5.1.1 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

5.1.2 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

5.1.3 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

6.1.1 The linear sampling method for data generated by small random scatterers

J. Garnier, H. Haddar, H. Montanelli

We present an extension of the linear sampling method for solving the sound-soft inverse scattering problem in two dimensions with data generated by randomly distributed small scatterers. The theoretical justification of our novel sampling method is based on a rigorous asymptotic model, a modified Helmholtz–Kirchhoff identity, and our previous work on the linear sampling method for random sources. Our numerical implementation incorporates boundary elements, Singular Value Decomposition, Tikhonov regularization, and Morozov’s discrepancy principle. We showcase the robustness and accuracy of our algorithms with a series of numerical experiments. dimensions 18.

6.1.2 Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator

L.Audibert, S. Meng

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter. 8

6.1.3 Exploring low-rank structure for an inverse scattering problem with far-field data

L.Audibert, S. Meng, B. Zhang, Y. Zhou

The inverse scattering problem exhibits an inherent low-rank structure due to its ill-posed nature; however developing low-rank structures for the inverse scattering problem remains challenging. In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm-Liouville problem. We first process the far-field data to obtain a post-processed data set within a disk domain. Subsequently, the post-processed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space, by dropping higher-order terms. The low-rank structure leads to a Hölder-logarithmic type stability estimate for arbitrary unknown functions, and a Lipschitz stability estimate for unknowns belonging to a finite dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability. 33

6.1.4 Fast Imaging of Local Perturbations in a Unknown Bi-Periodic Layered Medium

F. Cakoni, H. Haddar, T.P. Nguyen

We discuss a novel approach for imaging local faults inside an infinite bi-periodic layered medium in $R^3$ using acoustic measurements of scattered fields at the bottom or the top of the layer. The faulted area is represented by compactly supported perturbations with erroneous material properties. Our method reconstructs the support of perturbations without knowing or reconstructing the constitutive material parameters of healthy or faulty bi-period layer; only the size of the period is needed. This approach falls under the class of non-iterative imaging methods, known as the generalized linear sampling method with differential measurements. The advantage of applying differential measurements to our inverse problem is that instead of comparing the measured data against measurements due to healthy structures, one makes use of periodicity of the layer where the data operator restricted to single Floquet-Bloch modes plays the role of the one corresponding to healthy material. This leads to a computationally efficient and mathematically rigorous reconstruction algorithm. We present numerical experiments that confirm the viability of the approach for various configurations of defects 15.

6.2.1 Examples of non-scattering inhomogeneities

L. Chesnel, H. Haddar, H. Li, J. Xiao

We consider the scattering of waves by a penetrable inclusion embedded in some reference medium. We exhibit examples of materials and geometries for which non-scattering frequencies exist, i.e., for which at some frequencies there are incident fields which produce null scattered fields outside of the inhomogeneity. We show in particular that certain domains with corners or even cusps can support non-scattering frequencies. We relate the latter, for some inclusions, to resonance frequencies for Dirichlet or Neumann cavities. We also find situations where incident non-scattering fields solve the Helmholtz equation in a neighbourhood of the inhomogeneity and not in the whole space. In relation with invisibility, we give examples of inclusions of anisotropic materials which are non-scattering for all real frequencies. We prove that corresponding material indices must have a special structure on the boundary 31.

6.2.2 An algorithm for computing scattering poles based on dual characterization to interior eigenvalues

F. Cakoni, H. Haddar, D. Zilberberg

We present an algorithm for the computation of scattering poles for an impenetrable obstacle with Dirichlet or Robin boundary conditions in acoustic scattering. This paper builds upon our previous work titled ‘A duality between scattering poles and transmission eigenvalues in scattering theory’ (2020 Proc. A. 476 , 20200612 (doi:10.1098/rspa.2020.0612)), where we developed a conceptually unified approach for characterizing the scattering poles and interior eigenvalues corresponding to a scattering problem. This approach views scattering poles as dual to interior eigenvalues by interchanging the roles of incident and scattered fields. In this framework, both sets are linked to the kernel of the relative scattering operator that maps incident fields to scattered fields. This mapping corresponds to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Leveraging this dual characterization and inspired by the generalized linear sampling method for computing the interior eigenvalues, we present a novel numerical algorithm for computing scattering poles without relying on an iterative scheme. Preliminary numerical examples showcase the effectiveness of this computational approach 16.

