2023Activity reportProject-TeamIDEFIX

6.1.1 ECIP

• Name:
  Eddy Current Imaging for Pipes
• Keywords:
  Inverse problem, Partial differential equation, HPC, Domain decomposition
• Functional Description:
  This software identifies deposit on pipes from measurements of eddy current probes. It is based on finite elements and domain decomposition through the softwares HPDDM, PETSc and FreeFEM, for the resolution of the PDE model of the eddy current measurements. It uses an iterative algorithm to identify the deposit properties.
• Contact:
  Lorenzo Audibert
• Partner:
  Edf

6.1.2 SpinDoctor

• Name:
  SpinDoctor Diffusion MRI Simulation Toolbox
• Keywords:
  MRI, Simulation, Finite element modelling
• Functional Description:

  SpinDoctor can be used

  1. to solve the Bloch-Torrey PDE to obtain the dMRI signal (the toolbox provides a way of robustly fitting the dMRI signal to obtain the fitted Apparent Diffusion Coefficient (ADC)), 2. to solve the diffusion equation of the H-ADC model to obtain the ADC, 3. a short-time approximation formula for the ADC is also included in the toolbox for comparison with the simulated ADC.

• URL:
  https://­github.­com/­SpinDoctorMRI/
• Contact:
  Jing Rebecca Li

6.1.3 CASTOR

• Keyword:
  C++
• Functional Description:

  The objective of the castor library is to propose high-level semantics, inspired by the Matlab language, allowing fast software prototyping in a low-level compiled language. It is nothing more than a matrix management layer using the tools of the standard C++ library, in different storage formats (full, sparse and hierarchical). Indeed, the use of IDEs 1 such as Xcode, Visual studio, Eclipse, etc. allows today to execute compiled code (C, C++, fortran, etc.) with the same flexibility as interpreted languages (Matlab, Python, Julia, etc.).

  A header-only template library for matrix management has been developed based on the standard C++ library, notably the std::vector class. Many tools and algorithms are provided to simplify the development of scientific computing programs. Particular attention has been paid to semantics, for a simplicity of use “à la matlab”, but written in C++. This high-level semantic/low-level language coupling makes it possible to gain efficiency in the prototyping phase, while ensuring performance for applications. In addition, direct access to data allows users to optimize the most critical parts of their code in native C++. Finally, complete documentation is available, as well as continuous integration unit tests. All of this makes it possible to meet the needs of teaching, academic issues and industrial applications at the same time.

  The castor library provides tools to :

  create and manipulate dense, sparse and hierarchical matrices make linear algebra computations based on optimized BLAS library make graphical representations based on VTK library These tools are used by applicative projects :

  finite and boundary element method using Galerkin approximation analytical solutions for scattering problems

• URL:
  https://­leprojetcastor.­gitlab.­labos.­polytechnique.­fr/­castor/­#
• Contact:
  Matthieu Aussal

7.1.1 Ultrasonic imaging in highly heterogeneous backgrounds

H. Haddar, F. Pourahmadian

This work formally investigates the differential evolution indicators as a tool for ultrasonic tracking of elastic transformation and fracturing in randomly heterogeneous solids. Within the framework of periodic sensing, it is assumed that the background at time t◦ contains (i) a multiply connected set of viscoelastic, anisotropic, and piece-wise homogeneous inclusions, and (ii) a union of possibly disjoint fractures and pores. The support, material properties, and interfacial condition of scatterers in (i) and (ii) are unknown, while elastic constants of the matrix are provided. The domain undergoes progressive variations of arbitrary chemo-mechanical origins such that its geometric configuration and elastic properties at future times are distinct. At every sensing step t0, t1, ..., multi-modal incidents are generated by a set of boundary excitations, and the resulting scattered fields are captured over the observation surface. The test data are then used to construct a sequence of wavefront densities by solving the spectral scattering equation. The incident fields affiliated with distinct pairs of obtained wavefronts are analyzed over the stationary and evolving scatterers for a suit of geometric and elastic evolution scenarios entailing both interfacial and volumetric transformations. The main theorem establishes the invariance of pertinent incident fields at the loci of static fractures and inclusions between a given pair of time steps, while certifying variation of the same fields over the modified regions. These results furnish a basis for theoretical justification of differential evolution indicators for imaging in complex composites which, in turn, enable the exclusive tomography of evolution in a background endowed with many unknown features 19.

