particular the Willmore flow, with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow in both two and three dimensions. These algorithms rely on alternating diffusions of the signed distance function to the interface and a redistanciation step, and with careful choice of the applied diffusions, end up moving the zero level-set of the distance function by some geometrical quantity without resorting to any explicit transport equation. The constraints are enforced between the diffusion and redistanciation steps via a simple rescaling method. The energy globally decreases at the end of each global step. The algorithms feature the computational efficiency of thresholding methods without requiring any adaptive remeshing thanks to the use of a signed distance function to describe the interface. This opens their application to dynamic fluid-structure simulations for large and realistic cases. The methodology is validated by computing the equilibrium shapes of two- and three-dimensional vesicles, as well as the Clifford torus.

Published
2020
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24. Numerical computation of the cut locus via a variational approximation of the distance function

Author

François Générau, Edouard Oudet, Bozhidar Velichkov, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Dipartimento di Matematica [Pisa], University of Pisa - Università di Pisa, and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)

Subjects
cut locus, Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), relaxation, manifold, Calculus of variation, FOS: Mathematics, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, 49J45, 49Q10, 35R35, 49M05, 35J25, [MATH]Mathematics [math], Analysis of PDEs (math.AP)
Abstract

We propose a new method for the numerical computation of the cut locus of a compact submanifold of ℝ3 without boundary. This method is based on a convex variational problem with conic constraints, with proven convergence. We illustrate the versatility of our approach by the approximation of Voronoi cells on embedded surfaces of ℝ3.

Published
2020
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25. Inverse conductivity problem with internal data

Author

Faouzi Triki null, Tao Yin, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), California Institute of Technology (CALTECH), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Computational Mathematics, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35R30, 65N21, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.

Published
2020
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26. Stability for the multifrequency inverse medium problem

Author

Faouzi Triki, Gang Bao, Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Zhejiang University, Equations aux Dérivées Partielles (EDP), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Frequency band, FOS: Physical sciences, Inverse, Interval (mathematics), 01 natural sciences, Stability (probability), Mathematics - Analysis of PDEs, FOS: Mathematics, Stability estimates, Helmholtz equation, 0101 mathematics, [MATH]Mathematics [math], Mathematical Physics, Mathematics, 35, 74 35R30, 74B05, Secondary 47A52, 65J20, Scattering resonances, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Trace formula, Observable, Mathematical Physics (math-ph), Function (mathematics), Lipschitz continuity, 010101 applied mathematics, Bounded function, Inverse medium problem, MSC: 35L05, 35R30, 74B05, 47A52, 65J20, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; The solution of a multi-frequency 1d inverse medium problem consists of recovering the refractive index of a medium from measurements of the scattered waves for multiple frequencies. In this paper, rigorous stability estimates are derived when the frequency takes value in a bounded interval. It is showed that the ill-posedness of the inverse medium problem decreases as the width of the frequency interval becomes larger. More precisely, under certain regularity assumptions on the refractive index, the estimates indicate that the power in Hölder stability is an increasing function of the largest value in the frequency band. Finally, a Lipschitz stability estimate is obtained for the observable part of the medium function defined through a truncated trace formula.

Published
2020
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27. Identification of a boundary influx condition in a one-phase Stefan problem

Author

Faouzi Triki, Saloua Mani Aouadi, Chifaa Ghanmi, Université de Tunis El Manar (UTM), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
35, 65, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Phase (waves), Front (oceanography), Stefan problem, Boundary (topology), Inverse, Inverse problem, 16. Peace & justice, 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, Position (vector), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, 0101 mathematics, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; We consider a one-dimensional one-phase inverse Stefan problem for the heat equation. It consists in recovering a boundary influx condition from the knowledge of the position of the moving front, and the initial state. We derived a logarithmic stability estimate that shows that the inversion is ill-posed. The proof is based on integral equations and unique continuation for holomorphic functions. We also proposed a direct algorithm with a regularization term to solve the nonlinear inverse problem. Several numerical tests using noisy data are provided with relative errors analysis.