6.2.3 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

6.2.4 Averaged Steklov Eigenvalues, Inside OutsideDuality and Application to Inverse Scattering

L. Audibert, Houssem Haddar and Fabien Pourre

We introduce a new family of artificial backgrounds corresponding to averaged impedance boundary conditions formulated in an abstract framework. These backgrounds are used to define a finite number of averaged Steklov eigenvalues, which are associated with inverse scattering problems from inhomogeneous media. We prove that these special eigenvalues can be determined from full-aperture, fixed-frequency far-fields using the inside-outside duality method. We then show and numerically demonstrate how this method can be used to reconstruct averaged values of the refractive index. 30

6.3.1 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. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases ${\kappa }_1\ge 0$ and ${\kappa }_2\in [-{\kappa }_1;{\kappa }_1]$. 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.17

6.3.2 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.

6.4.1 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. 12

6.4.2 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 14.

6.5.1 SpinDoctor-IVIM: A virtual imaging framework for intravoxel incoherent motion MRI

Mojtaba Lashgari, Zheyi Yang, Miguel O. Bernabeu, Jing-Rebecca Li, Alejandro F. Frangi

Intravoxel incoherent motion (IVIM) imaging is increasingly recognised as an important tool in clinical MRI, where tissue perfusion and diffusion information can aid disease diagnosis, monitoring of patient recovery, and treatment outcome assessment. Currently, the discovery of biomarkers based on IVIM imaging, similar to other medical imaging modalities, is dependent on long preclinical and clinical validation pathways to link observable markers derived from images with the underlying pathophysiological mechanisms. To speed up this process, virtual IVIM imaging is proposed. This approach provides an efficient virtual imaging tool to design, evaluate, and optimise novel approaches for IVIM imaging. In this work, virtual IVIM imaging is developed through a new finite element solver, SpinDoctor-IVIM, which extends SpinDoctor, a diffusion MRI simulation toolbox. SpinDoctor-IVIM simulates IVIM imaging signals by solving the generalised Bloch–Torrey partial differential equation. The input velocity to SpinDoctor-IVIM is computed using HemeLB, an established Lattice Boltzmann blood flow simulator. Contrary to previous approaches, SpinDoctor-IVIM accounts for volumetric microvasculature during blood flow simulations, incorporates diffusion phenomena in the intravascular space, and accounts for the permeability between the intravascular and extravascular spaces. The above-mentioned features of the proposed framework are illustrated with simulations on a realistic microvasculature model19.

6.5.2 Alpha_Mesh_Swc: automatic and robust surface mesh generation from the skeleton description of brain cells

Alex McSweeney-Davis, Chengran Fang, Emmanuel Caruyer, Anne Kerbrat, Jing-Rebecca Li

In recent years, there has been a significant increase in publicly available skeleton descriptions of real brain cells from laboratories all over the world. In theory, this should make it possible to perform large scale realistic simulations on brain cells. However, currently there is still a gap between the skeleton descriptions and high quality simulation-ready surface and volume meshes of brain cells.

We propose and implement a tool called Alpha_Mesh_Swc to generate automatically and efficiently triangular surface meshes that are optimized for finite elements simulations. We use an Alpha Wrapping method with an offset parameter on component surface meshes to efficiently generate a global watertight mesh. Then mesh simplification and re-meshing are used to produce an optimal surface mesh. Our methodology limits the number of surface triangles while preserving geometrical accuracy, permits cutting and gluing of cell components, is robust to imperfect skeleton descriptions, and allows mixed cell descriptions (surface meshes combined with skeletons).

We compared the robustness, performance and accuracy of Alpha_Mesh_Swc against existing tools and found significant improvement in terms of mesh accuracy. We show, on average, we can generate fully automatically a brain cell (neurons or glia) surface mesh in a couple of minutes on a laptop computer resulting in a simplified surface mesh with only around 10k nodes. The resulting meshes were used to perform diffusion MRI simulations in neurons and microglia. The code and a number of sample brain cell surface meshes have been made publicly available.