7.1.2 Sampling method in linear elasticity for concrete: gravel honeycomb

L. Audibert, H. Haddar, JM Henault, F. Pourre

We consider the propagation of elastic waves at a given wavenumber in a penetrable medium with different levels of complexity ranging from an unbounded homogeneous medium to a partially bounded medium with or without a microstructure mimicking aggregates in concrete. In those various configurations we studied the Linear Sampling Method for imaging gravel honeycomb defects. The numerical solver is based on FreeFEM together with PETSc to perform parallel computations. Different source modeling has been investigated for point sources. The numerical results show satisfactory reconstruction in some idealistic setting. This is preparatory work before handling experimental data on mockup.

7.1.3 Differential imaging in periodic media

Y. Boukari, H. Haddar, N. Jenhani

We consider a problem of nondestructive testing of an infinite periodic penetrable layer using acoustic waves. This is an important problem with growing interest since periodic structures are part of many fascinating modern technological designs with applications in (bio)engineering and material sciences. In many sophisticated devises the periodic structure is complicated or difficult to model mathematically, hence evaluating its Green's function which is the fundamental tool of many imaging methods, is computationally expensive or even impossible. On the other hand, when looking for flows in such complex media, the option of reconstructing everything, i.e. both periodic structure and the defects, may not be viable. We proposed in an earlier work an approach which provides a criteria to reconstruct the support of local anomalies without knowing explicitly or reconstructing the periodic healthy background. We provide a theoretical justification of this method that avoids assuming that the local perturbation is also periodic. Our theoretical framework uses functional spaces with continuous dependence with respect to the Floquet-Bloch variable. The corner stone of the analysis is the justification of the Generalized Linear Sampling Method in this setting for a single Floquet-Bloch mode 10.

7.1.4 Time-vs. frequency-domain inverse elastic scattering: Theory and experiment

H. Haddar, X. Liu, F. Pourahmadian, J. Song

This study formally adapts the time-domain linear sampling method (TLSM) for ultrasonic imaging of stationary and evolving fractures in safety-critical components. The TLSM indicator is then applied to the laboratory test data and the obtained reconstructions are compared to their frequency-domain counterparts. The results highlight the unique capability of the time-domain imaging functional for high-fidelity tracking of evolving damage, and its relative robustness to sparse and reduced-aperture data at moderate noise levels. A comparative analysis of the TLSM images against the multifrequency LSM maps further reveals that thanks to the full-waveform inversion in time and space, the TLSM generates images of remarkably higher quality with the same dataset 18.

7.1.5 The linear sampling method for random sources

J. Garnier, H. Haddar, H. Montanelli

We present an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our sampling method is based on the Helmholtz–Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which we introduce and analyze. Implementations in MATLAB using boundary elements, the SVD, Tikhonov regularization, and Morozov's discrepancy principle are also discussed. We demonstrate the robustness and accuracy of our algorithms with several numerical experiments in two dimensions 16.

7.2.1 Asymptotic Expansion of Transmission Eigenvalues for Anisotropic Thin Layers

H. Boujlida, H. Haddar, M. Khenissi

We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [19]. We also provide formal derivation in more general cases validating the obtained theoretical result. 9.

7.2.2 Stability estimate for an inverse problem for the time harmonic magnetic schrödinger operator from the near and far field pattern

M. Bellassoued, H. Haddar, A. Labidi

We derive conditional stability estimates for inverse scattering problems related to time harmonic magnetic Schrödinger equation. We prove logarithmic type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near field or far field maps. Our approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions. 8.

7.2.3 Generalized impedance boundary conditions with vanishing or sign-changing impedance

L. Bourgeois, L. Chesnel

We consider a Laplace type problem with a generalized impedance boundary condition of the form ${\partial }_{\nu }u=-{\partial }_x\left(g{\partial }_{x}u\right)$ on a flat part $\Gamma$ of the boundary. Here $\nu$ is the outward unit normal vector to $\partial \Omega$, $g$ is the impedance parameter and $x$ is the coordinate along $\Gamma$. Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases $g=1$ or $g=-1$ have been investigated. In this work, we address situations where $\Gamma$ contains the origin and $g\left(x\right)=1_{x>0}\left(x\right)x^{\alpha }$ or $g\left(x\right)=-{\mathrm{sign}\left(x\right)\left|x\right|}^{\alpha }$ with $\alpha \ge 0$. In other words, we study cases where $g$ vanishes at the origin and changes its sign. The main message is that the well-posedness in the Fredholm sense of the corresponding problems depends on the value of $\alpha$. For $\alpha \in [0,1)$, we show that the associated operators are Fredholm of index zero while it is not the case when $\alpha =1$. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl's sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments which seem to corroborate the theorems. 32

