2.1 Introduction

The focus of our research is on the development of novel parallel numerical algorithms and tools appropriate for state-of-the-art mathematical models used in complex scientific applications, and in particular numerical simulations. The proposed research program is by nature multi-disciplinary, interweaving aspects of applied mathematics, computer science, as well as those of several specific applications, as porous media flows, elasticity, wave propagation in multi-scale media, molecular simulations, and inverse problems.

Our first objective is to develop numerical methods and tools for complex scientific and industrial applications that will enhance their scalable execution on the emergent heterogeneous hierarchical models of massively parallel machines. Our second objective is to integrate the novel numerical algorithms into a middle-layer that will hide as much as possible the complexity of massively parallel machines from the users of these machines.

3.1 Overview

The research described here is directly relevant to several steps of the numerical simulation chain. Given a numerical simulation that was expressed as a set of differential equations, our research focuses on mesh generation methods for parallel computation, novel numerical algorithms for linear algebra, as well as algorithms and tools for their efficient and scalable implementation on high performance computers. The validation and the exploitation of the results is performed with collaborators from applications and is based on the usage of existing tools. In summary, the topics studied in our group are the following:

• Numerical methods and algorithms
  • Mesh generation for parallel computation

  • Solvers for numerical linear algebra:

    domain decomposition methods, preconditioning for iterative methods

  • Computational kernels for numerical linear algebra
  • Tensor computations for high dimensional problems
• Validation on numerical simulations and other numerical applications

3.2 Domain specific language - parallel FreeFem

In the engineering, researchers, and teachers communities, there is a strong demand for simulation frameworks that are simple to install and use, efficient, sustainable, and that solve efficiently and accurately complex problems for which there are no dedicated tools or codes available. In our group we develop FreeFem++, a user-dedicated language for solving PDEs. The goal of FreeFem++ is not to be a substitute for complex numerical codes, but rather to provide an efficient and relatively generic tool for:

• getting a quick answer to a specific problem,
• prototyping the resolution of a new complex problem.

The current users of FreeFem++ are mathematicians, engineers, university professors, and students. In general for these users the installation of public libraries as MPI, MUMPS, Ipopt, Blas, lapack, OpenGL, fftw, scotch, PETSc, SLEPc is a very difficult problem. For this reason, the authors of FreeFem++ have created a user friendly language, and over years have enriched its capabilities and provided tools for compiling FreeFem++ such that the users do not need to have special knowledge of computer science. This leads to an important work on porting the software on different emerging architectures.

Today, the main components of parallel FreeFem++ are:

• definition of a coarse grid,
• splitting of the coarse grid,
• mesh generation of all subdomains of the coarse grid, and construction of parallel data structures for vectors and sparse matrices from the mesh of the subdomain,
• discretization by a chosen numerical method,
• call to a linear solver,
• analysis of the result.

All these components are parallel, except for the last point which is not in the focus of our research. However for the moment, the parallel mesh generation algorithm is very simple and not sufficient, for example it addresses only polygonal geometries. Having a better parallel mesh generation algorithm is one of the goals of our project. In addition, in the current version of FreeFem++, the parallelism is not hidden from the user, it is done through direct calls to MPI. Our goal is also to hide all the MPI calls in the specific language part of FreeFem++. In addition to these in-house domain decomposition methods, FreeFem++ is also linked to PETSc solvers which enables an easy use of third parties parallel multigrid methods.

3.3 Solvers for numerical linear algebra

Iterative methods are widely used in industrial applications, and preconditioning is the most important research subject here. Our research considers domain decomposition methods 6 and iterative methods and its goal is to develop solvers that are suitable for parallelism and that exploit the fact that the matrices are arising from the discretization of a system of PDEs on unstructured grids.

One of the main challenges that we address is the lack of robustness and scalability of existing methods as incomplete LU factorizations or Schwarz-based approaches, for which the number of iterations increases significantly with the problem size or with the number of processors. This is often due to the presence of several low frequency modes that hinder the convergence of the iterative method. To address this problem, we study different approaches for dealing with the low frequency modes as coarse space correction in domain decomposition or deflation techniques. We minimize the memory footprint and communication avoiding algorithms by leveraging mixed precision arithmetic.

We also focus on developing boundary integral equation methods that would be adapted to the simulation of wave propagation in complex physical situations, and that would lend themselves to the use of parallel architectures. The final objective is to bring the state of the art on boundary integral equations closer to contemporary industrial needs. From this perspective, we investigate domain decomposition strategies in conjunction with boundary element method as well as acceleration techniques (H-matrices, FMM and the like) that would appear relevant in multi-material and/or multi-domain configurations. Our work on this topic also includes numerical implementation on large scale problems, which appears as a challenge due to the peculiarities of boundary integral equations.