6.6.1 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 has been published 10.

6.6.2 Substructuring based FEM-BEM coupling for Helmholtz problems

A. Boisneault, M. Bonazzoli, X. Claeys, P. Marchand

This work concerns the solution of the Helmholtz equation in a medium composed of a bounded heterogeneous domain and an unbounded homogeneous one. Such problems can be expressed using classical FEM-BEM coupling techniques. We solve these coupled formulations using iterative solvers based on substructuring Domain Decomposition Methods (DDM), and aim to develop a convergence theory, with fast and guaranteed convergence. A recent article of Xavier Claeys proposed a substructuring Optimized Schwarz Method, with a nonlocal exchange operator, for Helmholtz problems on a bounded domain with classical conditions on its boundary (Dirichlet, Neumann, Robin). The variational formulation of the problem can be written as a bilinear application associated with the volume and another with the surface, for which, under certain sufficient assumptions, convergence of the DDM strategy is guaranteed. We have shown how some specific FEM-BEM coupling methods fit, or not, the previous framework, in which we consider Boundary Integral Equations (BIEs) instead of classical boundary conditions. In particular, we prove that the symmetric Costabel coupling satisfies the framework assumptions, implying that the convergence is guaranteed. This work is in the framework of OptiGPR3D Exploratory Action and has been presented at WAVES 2024 conference 22.

6.6.3 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 20.

6.6.4 High-order numerical integration on self-affine sets

P. Joly, M. Kachanovska, Z. Moitier

We construct an interpolatory high-order cubature rule to compute integrals of smooth functions over self-affine sets with respect to an invariant measure. The main difficulty is the computation of the cubature weights, which we characterize algebraically, by exploiting a self-similarity property of the integral. We propose an $h$-version and a $p$-version of the cubature, present an error analysis and conduct numerical experiments 25.

6.6.5 Cell seeding dynamics in a porous scaffold material designed for meniscus tissue regeneration

H. Jäger, E. Grosjean, S. Plunder, C. Redenbach, A. Keilmann , B. Simeon, C. Surulescu

We study the dynamics of a seeding experiment where a fibrous scaffold material is colonized by two types of cell populations. The specific application that we have in mind is related to the idea of meniscus tissue regeneration. In order to support the development of a promising replacement material, we discuss certain rate equations for the densities of human mesenchymal stem cells and chondrocytes and for the production of collagen-containing extracellular matrix. For qualitative studies, we start with a system of ordinary differential equations and refine then the model to include spatial effects of the underlying nonwoven scaffold structure. Numerical experiments as well as a complete set of parameters for future benchmarking are provided.

6.6.6 The non-intrusive reduced basis two-grid method applied to sensitivity analysis

E. Grosjean, B. Simeon

This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity analysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these two approaches may become too costly ans thus Reduced Basis Methods (RBMs) may be a viable option. We propose new NIRB two-grid algorithms for both the direct and adjoint state methods in the context of parabolic equations. The NIRB two-grid method uses the HF code solely as a “black-box”, requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline stage is time-consuming, but it is only executed once, whereas the online stage employs coarser grids and thus, is significantly less expensive than a fine HF evaluation. On the direct method, we prove on a classical model problem, the heat equation, that HF evaluations of sensitivities reach an optimal convergence rate in $L^{\infty }(0,T;H_0^1\left(\Omega \right))$, and then establish that these rates are recovered by the NIRB two-grid approximation. These results are supported by numerical simulations. We then propose a new procedure that further reduces the computational costs of the online step while only computing a coarse solution of the state equations. On the adjoint state method, we propose a new algorithm that reduces both the state and adjoint solutions. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system

6.7.1 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 has been published 11, and T. A. Vu has defended his PhD Thesis 29.