7.2.4 Reconstruction of averaging indicators for highly heterogeneous media

L. Audibert, H. Haddar, F. Pourre

We propose a new imaging algorithm capable of producing quantitative indicator functions for unknown and possibly highly oscillating media from multistatic far field measurements of scattered fields at a fixed frequency. The algorithm exploits the notion of modified transmission eigenvalues and their determination from measurements. We propose in particular the use of a new modified background obtained as the limit of a metamaterial background. It has the specificity of having a unique non trivial eigenvalue, which is particularly suited for the proposed imaging procedure. We show the efficiency of this new algorithm on some 2D experiments and emphasize its superiority with respect to some clasical approaches such as the Linear Sampling Method.

7.3.1 Acoustic waveguide with a dissipative inclusion

L. Chesnel, J. Heleine, S.A. Nazarov, J. Taskinen

We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient $\eta$, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when $\eta \to 0^{+}$ (small dissipation) and when $\eta \to +\infty$ (large dissipation). Surprisingly, at the limit $\eta \to +\infty$, we show that no energy is absorbed by the inclusion. This is due to a skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to $\eta$. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations. 12

7.3.2 Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer

L. Chesnel, S.A. Nazarov, J. Taskinen

We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector $\kappa =({\kappa }_1,{\kappa }_2)$ which characterizes the domain at the edges. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to $\kappa$. In particular, we show that for a given ${\kappa }_1>0$, there is some $h\left({\kappa }_1\right)>0$ such that discrete spectrum exists for ${\kappa }_2\in (-{\kappa }_1,0)\cup (h\left({\kappa }_1\right),{\kappa }_1)$ whereas it is empty for ${\kappa }_2\in [0,h\left({\kappa }_1\right)]$. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.37

7.3.3 On the breathing of spectral bands in periodic quantum waveguides with inflating resonators

L. Chesnel, S.A. Nazarov

We are interested in the lower part of the spectrum of the Dirichlet Laplacian $A^{\epsilon }$ in a thin waveguide ${\Pi }^{\epsilon }$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry $\Omega$ by a small factor $\epsilon >0$. The Floquet-Bloch theory ensures that the spectrum of $A^{\epsilon }$ has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to $+\infty$ as $O\left({\epsilon }^{-2}\right)$ when $\epsilon \to 0^{+}$. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in $\Omega$ that we denote by ${\mathrm{X}}_{†}$. Generically, i.e. for most $\Omega$, there holds ${\mathrm{X}}_{†}=\left\{0\right\}$ and the lower part of the spectrum of $A^{\epsilon }$ is very sparse, made of bands of length at most $O\left(\epsilon \right)$ as $\epsilon \to 0^{+}$. For certain $\Omega$ however, we have $\dim {\mathrm{X}}_{†}=1$ and then there are bands of length $O\left(1\right)$ which allow for wave propagation in ${\Pi }^{\epsilon }$. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing $\Omega$ around a particular ${\Omega }_{☆}$ where $\dim {\mathrm{X}}_{†}=1$. We show a breathing phenomenon for the spectrum of $A^{\epsilon }$: when inflating $\Omega$ around ${\Omega }_{☆}$, the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold ${\pi }^2/{\epsilon }^2$, stops breathing and becomes extremely short as $\Omega$ continues to inflate.35

7.3.4 Spectrum of the Dirichlet Laplacian in a thin cubic lattice

L. Chesnel, S.A. Nazarov

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $\epsilon \ll 1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+\infty$ as $\epsilon$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O\left(e^{-\delta /\epsilon }\right)$, $\delta >0$, while the length of the next spectral segments is $O\left(\epsilon \right)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^{\Omega }$ in the geometry $\Omega$ obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that $A^{\Omega }$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^{\Omega }$, that is no non trivial bounded solution at the threshold frequency for $A^{\Omega }$. This implies that the correct 1D model of the lattice for the next spectral segments is a graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis. 13

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

A.-S. Bonnet-Ben Dhia, L. Chesnel, M. Rihani

We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity $\epsilon$ and the permeability $\mu$ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in $\epsilon$ and in $\mu$, the corresponding scalar operators are not of Fredholm type in the usual $H^1$ spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical $L^2$ framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the T-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields. 31