Published
2020
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28. A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

Author

Laurent Thomann, Tadahiro Oh, University of Edinburgh, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013)

Subjects
Statistics and Probability, 02 engineering and technology, 01 natural sciences, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, Wick ordering, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Gibbs measure, nonlinear Schrödinger equation, Nonlinear Schrödinger equation, Mathematical physics, Mathematics, Hermite polynomials, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Riemannian manifold, 021001 nanoscience & nanotechnology, 35Q55, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear system, Laguerre polynomial, Hermite polynomial, Modeling and Simulation, Bounded function, symbols, Laguerre polynomials, white noise functional, 0210 nano-technology, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract

We consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure., Comment: 42 pages

Published
2018
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29. Recovering the initial condition in the one-phase Stefan problem

Author

Saloua Mani Aouadi, Chifaa Ghanmi, Faouzi Triki, University of Tunis El Manar, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
0209 industrial biotechnology, Applied Mathematics, Stefan problem, Boundary (topology), 02 engineering and technology, 01 natural sciences, Integral equation, 010101 applied mathematics, Mathematics - Analysis of PDEs, 020901 industrial engineering & automation, Position (vector), FOS: Mathematics, Free boundary problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, Applied mathematics, Initial value problem, Heat equation, Uniqueness, 0101 mathematics, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data., 22 pages, 6 figures, 6 tables. arXiv admin note: text overlap with arXiv:2002.09185

Published
2022
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30. Recovering point sources for the inhomogeneous Helmholtz equation *

Author

Faouzi Triki, Gang Bao, Yuantong Liu, Zhejiang University, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Helmholtz equation, Point source, Applied Mathematics, Mathematical analysis, Boundary (topology), 010103 numerical & computational mathematics, 01 natural sciences, Computer Science Applications, Theoretical Computer Science, 010101 applied mathematics, Mathematics - Analysis of PDEs, Signal Processing, Inverse scattering problem, FOS: Mathematics, Radiative transfer, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, Point (geometry), Uniqueness, 0101 mathematics, Mathematical Physics, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; The paper is concerned with an inverse point source problem for the Helmholtz equation. It consists of recovering the locations and amplitudes of a finite number of radiative point sources inside a given inhomogeneous medium from the knowledge of a single boundary measurement. The main result of the paper is a new Hölder type stability estimate for the inversion under the assumption that the point sources are well separated. The proof of the stability is based on a combination of Carleman estimates and a technique for proving uniqueness of the Cauchy problem.

Published
2021
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31. Construction of a Blow-Up Solution for the Complex Ginzburg–Landau Equation in a Critical Case

Author

Hatem Zaag, Nejla Nouaili, 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), 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 ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013)

Subjects
Complex Ginzburg-Landau equation, Mechanical Engineering, 010102 general mathematics, Mathematical analysis, Ginzburg landau equation, Blow-up profile, 01 natural sciences, Stability (probability), Interpretation (model theory), 010101 applied mathematics, Reduction (complexity), MSC: 35K57, 35K40, 35B44, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Dimension (vector space), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Point (geometry), 0101 mathematics, Finite time, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

We construct a solution for the Complex Ginzburg-Landau equation in some critical case, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution., arXiv admin note: text overlap with arXiv:1506.08306

Published
2017
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32. A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

Author

Pierre Raphaël, Oana Pocovnicu, Enno Lenzmann, Patrick Gérard, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Mathematisches Institut, University of Basel (Unibas), Heriot-Watt University [Edinburgh] (HWU), 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), Swiss National Foundation, NSF, ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013), European Project: 646650,H2020,ERC-2014-CoG,SINGWAVES(2015), Apollo - University of Cambridge Repository, Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), and 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)

Subjects
Multi-soliton, General Physics and Astronomy, Interval (mathematics), 01 natural sciences, Mathematics - Analysis of PDEs, cubic Szegö equation, Quantum mechanics, 0103 physical sciences, Saturation (graph theory), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, modulation theory, Real line, Mathematical Physics, Mathematics, MSC 35B40, 35L05, 35Q41, 35Q51, 37K40, half-wave equation, Partial differential equation, Turbulence, Applied Mathematics, 010102 general mathematics, 4901 Applied Mathematics, Norm (mathematics), growth of Sobolev norms, 49 Mathematical Sciences, wave turbulence, 010307 mathematical physics, Geometry and Topology, Transient (oscillation), Soliton, Analysis, Analysis of PDEs (math.AP)
Abstract

We consider the focusing cubic half-wave equation on the real line $$\begin{aligned} i \partial _t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(\xi )=|\xi |\hat{u}(\xi ), \ \ (t,x)\in {\mathbb {R}}_+\times {\mathbb {R}}. \end{aligned}$$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small $$L^2$$ -norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its $$H^1$$ -norm on a finite time interval, followed by (ii) a saturation regime in which the $$H^1$$ -norm remains stationary large forever in time.