6.7.2 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 a hydroelectric gravity dam and the underlying rock, using Full Waveform Inversion. Indeed, it appears that the roughness of the dam-rock interface has an effect on the sliding stability of gravity dams. We minimize a regularized misfit cost functional by computing its shape derivative and iteratively updating the interface shape by the gradient descent method. At each iteration, we simulate time-harmonic elasto-acoustic wave propagation models, coupling linear elasticity in the solid medium with acoustics in the reservoir. Numerical results using realistic noisy synthetic data demonstrate the method ability to accurately reconstruct the dam-rock interface with a limited number of measurements and in the presence of noise. This work has been presented at Journées Scientifiques AGAP Qualité 2024 23, ICIPE 2024 24, and WAVES 2024 21.

6.7.3 Silent sources on a surface for the Helmholtz equation and decomposition of $L^2$ vector fields

L. Bratchart, H. Haddar, C.V. Guillén

We study an inverse source problem with right hand side in divergence form for the Helmholtz equation, whose underlying model can be related to weak scattering from thin interfaces. This inverse problem is not uniquely solvable, as the forward operator has infinite-dimensional kernel. We present a decomposition of (not necessarily tangent) vector fields of $L^2$-class on a closed Lipschitz surface in $R^3$, which allows one to discuss an ansatz for the solution and constraints that restore uniqueness. This work can be seen as a generalization of results in the literature dealing with the Laplace equation, but in the Helmholtz case new ties arise between the observations from each side of the surface. Our proof is based on properties of the Calderón projector on the boundary of Lipschitz domains, that we establish in a $H^{-1}\times L^2$ setting 9.

8.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria International Program

8.2.1 Visits of international scientists

Action exploratoire OptiGPR3D

Participants: Lorenzo Audibert, Marcella Bonazzoli, Houssem Haddar, Frédéric Taillade.

• Title:
  Action exploratoire OptiGPR3D (Optimal direct and inverse modeling for 3D GPR imaging in complex environments)
• Partner Institution(s):
  IDEFIX (Inria, EDF, ENSTA Paris), POEMS (CNRS, Inria, ENSTA Paris)
• Duration:
  Start: 05/2022, 4 years
• Coordinators:
  Marcella Bonazzoli (IDEFIX, Inria), Pierre Marchand (POEMS, Inria)
• Administrator:
  Inria

"Biomedical Engineering Seed Grant Program" funded by the Fondation Bettencourt Schueller

Participants: J.-R. Li, A. McSweeney-Davis, S. Sedlar.

• Title:
  Investigation of potential biomarkers to detect chronic inflammation in Multiple Sclerosis through diffusion MRI
• Partner Institution(s):
  IDEFIX (Inria, EDF, ENSTA Paris), CHU de Rennes, Univ Rennes
• Duration:
  09/2023 - 12/2025
• Coordinators:
  J.-R. Li (IDEFIX, Inria), Anne Kerbrat (CHU de Rennes)
• Administrator:
  Inria

9.1.1 Scientific events: organisation

• L. Chesnel co-organizes the seminar common to the three teams IDEFIX-MEDISIM-POEMS.
• L. Chesnel participates to the organization of the "Journées Ondes des Poètes 2024" to celebrate the 60th birthday of A.-S. Bonnet-BenDhia, E. Bécache, C. Hazard and E. Lunéville. More than 100 participants.
• M. Bonazzoli (with Christian Glusa, Pierre Marchand) organized a minisymposium at SIAM LA 2024 (Paris, May 2024).
• M. Bonazzoli and H. Montanelli organized IDEFIX days (Palaiseau, December 2024).

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.
• H. Haddar is memeber of the scientifi committee of the Conference on Mathematics of Wave Phenomena (Karlsruhe) 2025