7.5.1 A simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI

Chengran Fang, Zheyi Yang, Demian Wassermann, Jing-Rebecca Li

We propose a framework to train supervised learning models on synthetic data to estimate brain microstructure parameters using diffusion magnetic resonance imaging (dMRI). Although further validation is necessary, the proposed framework aims to seamlessly incorporate realistic simulations into dMRI microstructure estimation. Synthetic data were generated from over 1,000 neuron meshes converted from digital neuronal reconstructions and linked to their neuroanatomical parameters (such as soma volume and neurite length) using an optimized diffusion MRI simulator that produces intracellular dMRI signals from the solution of the Bloch–Torrey partial differential equation. By combining random subsets of simulated neuron signals with a free diffusion compartment signal, we constructed a synthetic dataset containing dMRI signals and 40 tissue microstructure parameters of 1.45 million artificial brain voxels. To implement supervised learning models we chose multilayer perceptrons (MLPs) and trained them on a subset of the synthetic dataset to estimate some microstructure parameters, namely, the volume fractions of soma, neurites, and the free diffusion compartment, as well as the area fractions of soma and neurites. The trained MLPs perform satisfactorily on the synthetic test sets and give promising in-vivo parameter maps on the MGH Connectome Diffusion Microstructure Dataset (CDMD). Most importantly, the estimated volume fractions showed low dependence on the diffusion time, the diffusion time independence of the estimated parameters being a desired property of quantitative microstructure imaging. The synthetic dataset we generated will be valuable for the validation of models that map between the dMRI signals and microstructure parameters. The surface meshes and microstructures parameters of the aforementioned neurons have been made publicly available.15

7.5.2 A second order asymptotic model for diffusion MRI in permeable media

Marwa Kchaou, Jing-Rebecca Li

Starting from a reference partial differential equation model of the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses, one can use periodic homogenization theory to establish macroscopic models. A previous work introduced an asymptotic model that accounted for permeable interfaces in the imaging medium. In this paper we formulate a higher order asymptotic model to treat higher values of permeability. We explicitly solved this new asymptotic model to obtain a system of ordinary differential equations that can model the diffusion MRI signal and we present numerical results showing the improved accuracy of the new model in the regime of higher permeability.17

7.5.3 Incorporating interface permeability into the diffusion MRI signal representation using impermeable Laplace eigenfunctions

Zheyi Yang, Chengran Fang, Jing-Rebecca Li

The complex-valued transverse magnetization due to diffusion-encoding magnetic field gradients acting on a permeable medium can be modeled by the Bloch–Torrey partial differential equation. The diffusion magnetic resonance imaging (MRI) signal has a representation in the basis of the Laplace eigenfunctions of the medium. However, in order to estimate the permeability coefficient from diffusion MRI data, it is desirable that the forward solution can be calculated efficiently for many values of permeability. Approach. In this paper we propose a new formulation of the permeable diffusion MRI signal representation in the basis of the Laplace eigenfunctions of the same medium where the interfaces are made impermeable. Main results. We proved the theoretical equivalence between our new formulation and the original formulation in the case that the full eigendecomposition is used. We validated our method numerically and showed promising numerical results when a partial eigendecomposition is used. Two diffusion MRI sequences were used to illustrate the numerical validity of our new method. Significance. Our approach means that the same basis (the impermeable set) can be used for all permeability values, which reduces the computational time significantly, enabling the study of the effects of the permeability coefficient on the diffusion MRI signal in the future.20

7.6.1 Influence of the partition of unity on SORAS preconditioner

M. Bonazzoli, X. Claeys, F. Nataf, P.-H. Tournier

We have investigated numerically the influence of the choice of the partition of unity on the convergence of the Symmetrized Optimized Restricted Additive Schwarz (SORAS) preconditioner for the heterogeneous reaction-convection-diffusion equation. Previously, we had analyzed the convergence of this overlapping domain decomposition preconditioner for generic non self-adjoint or indefinite problems, like the reaction-convection-diffusion equation. In the numerical experiments, we had noticed that the number of iterations for convergence of preconditioned GMRES appeared not to vary significantly when increasing the overlap width. In this work we show that actually this is due to the particular choice of the partition of unity for the preconditioner, and we study the dependence on the overlap and on the number of subdomains for two kinds of partition of unity. Our numerical investigation shows that the second kind of partition of unity, which is non-zero in the interior of the whole overlapping region, generally improves the iteration counts obtained with the first kind of partition of unity, whose gradient is zero on the subdomain interfaces. Moreover, the first kind of partition of unity, which would be the natural choice for ORAS solver instead, yields for SORAS preconditioner iterations counts that do not vary significantly when increasing the overlap width. A proceedings on this topic 22 have been published.