Published
2020
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33. Geometric optics for surface waves in nonlinear elasticity

Author

Coulombel, Jean-Francois, Williams, Mark, Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), University of North Carolina [Chapel Hill] (UNC), University of North Carolina System (UNC), and ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013)

Subjects
Mathematics - Analysis of PDEs, Applied Mathematics, General Mathematics, 010102 general mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 01 natural sciences, Analysis of PDEs (math.AP)
Abstract

This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an \emph{approximate} Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which we refer to as "the amplitude equation", is an integrodifferential equation of nonlocal Burgers type. We begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^\eps$ to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength $\eps$, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to $u^\eps$ on a time interval independent of $\eps$. The paper focuses mainly on the case of Rayleigh waves that are \emph{pulses}, which have profiles with continuous Fourier spectrum, but our method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.

Published
2020

34. A Global stability estimate for the photo-acoustic inverse problem in layered media

Author

Faouzi Triki, Kui Ren, University of Texas at Austin [Austin], 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]), and ANR-11-LABX-0025,PERSYVAL-lab,Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011)

Subjects
Physics, Diffusion equation, business.industry, Applied Mathematics, 010102 general mathematics, Photoacoustic imaging in biomedicine, Inversion (meteorology), Acoustic wave, Inverse problem, Wave equation, 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, Optics, Optimization and Control (math.OC), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Photo acoustic, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, business, Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

International audience; This paper is concerned with the stability issue in determining absorption and diffusion coefficients in photoacoustic imaging. Assuming that the medium is layered and the acoustic wave speed is known we derive global Hölder stability estimates of the photo-acoustic inversion. These results show that the reconstruction is stable in the region close to the optical illumination source, and deteriorate exponentially far away. Several experimental pointed out that the resolution depth of the photo-acoustic modality is about tens of millimeters. Our stability estimates confirm these observations and give a rigorous quantification of this depth resolution.

Published
2019
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35. Reconstruction of domains with algebraic boundaries from generalized polarization tensors

Author

Mihai Putinar, Andries Steenkamp, Habib Ammari, Faouzi Triki, Seminar for Applied Mathematics (ETH-SAM), University of California [Santa Barbara] (UC Santa Barbara), University of California (UC), 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]), ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017), ANR-11-LABX-0025,PERSYVAL-lab,Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011), University of California [Santa Barbara] (UCSB), and University of California

Subjects
35R30, 35C20, General Mathematics, 01 natural sciences, Planar, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Artificial Intelligence & Image Processing, 0101 mathematics, Algebraic number, Finite set, generalized polarization tensors, math.AP, Physics, 35C20, algebraic domains, inverse problems, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Inverse problem, Polarization (waves), shape classification, 010101 applied mathematics, Generalized polarization tensors, Algebraic domains, Inverse problems, Shape classification, 35R30, Analysis of PDEs (math.AP)
Abstract

This paper aims at showing the stability of the recovery of a smooth planar domain with a real algebraic boundary from a finite number of its generalized polarization tensors. It is a follow-up of the work [H. Ammari, M. Putinar, A. Steenkamp, and F. Triki, Math. Ann., 375 (2019), pp. 1337-1354], where it is proved that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one-dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The recovery procedure is implemented without any assumption on the regularity of the domain to be reconstructed, and its performance and limitations are illustrated. © 2019 Society for Industrial and Applied Mathematics. ISSN:1936-4954

Published
2019
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36. Long time dynamics for damped Klein-Gordon equations

Author

Geneviève Raugel, Wilhelm Schlag, Nicolas Burq, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, University of Chicago, ANR: ANR-13-BS01-0010-03 (ANA\'E).NSF: DMS-1160817, and ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013)

Subjects
Dynamical systems theory, General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Invariant manifold, Mathematics::Analysis of PDEs, Banach space, Dynamical Systems (math.DS), Type (model theory), 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Dynamical Systems, 0101 mathematics, Klein–Gordon equation, Mathematical physics, Physics, 010102 general mathematics, Nonlinear system, Bounded function, symbols, MSC: 35L05, 74J20, 35H15, 37L30, 37L50, 010307 mathematical physics, Analysis of PDEs (math.AP), Sign (mathematics)
Abstract

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, $1\textless{}p\textless{}(d+2)/(d-2)$ as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).