9.1.3 Journal

9.1.4 Invited talks

• L. Chesnel, Analysis seminar of the CMAP, École Polytechnique, Palaiseau, March 2024.
• L. Chesnel, EDP seminar, UVSQ, Versailles, September 2024.
• M. Bonazzoli, CMAM-10, 10th International Conference on Computational Methods in Applied Mathematics 2024, Bonn, Germany, June 2024.
• M. Bonazzoli, DD28, 28th International Domain Decomposition Conference 2024, KAUST, Saudi Arabia, January 2024.
• M. Bonazzoli, Seminar at the Chair for Scientific Computing (SciComp), University of Kaiserslautern-Landau, Kaiserslautern, Germany, November 2024.
• M. Bonazzoli, Seminar at LAMSIN, ENIT, Tunis El-manar University, Tunis, Tunisia, June 2024.
• J.-R. Li, Numerical Methods for PDEs and Their Applications, Institut Mittag-Leffler, Djursholm, Sweden, June 2024.
• H. Haddar, Invited minisymposium talk, AIMS Conference, Abu Dhabi, 2024.
• H. Haddar, Invited Talk, Seminar EDP-CS of LMRS University Rouen, 2024
• H. Haddar, Invited talk, Conference on Inverse Problems for PDEs, honouring D. Colton, Rutgers University, 2024

9.1.5 Research administration

• M. Bonazzoli is the International partnerships Scientific Correspondent for Inria Saclay.
• M. Bonazzoli took part in Feb. 2024 to the prize committee for SMAI-GAMNI PhD Award 2024.
• M. Bonazzoli took part in Jul. 2024 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).

9.2.1 Teaching

• Doctorat: H. Haddar et L. Audibert, Inverse problems: Algorithms and Applications. Executive Education, Ecole Polytechnique.
• Bachelor: H. Montanelli, Optmization and Control, 2nd year of École Polytechnique, 40 TD hours.
• Bachelor: H. Montanelli, Introduction to Statistics, 1st year of ENSTA Paris, 12 TD hours.
• Bachelor: H. Montanelli, Introduction to Probability, 1st year of ENSTA Paris, 12 TD hours.
• Master: L. Chesnel, Analyse variationnelle des équations aux dérivées partielles, 2nd year of Ecole Polytechnique, 40 TD hours.
• Master: L. Chesnel, Modal - Modélisation mathématique par la démarche expérimentale, 2nd year of École Polytechnique, creation and supervision of projects for two groups of three 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.
• Bachelor: L. Audibert, Optimisation quadratique sous contrainte linéaire, 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.
• Master: E. Grosjean, Continuous Optimization (ENT305), 3rd year (major energy) of ENSTA Paris, 21 hours.
• Master: E. Grosjean, Continuous Optimization project, 3rd year (major energy) of ENSTA Paris (ENT306), 21 hours.

9.2.2 Supervision

• 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 defended in July 2024: T.A. Vu, One-shot inversion methods and domain decomposition (2020-2024), 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, ENSTA Paris, and P. Marchand, Inria).
• PhD in progress: V. Chenu, Machine learning for inverse scattering problems, (2024-), H. Haddar and H. Montanelli.
• MSc thesis: H. Negrel, Machine learning for inverse scattering problems, (Apr.–Sep. 2024), H. Haddar and H. Montanelli.
• 2nd year Engineer School internship: E. Jung, The LSM for random sources and penetrable objects, (May–Aug. 2024), H. Montanelli.
• 2nd year Engineer School internship: M. Leboulanger, Méthodes de décomposition de domaine pour la résolution de problèmes inverses, (May–Aug. 2024), M. Bonazzoli.
• Postdoc: M.A. Boukraa, Inverse problem methods for interface imaging: application to a concrete-rock interface for a hydroelectric dam, (2022-2024), supervisors: M. Bonazzoli, D. Vautrin, collaborators: L. Audibert, H. Haddar, F. Taillade.
• PhD in progress: M. Mathevet, Eddy current imaging with inverse problems methods. (2022-), L. Audibert and H. Haddar.
• Postdoc in progress: D.Q. Bui, Optimal inverse modeling for GPR imaging, (2023-), L. Audibert, M. Bonazzoli, H. Haddar, P. Marchand, F. Taillade
• Phd defended in November 2024: D. Lerévérend, Étude analytique et numérique de problèmes inverses en diffraction acoustique pour la conception de microphones spatiaux, H. Haddar.
• Phd defended in December 2024: F. Pourre, Using spectral signature for imaging, L. Audibert and H. Haddar.
• PhD in progress: M. Chavanne, Signature spectrale pour l'électromagnétisme, (2024-), L. Audibert and H. Haddar.
• PhD in progress: A. McSweeney, Measuring Fluid Debits in Ducts using ultrasounds, (2024-), L. Audibert and H. Haddar.
• PhD in progress: C. Hivart, LSM for inverse elastodynamic problems in concrete, (2024-), L. Audibert and H. Haddar.