7.6.2 Multi-domain FEM-BEM coupling for acoustic scattering

M. Bonazzoli, X. Claeys

We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with domain variational formulations for the heterogeneous subdomain. This is an extension of Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains abut. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects; here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a Gårding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances. An article on this topic is under review 29.

7.6.3 Computing singular and near-singular integrals over curved boundary elements: The strongly singular case

H. Montanelli, F. Collino, H. Haddar

We present algorithms for computing strongly singular and near-singular surface integrals over curved triangular patches, based on singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy and robustness of our method for quadratic basis functions and quadratic triangles by integrating it into a boundary element code and solving several scattering problems in 3D. We also give numerical evidence that the utilization of curved boundary elements enhances computational efficiency compared to conventional planar elements 38.

7.7.1 On the convergence analysis of one-shot inversion methods

M. Bonazzoli, H. Haddar, T. A. Vu

When an inverse problem is solved by a gradient-based optimization algorithm, the corresponding forward and adjoint problems, which are introduced to compute the gradient, can be also solved iteratively. The idea of iterating at the same time on the inverse problem unknown and on the forward and adjoint problem solutions yields to the concept of one-shot inversion methods. We are especially interested in the case where the inner iterations for the direct and adjoint problems are incomplete, that is, stopped before achieving a high accuracy on their solutions. Here, we focus on general linear inverse problems and generic fixed-point iterations for the associated forward problem. We analyze variants of the so-called multi-step one-shot methods, in particular semi-implicit schemes with a regularization parameter. We establish sufficient conditions on the descent step for convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical gradient descent, where the forward and adjoint problems are solved exactly by a direct solver instead. We observe that very few inner iterations are enough to guarantee good convergence of the inversion algorithm, even in the presence of noisy data. An article on this topic is under review 30.

7.7.2 Imaging a dam-rock interface with inversion of a full elastic-acoustic model

L. Audibert, M. Bonazzoli, M. A. Boukraa, H. Haddar, D. Vautrin

We are interested in imaging the interface between the concrete structure of hydroelectric dams and the rock foundation using non-destructive seismic waves. We present a geophysical technique for processing seismic measurements to obtain an image of the interface with metric resolution. The proposed technique is based on "Full Waveform Inversion" with a shape optimization approach. Numerical results using synthetic measurements demonstrate the method ability to accurately recover the interface with a limited number of measurement points and in the presence of noise. A proceedings has been submitted to ICIPE2024.

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

S. Chaabane, H. Haddar, R. Jerbi We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simplified case and numerically show its efficiency for some two dimensional experiments 11.

9.1.1 Scientific events: organisation

9.1.2 Scientific events: selection

• L. Chesnel and H. Haddar are members of the scientific committee of the Waves conference.
• L. Audibert (2023 - ) is president of the scientific committee of the school CEA-EDF-INRIA.

9.1.3 Invited talks

• H. Montanelli, New Jersey Institute of Technology, Fluid Mechanics and Waves Seminar, November 2023.
• H. Montanelli, Congress of Young Researchers in Mathematics and Applications, Gif-sur-Yvette, September 2023.
• H. Montanelli, Numerical Analysis in the 21st Century, Oxford, August 2023.
• H. Montanelli, Inria & Côte d'Azur University, FACTAS Seminar, June 2023.
• H. Montanelli, SIAM CSE 2023, Amsterdam, February 2023.
• H. Montanelli, Orléans University, Applied Analysis Seminar, February 2023.
• H. Montanelli, University of Pau, Mathematics and Applications Seminar, February 2023.
• H. Montanelli, Aix-Marseille University, Applied Analysis Seminar, January 2023.
• L. Chesnel, ANEDP seminar, Lille, December 2023.
• L. Chesnel, thematic day PLASMON 2023, Marseille, December 2023.
• L. Chesnel, LMA seminar, Marseille, October 2023.
• L. Chesnel, LMS-Bath Symposium "Operators, Asymptotics, Waves", Bath, July 2023.
• L. Chesnel, INI Programe Mathematical theory and applications of multiple wave scattering, Cambridge, May 2023.
• L. Chesnel, Maxwell institute applied and computational mathematics seminar, Edinburgh, May 2023.
• M. Bonazzoli, CSE23, SIAM Conference on Computational Science and Engineering 2023, Amsterdam, The Netherlands, February 2023.
• H. Haddar, Invited plenary talk at AIP 2023, Gottingen, Germany.
• H. Haddar, LAMHA seminar, June 2023, Sfax, Tunisia.
• H. Haddar, Invited talk at a minisymposium in IEEE CAMA, 2023, Genova, Italy.
• L. Audibert, AIP 2023, Gottingen, Germany, September 2023.