3.4 Computational kernels for numerical linear and multilinear algebra

The design of new numerical methods that are robust and that have well proven convergence properties is one of the challenges addressed in Alpines. Another important challenge is the design of parallel algorithms for the novel numerical methods and the underlying building blocks from numerical linear algebra. The goal is to enable their efficient execution on a diverse set of node architectures and their scaling to emerging high-performance clusters with an increasing number of nodes.

Increased communication cost is one of the main challenges in high performance computing that we address in our research by investigating algorithms that minimize communication, as communication avoiding algorithms. The communication avoiding algorithmic design is an approach originally developed in our group since more than ten years (initially in collaboration with researchers from UC Berkeley and CU Denver). While our first results concerned direct methods of factorization as LU or QR factorizations, our recent work focuses on designing robust algorithms for computing the low rank approximation of a matrix or a tensor. We focus on both deterministic and randomized approaches.

Our research also focuses on solving problems of large size that feature high dimensions as in molecular simulations. The data in this case is represented by objects called tensors, or multilinear arrays. The goal is to design novel tensor techniques to allow their effective compression, i.e. their representation by simpler objects in small dimensions, while controlling the loss of information. The algorithms are aiming to being highly parallel to allow to deal with the large number of dimensions and large data sets, while preserving the required information for obtaining the solution of the problem.

4.1 Radiative transfer in the atmosphere

The Hierarchical matrix acceleration techniques implemented in our parallel Htool 7.1.3 library, which is interfaced with FreeFEM, allows to efficiently solve the radiative transfer equations with complexity n log(n), where n is the number of vertices of the physical domain. The method can be used to study the change of temperature in the atmosphere, and in particular the effect of clouds and greenhouse gases 7, 13.

4.2 Inverse problems

4.3 Numerical methods for time harmonic wave propagation problems

Acoustic, electromagnetic and elastic linear waves are ubiquitous phenomena in science and engineering. The numerical simulation of their propagation and interaction is a core task in areas like medical imaging, seismic remote sensing, design of electronic devices, atmospheric particle scattering, radar and sonar modelling, etc. The design of appropriate numerical schemes is challenging because of approximation issues arising from the fact that the solutions are highly oscillatory, which usually require a large computational effort to produce a meaningful numerical approximation. In addition, common formulations suffer from stability issues, such as the numerical dispersion (also known as pollution effect), and sign-indefiniteness, which typically limit the achievable accuracy. The variety of different applications and the challenges of their numerical simulations produces a great research effort in trying to design new efficient schemes: new approximation spaces using degrees of freedom sparingly (wave based and Trefftz methods), preconditionners for solving large linear systems (e.g. by domain decomposition), new formulations, acceleration techniques, etc.

4.4 Molecular simulations

Molecular simulation is one of the most dynamic areas of scientific computing. Its field of application is very broad, ranging from theoretical chemistry and drug design to materials science and nanotechnology. It provides many challenging problems to mathematicians, and computer scientists.

In the context of the ERC Synergy Grant EMC2 we address several important limitations of state of the art molecular simulation. In particular, the simulation of very large molecular systems, or smaller systems in which electrons interact strongly with each other, remains out of reach today. In an interdisciplinary collaboration between chemists, mathematicians and computer scientists, we focus on developing a new generation of reliable molecular simulation algorithms and software.

5.1 Impact of research results

Our activities in scientific computing help in the design and optimization of non fossil energy resources. Here are three examples we are aware of and there are many others:

• the startup Airthium specialized in decarbonized energy uses our free parallel finite element software FreeFEM.
• The French electricity company EDF has integrated our domain decomposition methods GenEO via the open source library HPDDM in its simulation software Salomé in order to be able to perform large scale computations related to nuclear reactors.
• In the context of climate change, our work on radiative transfer studies the change of temperature in the atmosphere, in particular due to the effect of clouds and greenhouse gases.

6.1 Awards

ERC Synergy 2024

PSINumScat (Phase Space Inspired Numerical Methods for High-Frequency Scattering)

This Synergy project has been granted to Pierre-Henri Tournier, (the other leaders are Euan A. Spence, professor at the University of Bath, and Jeffrey Galkowski, professor at University College London), aims to design, analyze and implement in free software new methods for the numerical resolution of high-frequency acoustic and electromagnetic wave propagation problems. These methods will be faster, more reliable, and more widely applicable than current state-of-the-art methods, leveraging techniques from fundamental mathematics specifically designed to study high-frequency problems. These techniques come from the field of semi-classical analysis and are based on the study of functions in phase space (the combination of physical space and frequency space). Since numerical analysis and semi-classical analysis of high-frequency wave propagation problems are mature areas of mathematics, it is surprising that these two areas operate largely in isolation from each other. None of the extensive knowledge of high-frequency waves gained from semi-classical analysis has been used in the design of numerical methods for these problems. The project will take advantage of this untapped potential and make the resulting methods available to the scientific community in the free software for multi-physics numerical simulation FreeFEM developed at the Laboratory J.L. Lions-INRIA team Alpines.