9.2.3 Juries

• J.-R. Li, President of PhD jury of Deneb Boito (Biomedical Engineering, Linköping University, Sweden). 01/2024.

9.3.1 Participation in Live events

• M. Bonazzoli and L. Chesnel were volunteers at Inria stand at Fête de la Science (Institut Polytechnique de Paris), for middle and high school classes (1st day) and open to the public (2nd day), Oct. 2024.
• L. Chesnel went to an elementary school class to make an initiation to graph theory.
• M. Bonazzoli gave a science popularization talk for high school students who competed in the Paris-Saclay regional phase of Tournois Français des Jeunes Mathématiciennes et Mathématiciens (ENSTA Paris).
• M. Bonazzoli participated to speed-meetings with female high school students (Rendez-vous des Jeunes Mathématiciennes et Informaticiennes at Inria Saclay, Feb. 2024, and at ENSTA Paris, Feb. 2024), to answer their questions about the studies and career as a mathematician.

International journals

• 7 articleL.Lorenzo Audibert, H.Houssem Haddar and F.Fabien Pourre. Reconstruction of averaging indicators for highly heterogeneous media.Inverse Problems404March 2024, 045028HALDOIback to text
• 8 articleL.Lorenzo Audibert and S.Shixu Meng. Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator.Inverse Problems409July 2024, 095007HALDOIback to text
• 9 articleL.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.SIAM Journal on Mathematical Analysis5712025, 682 -- 713HALDOIback to text
• 10 articleM.Marcella Bonazzoli and X.Xavier Claeys. Multidomain FEM-BEM coupling for acoustic scattering.Journal of Integral Equations and Applications362June 2024, 129-167HALDOIback to text
• 11 articleM.Marcella Bonazzoli, H.Houssem Haddar and T.-A.Tuan-Anh Vu. On the convergence analysis of one-shot inversion methods.SIAM Journal on Applied Mathematics8462024, 2440-2475HALDOIback to text
• 12 articleA.-S.Anne-Sophie Bonnet-Ben, L.Lucas Chesnel and M.Mahran Rihani. Maxwell's equations with hypersingularities at a negative index material conical tip.Pure and Applied AnalysisDecember 2024. In press. HALback to text
• 13 articleM. A.Mohamed Aziz Boukraa, L.Laëtitia Caillé and F.Franck Delvare. Fading regularization method for an inverse boundary value problem associated with the biharmonic equation.Journal of Computational and Applied Mathematics457September 2024, 116285HALDOI
• 14 articleL.Laurent Bourgeois and L.Lucas Chesnel. Generalized impedance boundary conditions with vanishing or sign-changing impedance.SIAM Journal on Mathematical Analysis563June 2024, 4223-4251HALDOIback to text
• 15 articleF.Fioralba Cakoni, H.Houssem Haddar and T.-P.Thi-Phong Nguyen. Fast Imaging of Local Perturbations in a Unknown Bi-Periodic Layered Medium.Journal of Computational Physics501March 2024, 112773HALDOIback to text
• 16 articleF.Fioralba Cakoni, H.Houssem Haddar and D.Dana Zilberberg. An algorithm for computing scattering poles based on dual characterization to interior eigenvalues.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences4802292June 2024HALDOIback to text
• 17 articleL.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.Journal of Spectral TheoryApril 2024HALback to text
• 18 articleJ.Josselin Garnier, H.Houssem Haddar and H.Hadrien Montanelli. The linear sampling method for data generated by small random scatterers.SIAM Journal on Imaging Sciences1742024, 2142-2173HALDOIback to text
• 19 articleM.Mojtaba Lashgari, Z.Zheyi Yang, M. O.Miguel O Bernabeu, J.-R.Jing-Rebecca Li and A. F.Alejandro F Frangi. SpinDoctor-IVIM: A virtual imaging framework for intravoxel incoherent motion MRI.Medical Image Analysis99October 2024, 103369HALDOIback to text
• 20 articleH.Hadrien Montanelli, F.Francis Collino and H.Houssem Haddar. Computing singular and near-singular integrals over curved boundary elements: The strongly singular case.SIAM Journal on Scientific Computing4662024, A3756-A3778HALDOIback to text