Published
2017
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37. Trend to the equilibrium for the Fokker-Planck system with a strong external magnetic field

Author

Karaki, Zeinab, Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
FOS: Physical sciences, Mathematical Physics (math-ph), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Functional Analysis (math.FA), Mathematics - Functional Analysis, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Spectral Theory (math.SP), Mathematical Physics, Analysis of PDEs (math.AP), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We consider the Fokker-Planck equation with a strong external magnetic field. Global-in-time solutions are built near the Maxwellian, the global equilibrium state for the system. Moreover, we prove the convergence to equilibrium at exponential rate. The results are first obtained on spaces with an exponential weight. Then they are extended to larger functional spaces, like the Lebesgue space and the Sobolev space with polynomial weight, by the method of factorization and enlargement of the functional space developed in [Gualdani, Mischler, Mouhot, 2017].

Published
2019

38. HIGH ORDER PARACONTROLLED CALCULUS

Author

Frédéric Bernicot, I. Bailleul, 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), Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), European Project: 637510,H2020,ERC-2014-STG,FAnFArE(2015), 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 Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0020,SINGULAR,Equations aux dérivées partielles singulières(2016), 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 ), Laboratoire de Mathématiques Jean Leray ( LMJL ), Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ), and European Project : 637510,H2020,ERC-2014-STG,FAnFArE ( 2015 )

Subjects
Statistics and Probability, [ MATH.MATH-CA ] Mathematics [math]/Classical Analysis and ODEs [math.CA], [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], 01 natural sciences, Theoretical Computer Science, 010104 statistics & probability, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Classical Analysis and ODEs (math.CA), FOS: Mathematics, Calculus, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, 60H15, 35R60, 35R01, 0101 mathematics, Algebraic number, High order, Anderson impurity model, Mathematical Physics, Mathematics, Paracontrolled calculus, Algebra and Number Theory, Partial differential equation, generalised Parabolic Anderson Model equation, Probability (math.PR), 010102 general mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Computational Mathematics, Mathematics - Classical Analysis and ODEs, stochastic singular PDEs, Geometry and Topology, Generalised KPZ equation, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability, Analysis, Analysis of PDEs (math.AP)
Abstract

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators. We illustrate the efficiency of this elementary approach on the examples of the 3-dimensional generalised parabolic Anderson model equation and the generalised KPZ equation driven by a (1+1)-dimensional space/time white noise., Comment: v5, 67 pages. Refined correctors introduced, and an improved continuity result for the merging operator proved. Final version

Published
2019
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39. Dirac brushes (or, the fractional Fourier transform of Dirac combs)

Author

Viola, Joe, Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and Région Pays de la Loire through the project EONE (Évolution des Opérateurs Non-Elliptiques)

Subjects
42A16, 81S30, 11F27, 11F30, poisson summation, Mathematics - Number Theory, Mathematics - Classical Analysis and ODEs, MSC 42A16, 81S30, 11F27, 11F30, Classical Analysis and ODEs (math.CA), FOS: Mathematics, theta function, Number Theory (math.NT), metaplectic representation, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], fractional Fourier transform, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract

In analogy with the Poisson summation formula, we identify when the fractional Fourier transform, applied to a Dirac comb in dimension one, gives a discretely supported measure. We describe the resulting series of complex multiples of delta functions, and through either the metaplectic representation of $SL(2,\mathbb{Z})$ or the Bargmann transform, we see that the the identification of these measures is equivalent to the functional equation for the Jacobi theta functions. In tracing the values of the antiderivative in certain small-angle limits, we observe Euler spirals, and on a smaller scale, these spirals are made up of Gauss sums which give the coefficient in the aforementioned functional equation., 42 pages, 7 figures