9.1.4 Research administration

• M. Bonazzoli is the International partnerships Scientific Correspondent for Inria Saclay.
• M. Bonazzoli took part in Jul. 2023 to the prize committee for Prix Junior Maryam Mirzakhani awarded by Fondation Mathématique Jacques-Hadamard (FMJH) to two young female students for a mathematics project.
• M. Bonazzoli is a volunteer member of Opération Postes (newsletter and website, which gathers detailed information about the French competitive selections for permanent positions in Mathematics and Informatics, supported by the French academic societies SMAI, SAGIP, SFdS, SIF, and SMF).
• J.-R. Li is a member of INRIA Commission d'Evaluation, 2015-2023.
• H. Haddar is a member of the BCEP of INRIA Sacaly.

9.2.1 Teaching

• Doctorat: H. Haddar et L. Audibert, Inverse problems: Algorithms and Applications. Executive Education, Ecole Polytechnique.
• Master: H. Montanelli, Modal - Modélisation mathématique par la démarche expérimentale, 2nd year of École Polytechnique, creation and supervision of a project for five students.
• Bachelor: H. Montanelli, Commande des systèmes dynamiques, 1st year of ENSTA Paris, 12 TD hours.
• Master: L. Chesnel, Analyse variationnelle des équations aux dérivées partielles, 2nd year of Ecole Polytechnique, 20 TD hours.
• Master: L. Chesnel, Modal - Modélisation mathématique par la démarche expérimentale, 2nd year of École Polytechnique, creation and supervision of a project for two students.
• Bachelor: L. Chesnel, Numerical Methods for ODEs, 3rd year of the Bachelor of Ecole Polytechnique, 20 TD hours.
• Master: M. Bonazzoli, Mathematics for data science, 1st year of Computer Science Master, Université Paris-Saclay, 21 hours (lessons and TD).
• Master: M. Bonazzoli, La méthode des éléments finis, 2nd year of Engineering School, ENSTA Paris, 12 TD hours.
• Bachelor: M. Bonazzoli, Fonctions de variable complexe, 1st year of Engineering School, ENSTA Paris, 12 TD hours.
• Bachelor: L. Audibert, Introduction à la discrétisation des équations aux dérivées partielles, 1st year of Engineering School, ENSTA Paris, 12 TD hours.
• Licence: H. Haddar, Complex analysis and Elementary tools of analysis for partial differential equations, for students in the first year of Ensta ParisTech curriculum. 37 equivalent TD hours. 2021-present.
• Master: J.-R. Li, Refresher Course in Math and a Project on Numerical Modeling Done in Pairs, The Energy Environment: Science Technology and Management (STEEM) Master Program, Ecole Polytechnique.
• Bachelor: J.-R. Li, Introduction à Matlab, ENSTA Paris.

9.2.2 Supervision

• PhD in progress: F. Pourre, Using spectral signature for imaging. (2021-), L. Audibert and H. Haddar.
• PhD in progress: M. Mathevet, Eddy current imaging with inverse problems methods. (2022-), L. Audibert and H. Haddar.
• PhD in progress: A. Parigaux, (2022-), Construction of transparent conditions for electromagnetic waveguides, analysis and applications. (2022-), L. Chesnel and A.S. Bonnet Ben Dhia.
• PhD in progress: T.A. Vu, One-shot inversion methods and domain decomposition (2020-), M. Bonazzoli and H. Haddar.
• PhD in progress: A. Boisneault, Numerical methods and high performance simulation for 3D imaging in complex media, (2023-), M. Bonazzoli (with X. Claeys, Sorbonne Université, and P. Marchand, Inria).
• Master's degree research internship: A. Boisneault, Numerical methods and high performance simulation for 3D imaging in complex media, (Apr.–Aug. 2023), M. Bonazzoli (with X. Claeys, Sorbonne Université, and P. Marchand, Inria).
• Postdoc in progress: M.A. Boukraa, Inverse problem methods for interface imaging: application to a concrete-rock interface for a hydroelectric dam, (2022-), supervisors: M. Bonazzoli, D. Vautrin, collaborators: L. Audibert, H. Haddar, F. Taillade.
• Postdoc in progress: D.Q. Bui, Optimal inverse modeling for GPR imaging, (2023-), L. Audibert, M. Bonazzoli, H. Haddar, P. Marchand, F. Taillade.
• PhD: Zheyi Yang (10/2020 – 12/2023). Numerical methods to estimate brain micro-structure from diffusion MRI data. J.-R. Li.
• PhD: Chengran Fang (10/2019 – 2/2023). Neuron modeling, Bloch-Torrey equation, and their application to brain microstructure estimation using diffusion MRI. J.-R. Li, Co-advisor: Demian Wassermann(MIND, Inria Saclay).
• Master's degree research internship: Mohamed Ahmane, Resolution of the inverse problem of electrical impedance tomography by Bayesian inference. J.-R. Li, Co-advisor: Lisl Weynans (CARMEN, Inria Bordeaux).
• PhD: N. Jenhani, Generalized linear sampling methods for locally perturbed periodic layers, (2020-12/2023) H. Haddar and Y. Boukari.
• PhD: A. Labidi, Inverse scattering problems for the time harmonic magnetic Schrödinger operator. (2020-12/2023) H. Haddar and M. Bellassoued.
• PhD: R. Jerbi, Combined inversion methods for inverse conductivity problems (2020-11/2023) H. Haddar and S. Chaabane.