International peer-reviewed conferences

• 21 inproceedingsM.Mohamed Aziz Boukraa, L.Lorenzo Audibert, M.Marcella Bonazzoli, H.Houssem Haddar and D.Denis Vautrin. High-Resolution Seismic Imaging for Dam-Rock Interface using Full-Waveform Inversion.Proceedings of The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation.WAVES 2024 - 16th International Conference on Mathematical and Numerical Aspects of Wave PropagationBerlin, GermanyJune 2024HALDOIback to text
• 22 inproceedingsA.Antonin Boisneault, M.Marcella Bonazzoli, X.Xavier Claeys and P.Pierre Marchand. Substructuring based FEM-BEM coupling for Helmholtz problems.Proceedings of The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation.WAVES 2024 - 16th International Conference on Mathematical and Numerical Aspects of Wave PropagationBerlin, Germany2024HALDOIback to text
• 23 inproceedingsM. A.Mohamed Aziz Boukraa, L.Lorenzo Audibert, M.Marcella Bonazzoli, H.Houssem Haddar and D.Denis Vautrin. Imaging of Dam-Foundation contact by Full Waveform Inversion with a shape optimization approach.E3S Web Conf.Journées Scientifiques AGAP Qualité 2024504Poitiers, FranceMarch 2024, 04002HALDOIback to text
• 24 inproceedingsM. A.Mohamed Aziz Boukraa, M.Marcella Bonazzoli, H.Houssem Haddar, L.Lorenzo Audibert and D.Denis Vautrin. Imaging a dam-rock interface with inversion of a full elastic-acoustic model.ICIPE 2024 - 11th International Conference on Inverse Problems in Engineering: Theory and PracticeBuzios - Rio de Janeiro, BrazilJune 2024HALback to text
• 25 inproceedingsP.Patrick Joly, M.Maryna Kachanovska and Z.Zoïs Moitier. Efficient methods for the solution of boundary integral equations on fractal antennas.https://doi.org/10.17617/3.MBE4AAWAVES 2024 - 16th International Conference on Mathematical and Numerical Aspects of Wave PropagationBerlin, GermanyJune 2024HALDOIback to text

Conferences without proceedings

• 26 inproceedingsM.Monique Dauge, S.Stéphane Balac, G.Gabriel Caloz and Z.Zoïs Moitier. Whispering Gallery Modes and Frequency Combs: Two excursions in the world of photonic resonators.Book of Abstracts, The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2024)Waves 2024 - The 16th International Conference on Mathematical and Numerical Aspects of Wave PropagationBerlin, GermanyEdmond2024HALDOI

Scientific book chapters

• 27 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, 61-68HALDOI

Doctoral dissertations and habilitation theses

• 28 thesisF.Fabien Pourre. Construction and analysis of spectral signatures for defects in complex media.Institut polytechnique de ParisDecember 2024HAL
• 29 thesisT.-A.Tuan-Anh Vu. One-shot inversion methods and domain decomposition.Institut Polytechnique de ParisJuly 2024HALback to text

Reports & preprints

• 30 miscL.Lorenzo Audibert, H.Houssem Haddar and F.Fabien Pourre. Averaged Steklov Eigenvalues, Inside Outside Duality and Application to Inverse Scattering.January 2025HALback to text
• 31 miscL.Lucas Chesnel, H.Houssem Haddar, H.Hongjie Li and J.Jingni Xiao. Examples of non-scattering inhomogeneities.December 2024HALback to text
• 32 miscP.Patrick Joly, M.Maryna Kachanovska and Z.Zoïs Moitier. High-order numerical integration on self-affine sets.September 2024HAL
• 33 miscY.Yuyuan Zhou, L.Lorenzo Audibert, S.Shixu Meng and B.Bo Zhang. Exploring low-rank structure for an inverse scattering problem with far-field data.December 2024HALback to text