Published
2018

40. On the Sendov conjecture for polynomials with simple zeros

Author

Dalmasso, Robert, Equations aux Dérivées Partielles (EDP ), Laboratoire Jean Kuntzmann (LJK ), and 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])

Subjects
Sendov's conjecture, Mathematics - Analysis of PDEs, polynomial, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 2010 MSC: 30C10, 30C15, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Walsh's Coincidence Theorem, Analysis of PDEs (math.AP), simple zeros
Abstract

The Sendov conjecture asserts that if all the zeros of a polynomial p lie in the closed unit disk then there must be a zero of p ' within unit distance of each zero. In this paper we give a partial result when p has simple zeros.

Published
2018

41. A Pseudo-local Property of Gravity Water Waves System

Author

Quang-Huy Nguyen, Laboratoire de Mathématiques d'Orsay (LMO), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), This work was partially supported by the labex LMH through the grant no ANR-11-LABX-0056-LMH in the 'Programme des Investissements d'Avenir', and ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013)

Subjects
gravity water waves, spatial decay, pseudo-local, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Local property, 01 natural sciences, 010101 applied mathematics, Sobolev space, Computational Mathematics, Mathematics - Analysis of PDEs, Bounded function, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Contraction (operator theory), Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

By proving a weighted contraction estimate in uniformly local Sobolev spaces for the flow of gravity water waves, we show that this nonlocal system is in fact pseudo-local in the following sense: locally in time, the dynamic far away from a given bounded region has a small effect on that region (again, in a sense that we will make precise in the article). Our estimate on the flow also implies a new spatial decay property of the waves. To prove this result, we establish a paradifferential calculus theory in uniformly local Sobolev spaces with weights., We improved the main results and the paradifferential calculus for a general class of weights. Typos fixed, explanations and details added

Published
2016
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42. Sparse domination via the helicoidal method

Author

Cristina Benea, Camil Muscalu, Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Cornell University [New York], European Project: 637510,H2020,ERC-2014-STG,FAnFArE(2015), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, General Mathematics, 010102 general mathematics, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], 01 natural sciences, Multiplier (Fourier analysis), symbols.namesake, Operator (computer programming), Fourier transform, Mathematics - Classical Analysis and ODEs, 0103 physical sciences, symbols, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 010307 mathematical physics, 0101 mathematics, Symbol (formal), Subspace topology, Mathematics
Abstract

Using exclusively the localized estimates upon which the helicoidal method was built, we show how sparse estimates can also be obtained. This approach yields a sparse domination for multiple vector-valued extensions of operators as well. We illustrate these ideas for an $n$-linear Fourier multiplier whose symbol is singular along a $k$-dimensional subspace of $\Gamma=\lbrace \xi_1+\ldots+\xi_{n+1}=0 \rbrace$, where $k, Comment: 60 pages

Published
2018

43. Stability estimates for the fault inverse problem

Author

Darko Volkov, Faouzi Triki, 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]), Department of Computer Science (WPI CS), Worcester Polytechnic Institute, and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Plane (geometry), Applied Mathematics, Mathematical analysis, Boundary (topology), 010103 numerical & computational mathematics, Slip (materials science), Inverse problem, Half-space, Lipschitz continuity, 01 natural sciences, Computer Science Applications, Theoretical Computer Science, 010101 applied mathematics, Mathematics - Analysis of PDEs, Signal Processing, Displacement field, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Mathematical Physics, Mathematics, Analysis of PDEs (math.AP)
Abstract

International audience; We study in this paper stability estimates for the fault inverse problem. In this problem, faults are assumed to be planar open surfaces in a half space elastic medium with known Lam\'e coefficients. A traction free condition is imposed on the boundary of the half space. Displacement fields present jumps across faults, called slips, while traction derivatives are continuous. It was proved in \cite{volkov2017reconstruction} that if the displacement field is known on an open set on the boundary of the half space, then the fault and the slip are uniquely determined. In this present paper, we study the stability of this uniqueness result with regard to the coefficients of the equation of the plane containing the fault. If the slip field is known we state and prove a Lipschitz stability result. In the more interesting case where the slip field is unknown, we state and prove another Lipschitz stability result under the additional assumption, which is still physically relevant, that the slip field is one directional.