9.2.3 Interventions

• M. Bonazzoli and L. Chesnel were volunteers at Inria stand at Fête de la Science (Institut Polytechnique de Paris), Oct. 2023.
• M. Bonazzoli participated to several speed-meetings with high/middle school students (Rendez-vous des Jeunes Mathématiciennes et Informaticiennes at Inria Saclay, Feb. 2023, and at École Normale Supérieure Paris, Dec. 2023; Journée Filles, maths et informatique : une équation lumineuse at École Polytechnique, Mar. 2023) to answer their questions about the studies and career as a mathematician.
• L. Chesnel went to an elementary school class to make an initiation to geometry.
• J.-R. Li participated in Rencontre avec les Bachelor de l'X autour des mathématiques, Nov. 2023, Institut des Hautes Études Scientifiques (IHES).

International journals

• 7 articleL.Lorenzo Audibert, H.Hugo Girardon, H.Houssem Haddar and P.Pierre Jolivet. Inversion of Eddy-Current Signals Using a Level-Set Method and Block Krylov Solvers.SIAM Journal on Scientific Computing4532023, B366-B389HALDOI
• 8 articleM.Mourad Bellassoued, H.Houssem Haddar and A.Amal Labidi. Stability estimate for an inverse problem for the time harmonic magnetic schrödinger operator from the near and far field pattern.SIAM Journal on Mathematical Analysis5542023, 2475-2504HALDOIback to text
• 9 articleH.Hanen Boujlida, H.Houssem Haddar and M.Moez Khenissi. Asymptotic Expansion of Transmission Eigenvalues for Anisotropic Thin Layers.Applicable Analysis2023, 1-22HALDOIback to text
• 10 articleY.Yosra Boukari, H.Houssem Haddar and N.Nouha Jenhani. Analysis of sampling methods for imaging a periodic layer and its defects.Inverse Problems395March 2023, 055001HALDOIback to text
• 11 articleS.Slim Chaabane, H.Houssem Haddar and R.Rahma Jerbi. A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem.Inverse Problems399July 2023, 095001HALDOIback to text
• 12 articleL.Lucas Chesnel, J.Jérémy Heleine, S. A.Sergei A. Nazarov and J.Jari Taskinen. Acoustic waveguide with a dissipative inclusion.ESAIM: Mathematical Modelling and Numerical Analysis576November 2023, 3585-3613HALDOIback to text
• 13 articleL.Lucas Chesnel and S. A.Sergei A. Nazarov. Spectrum of the Dirichlet Laplacian in a thin cubic lattice.ESAIM: Mathematical Modelling and Numerical Analysis576November 2023, 3251-3273HALDOIback to text
• 14 articleC.Chengran Fang, D.Demian Wassermann and J.-R.Jing-Rebecca Li. Fourier representation of the diffusion MRI signal using layer potentials.SIAM Journal on Applied Mathematics831February 2023, 99-121HALDOI
• 15 articleC.Chengran Fang, Z.Zheyi Yang, D.Demian Wassermann and J.-R.Jing-Rebecca Li. A simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI.Medical Image Analysis90December 2023, 102979HALDOIback to text
• 16 articleJ.Josselin Garnier, H.Houssem Haddar and H.Hadrien Montanelli. The linear sampling method for random sources.SIAM Journal on Imaging Sciences1632023, 1572-1593HALDOIback to text
• 17 articleM.Marwa Kchaou and J.-R.Jing-Rebecca Li. A second order asymptotic model for diffusion MRI in permeable media.ESAIM: Mathematical Modelling and Numerical Analysis574July 2023, 1953-1980HALDOIback to text
• 18 articleX.Xiaoli Liu, J.J Song, F.Fatmeh Pourahmadian and H.Houssem Haddar. Time-vs. frequency-domain inverse elastic scattering: Theory and experiment.SIAM Journal on Applied Mathematics8332023HALDOIback to text
• 19 articleF.Fatemeh Pourahmadian and H.Houssem Haddar. Ultrasonic imaging in highly heterogeneous backgrounds.Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences47922712023HALDOIback to text
• 20 articleZ.Zheyi Yang, C.Chengran Fang and J.-R.Jing-Rebecca Li. Incorporating interface permeability into the diffusion MRI signal representation while using impermeable Laplace eigenfunctions.Physics in Medicine and Biology6817August 2023, 175036HALDOIback to text
• 21 articleZ.Zheyi Yang, I.Imen Mekkaoui, J.Jan Hesthaven and J.-R.Jing-Rebecca Li. Asymptotic models of the diffusion MRI signal accounting for geometrical deformations.MathematicS In Action1212023, 65-85HALDOI