7.1 New software

8.1 A robust and adaptive GenEO-type domain decomposition preconditioner for H(curl) problems in general non-convex three-dimensional geometries

In 1, we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for H(curl) problems. It is well known that convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement a near-kernel space made from the gradient of scalar functions. The new class of preconditioner is inspired by the idea of subspace decomposition, but based on spectral coarse spaces, and is specially designed for curl-conforming discretisations of Maxwell's equations in heterogeneous media on general domains which may have holes. Our approach has wider applicability and theoretical justification than the well-known Hiptmair-Xu auxiliary space preconditioner, with results extending to the variable coefficient case and non-convex domains at the expense of a larger coarse space.

8.2 Coarse spaces for non-symmetric two-level preconditioners based on local generalized eigenproblems

In 27, we present an analysis of adaptive coarse spaces which is quite general since it applies to symmetric and non symmetric problems, to symmetric preconditioners such the additive Schwarz method (ASM) and to the non-symmetric preconditioner restricted additive Schwarz (RAS), as well as to exact or inexact subdomain solves. The coarse space is built by solving generalized eigenvalues in the subdomains and applying a well-chosen operator to the selected eigenvectors.

8.3 A Robust Two-Level Schwarz Preconditioner For Sparse Matrices

In 24 daas:hal-04381509, we introduce a fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices. The preconditioner is analyzed for symmetric positive definite (SPD) matrices. For those matrices, the coarse space is constructed based on approximating two local subspaces in each subdomain. These subspaces are obtained by approximating a number of eigenvectors corresponding to dominant eigenvalues of two judiciously posed generalized eigenvalue problems. The number of eigenvectors can be chosen to control the condition number. For general sparse matrices, the coarse space is constructed by approximating the image of a local operator that can be defined from information in the coefficient matrix. The connection between the coarse spaces for SPD and general matrices is also discussed. Numerical experiments show the great effectiveness of the proposed preconditioners on matrices arising from a wide range of applications. The set of matrices includes SPD, symmetric indefinite, nonsymmetric, and saddle-point matrices. In addition, we compare the proposed preconditioners to the state-of-the-art domain decomposition preconditioners.

8.4 Microwave tomographic imaging of shoulder injury

One of the most challenging shoulder injuries is rotator cuff tear, which increases with aging and particularly happens among athletes. These tears cause pain and highly affect the functionality of the shoulder. The motivation of this work 17 is to detect these tears by microwave tomographic imaging. This imaging method requires the solution of an inverse problem based on a minimization algorithm, with successive solutions of a direct problem. The direct problem consists in solving the time-harmonic Maxwell’s equations discretized with a Nédélec edge finite element method, and we make use of an ORAS Schwarz domain decomposition preconditioner to solve it efficiently in parallel. We propose a wearable imaging system design with 96 antennas distributed over two fully-circular and two half-circular layers, adapted to the real shoulder structure and designed to surround it partially. We test the imaging system numerically using realistic CAD models for shoulder profile and bones, including humerus and scapula, and taking into account the dielectric properties of the different tissues (bone, tendon, muscle, skin, synovial fluid). The reconstructed images of the shoulder joint obtained using noisy synthetic data show that the proposed imaging system is capable of accurately detecting and localizing large (5mL) and partial (1mL) rotator cuff tears. The reconstruction takes 20 minutes on 480 computing cores and shows great promise for rapid diagnosis or medical monitoring. In a second step, we take advantage of numerical modeling to try and optimize the number of antennas in the imaging system, achieving a drastic reduction from 96 to 32 antennas while still being able to detect the injury in the difficult partial tear case, reducing the computing time from 20 to 12 minutes.

In 16, 21, we use the previous forward numerical modeling techniques to generate a generalizable dataset of scattering parameters in order to bypass real-world data collection challenges. We then explore the use of machine learning (ML) algorithms for the fast detection of shoulder tendon injuries. The corresponding data of various healthy and injured models are categorized into two classes. We use a support vector machine (SVM) to differentiate between injured and healthy shoulder models. This approach is more efficient in terms of required memory resources and computing time compared with traditional imaging methods, and we manage to achieve an accuracy of 100%.

8.5 New developments in FreeFEM

Work has been started to ease the parallelization of a user's sequential code. In this setting, the parallelization of the assembly and solution steps is entirely hidden to the user, and a sequential code can be parallelized with only very few changes in the script. For example, PETSc can easily be used to solve the linear system stemming from a coupled problem with a fieldsplit preconditioner.