Published
2018
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44. Weakly nonlinear surface waves in magnetohydrodynamics

Author

Pierre, Olivier, Coulombel, Jean-François, Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), 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), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-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
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

International audience; This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the incompressible magnetohydrodynamics equations. Current vortex sheets are piecewise smooth solutions to the incompressible magnetohydrodynamics equations that satisfy suitable jump conditions for the velocity and magnetic field on the (free) discontinuity surface. In this work, we complete an earlier work by Ali and Hunter and construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to `surface waves' on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime that we consider here, their leading amplitude is governed by a nonlocal Hamilton-Jacobitype equation known as the `HIZ equation' (standing for Hamilton-Il'insky-Zabolotskaya) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. Based on a suitable duality formula, we exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. Theverification of these solvability conditions is based on a combination of mere algebra and arguments of combinatorial analysis. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the free boundary problem. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves does not arise in the same way for the current vortex sheet problem.

Published
2018
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45. Dimension reduction and optimality of the uniform state in a Phase-Field-Crystal model involving a higher order functional

Author

Hamdi Zorgati, Radu Ignat, 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 III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, Laboratoire Equations aux Dérivées Partielles (LEDP), Université de Tunis El Manar (UTM), Faculté des Sciences Mathématiques, Physiques et Naturelles de Tunis (FST), Université de Tunis - El Manar II, 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 Toulouse (UT), and University of Tunis El Manar

Subjects
01 natural sciences, Reduced model, 010305 fluids & plasmas, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Energy functional, fixed mass constraint, Applied Mathematics, Dimensionality reduction, General Engineering, Order (ring theory), State (functional analysis), Ohta-Kawasaki, 010101 applied mathematics, Constraint (information theory), Γ-convergence, Modeling and Simulation, global minimality, Phase-Field-Crystal, Phase field crystal model, Analysis of PDEs (math.AP)
Abstract

International audience; We study a Phase-Field-Crystal model described by a free energy functional involving second order derivatives of the order parameter in a periodic setting and under a fixed mass constraint. We prove a Γ-convergence result in an asymptotic thin-film regime leading to a reduced 2-dimensional model. For the reduced model, we prove necessary and sufficient conditions for the global minimality of the uniform state. We also prove similar results for the Ohta-Kawasaki model.

Published
2018
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46. Stabilisation of wave equations on the torus with rough dampings

Author

Patrick Gérard, Nicolas Burq, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013)

Subjects
Pure mathematics, Ocean Engineering, 93D15, 01 natural sciences, 35L05, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, Geometric control, controle, ondes, tores, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], wave equations, 35S05, 0101 mathematics, Physics, 010102 general mathematics, 58J40, Torus, second microlocalization, Wave equation, stabilization, geometric control, 34L20, 35Q93, 010307 mathematical physics, Stabilisation, control, Analysis of PDEs (math.AP)
Abstract

For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x) = \sum_{j=1}^N a_j 1_{x\in R_j}$ ($R_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (for any compact (smooth) manifold and any $L^\infty$ damping), and we prove that it is sufficient in our particular case on $\mathbb{T}^2$ (and for our particular dampings).; Pour l'\'equation des ondes amortie sur une vari\'et\'e compacte, dans le cas d'un amortissement {\em continu}, la condition de contr\^ole g\'eom\'etrique est n\'ecessaire et suffisante pour la stabilisation uniforme. Dans cet article, sur le tore $\mathbb{T}^2$ et dans le cas o\`u $a(x) = \sum_{j=1}^N a_j 1_{x\in R_j}$ ($R_j$ sont des polygones), nous exhibons une condition g\'eom\'etrique n\'ecessaire et suffisante tr\`es simple. Nous proposons aussi une g\'en\'eralisation naturelle de la condition de contr\^ole g\'eom\'etrique, pour un amortissement seulement $L^\infty$. Cette g\'en\'eralisation est toujours n\'ecessaire pour la stabilisation uniforme (sur toute vari\'et\'e compacte r\'eguli\`ere), et nous d\'emontrons dans cet article qu'elle est suffisante dans notre cas particulier du tore $\mathbb{T}^2$ (et pour nos fonctions d'amortissement particuli\`eres).