Scientific book chapters

• 22 inbookM.Marcella Bonazzoli, X.Xavier Claeys, F.Frédéric Nataf and P.-H.Pierre-Henri Tournier. How does the partition of unity influence SORAS preconditioner?149Domain Decomposition Methods in Science and Engineering XXVIILecture Notes in Computational Science and EngineeringSpringer Nature SwitzerlandJanuary 2024, 61-68HALDOIback to text

Doctoral dissertations and habilitation theses

• 23 thesisC.Chengran Fang. Neuron modeling, Bloch-Torrey equation, and their application to brain microstructure estimation using diffusion MRI.Université Paris-SaclayFebruary 2023HAL
• 24 thesisN.Nouha Jenhani. Generalized linear sampling methods for locally perturbed periodic layers.Ecole Nationale d'ingénieur de TunisDecember 2023HAL
• 25 thesisR.Rahma Jerbi. Combined inversion methods for inverse conductivity problems.University of SfaxOctober 2023HAL
• 26 thesisA.Amal LABIDI. Inverse scattering problems for the time harmonic magnetic Schrödinger operator.Université Tunis El Manar (Tunisie)December 2023HAL

Reports & preprints

• 27 miscL.Lorenzo Audibert, H.Houssem Haddar and F.Fabien Pourre. Reconstruction of averaging indicators for highly heterogeneous media.2023HAL
• 28 miscL.Laurent Baratchart, H.Houssem Haddar and C.Cristóbal Villalobos Guillén. Silent sources on a surface for the Helmholtz equation and decomposition of L² vector fields.December 2023HAL
• 29 miscM.Marcella Bonazzoli and X.Xavier Claeys. Multi-domain FEM-BEM coupling for acoustic scattering.May 2023HALback to text
• 30 miscM.Marcella Bonazzoli, H.Houssem Haddar and T. A.Tuan Anh Vu. On the convergence analysis of one-shot inversion methods.July 2023HALback to text
• 31 miscA.-S.Anne-Sophie Bonnet-Ben, L.Lucas Chesnel and M.Mahran Rihani. Maxwell's equations with hypersingularities at a negative index material conical tip.May 2023HALback to text
• 32 miscL.Laurent Bourgeois and L.Lucas Chesnel. Generalized impedance boundary conditions with vanishing or sign-changing impedance.September 2023HALback to text
• 33 miscF.Fioralba Cakoni, H.Houssem Haddar and T.-P.Thi-Phong Nguyen. Fast Imaging of Local Perturbations in a Unknown Bi-Periodic Layered Medium.2023HAL
• 34 miscL.Lucas Chesnel, J.Jérémy Heleine, S. A.Sergei A Nazarov and J.Jari Taskinen. Acoustic waveguide with a dissipative inclusion.May 2023HAL
• 35 miscL.Lucas Chesnel and S. A.Sergei A Nazarov. On the breathing of spectral bands in periodic quantum waveguides with inflating resonators.December 2023HALback to text
• 36 miscL.Lucas Chesnel and S. A.Sergei A Nazarov. Spectrum of the Dirichlet Laplacian in a thin cubic lattice.January 2023HAL
• 37 miscL.Lucas Chesnel, S. A.Sergei A. Nazarov and J.Jari Taskinen. Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer.March 2023HALback to text
• 38 miscH.Hadrien Montanelli, F.Francis Collino and H.Houssem Haddar. Computing singular and near-singular integrals over curved boundary elements: The strongly singular case.September 2023HALback to text