Published
2017

47. On the Kramers-Fokker-Planck equation with decreasing potentials in dimension one

Author

Novak, Radek, Wang, Xue Ping, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Equations aux dérivées partielles, and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST)

Subjects
Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, return to local equilibrium, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Kramers-Fokker-Planck equation, pseudo-spectral estimates, threshold spectral analysis, Spectral Theory (math.SP), Analysis of PDEs (math.AP), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

For quickly decreasing potentials with one position variable, the first threshold zero is always a resonance of the Kramers-Fokker-Planck operator. In this article we study low-energy spectral properties of the operator and calculate large time asymptotics of solutions in terms of the Maxwellian.

Published
2017

48. Small perturbations of an interface for elastostatic problems

Author

Lagha, Jihene, Triki, Faouzi, Zribi, Habib, Faculté des Sciences Mathématiques, Physiques et Naturelles de Tunis (FST), Université de Tunis El Manar (UTM), 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]), Department of Mathematics, College of Sciences, Saudi Arabia, University of Hafr Al Batin, Saudi Arabia, ANR-11-LABX-0025,PERSYVAL-lab,Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011), and University of Hafr Al Batin [Saudi Arabia] (UHB)

Subjects
Mathematics - Analysis of PDEs, elastic moment tensors, boundary integral method, FOS: Mathematics, Small perturbations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35B30, 35C20, 31B10, interface problem, asymptotic expansions, Lamé system, MSC2000: 35B30, 35C20, 31B10, Analysis of PDEs (math.AP)
Abstract

We consider solutions to the Lam\'e system in two dimensions. By using systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation), we establish a rigorous asymptotic expansion for the perturbations of the displacement field caused by small perturbations of the shape of an elastic inclusion with C2-boundary. We extend these techniques to determine a relationship between traction-displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moments tensors (EMTs) due to the presence of small changes in the interface of the inclusion., Comment: 42pages,0 figures. arXiv admin note: text overlap with arXiv:1601.06774

Published
2017
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49. Stability of finite difference schemes for hyperbolic initial boundary value problems: numerical boundary layers

Author

Benjamin Boutin, Jean-François Coulombel, 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 ), Laboratoire de Mathématiques Jean Leray ( LMJL ), Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion. ( 2013 ), ANR-14-CE25-0001,ACHYLLES,Capture de l'Asymptotique pour des Systèmes Hyperboliques de Lois de Conservation avec Termes Source ( 2014 ), 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), Equations aux dérivées partielles, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), ANR-14-CE25-0001,ACHYLLES,Capture de l'Asymptotique pour des Systèmes Hyperboliques de Lois de Conservation avec Termes Source(2014), 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 Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
65M12, 65M06, 65M20, Control and Optimization, Discretization, Boundary (topology), 010103 numerical & computational mathematics, boundary layers, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], numerical schemes, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Boundary value problem, 0101 mathematics, Mathematics, Dirichlet boundary condition, Semigroup, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Finite difference, transport equations, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], stability, Computational Mathematics, Boundary layer, Modeling and Simulation, symbols, Cover (algebra), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; In this article, we give a unified theory for constructing boundary layer expansions for dis-cretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels, while semigroup estimates were restricted, up to now, to numerical schemes with two time levels only.

Published
2017
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50. Logarithmic stability of parabolic Cauchy problems

Author

Choulli, Mourad, Yamamoto, Masahiro, Equations aux dérivées partielles ( EDP ), Institut Élie Cartan de Lorraine ( IECL ), Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ), Graduate School of Mathematical Sciences, The University of Tokyo, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and The University of Tokyo (UTokyo)

Subjects
35R25, 35K99, 58J35, Carleman inequality, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, Parabolic Cauchy problems, Hardy inequality, logarithmic stability, FOS: Mathematics, Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

The uniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard \cite{Ha} these kind of problems are known to be ill-posed and even severely ill-posed. Until now there are only few partial results concerning the quantification of the stability for parabolic Cauchy problems. In the present article, we bring the complete answer to this issue, provided that the space domain has finite diameter with respect to the geodesic distance and assuming that solutions are sufficiently smooth.

Published
2017

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