The FreeFEM software now includes the GenEO for saddle-point 11 examples with PCHPDDM in PETSc.

FreeFEM can now run Markdown (.md) files as well as .edp files. The Markdown syntax can be used to document FreeFEM scripts. When running a .md file, FreeFEM will execute the code contained in the FreeFEM code blocks. Documented Markdown examples in the distribution can be viewed on the documentation website. Markdown can also be previewed with Visual Studio Code, with the FreeFEM VS Code extension providing syntax highlighting for FreeFEM code blocks.

8.6 Reduced models for highly oscillating differential equations

In this part we are concerned with solving Vlasov equation involving several time scales. This work shows that finding and solving reduced models, though complicated to derive, turns out to be very useful for reducing the computational cost of an equation. In addition, using the parareal algorithm, a method performing parallel computing in time, allows to accelerate the computations.

Thus, in 19 we derive a reduced first-order model from a two-scale asymptotic expansion in a small parameter, in order to approximate the solution of a stiff differential equation. The problem of interest is a multi-scale Newton-Lorentz equation modeling the dynamics of a charged particle under the influence of a linear electric field and of a perturbed strong magnetic field. First, we show that in short times, the first-order model provides a much better approximation than the zero-order one. Then, we propose a volume-preserving method using a particular splitting technique to solve numerically the first-order model. Finally, it turns out that the first-order model does not systematically provide a satisfactory approximation in long times. To overcome this issue, we implement a recent strategy based on the Parareal algorithm, in which the first-order approximation is used for the coarse solver. This approach allows to perform efficient and accurate long-time simulations for any small parameter. Numerical results for two realistic Penning traps are provided to support these statements.

Second, we extended the previous idea to a more general equation, which models the motion of a charged particle in a tokamak-like magnetic field, dependent on space and time. Thus, by symbolic computations, we obtained first-order equations approximating stiff differential equations in the general case. This is a work in progress.

Finally, during the internship of Crédo Fanou (stage de M2) we studied theoretical results on the error estimation of reduced models, obtained by averaging, for generic stiff differential equations. We applied these results to Newton-Lorentz equation in order to obtain such error estimation. Elarif Djabir (stage de M1) performed, during his internship, numerical simulations of such stiff equations, in order to study the error bounds and their optimality. This is ongoing work.

8.7 Stable Trefftz approximation of Helmholtz solutions using evanescent plane waves

Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic.

In 26 we extend the analysis of 12 from two to three space dimensions. We show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. A non-trivial challenge in 3D was the parametrization of the complex direction set. Our approach involves defining a complex-valued reference direction and then consider its rigid-body rotations via Euler angles. Then, by generalizing the Jacobi–Anger identity to complex-valued directions, we prove that any solution of the Helmholtz equation on a three-dimensional ball can be written as a continuous superposition of evanescent plane waves in a stable way. Our numerical results showcase a great improvement of our proposed method on the achievable accuracy compared to standard method.

We are currently exploiting these new results to construct discontinuous Trefftz schemes based on plane-wave approximation spaces, which was the subject of the internship of Vincent Robert . In addition, and this is very novel with respect to the current literature, we are able to obtain conforming Galerkin discretization from plane-wave approximation spaces, which opens up to a whole new approach for Trefftz methods. Finally, we are working on new integral representations of any Helmholtz solution in a smooth convex domain, thereby generalyzing the above works on disk and balls.

8.8 Domain decomposition methods for random multi-scale Helmholtz problems arising in ultrasound imaging

The development of new quantitative ultrasound imaging algorithms, which aim to reconstruct a map of the local ultrasound speed in the medium, requires a validation process that can be achieved through numerical simulation. With this application in mind, we consider the scattering of plane waves by a tissue-mimicking medium where up to a hundred unresolved scatterers per wavelength are randomly distributed throughout the medium. The domains (about a hundred wavelengths in size) require billion degrees of freedom in a simulation, which corresponds to the state of the art in terms of direct numerical simulation capacity.

In the internship of Chris Teku , we investigated the efficiency and scalability of one-level and two-levels domain decomposition techniques to accurately solve the full scale model. The primary objective was to validate quantitative stochastic homogenization results obtained in 31, particularly the asymptotic expansions of the scattered field with respect to the size of the scatterers. This is still on-going work.

8.9 Parallel approximation of the exponential of Hermitian matrices

In 9, we consider a rational approximation of the exponential function to design an algorithm for computing matrix exponential in the Hermitian case. Using partial fraction decomposition, we obtain a parallelizable method, where the computation reduces to independent resolutions of linear systems. We analyze the effects of rounding errors on the accuracy of our algorithm. We complete this work with numerical tests showing the efficiency of our method and a comparison of its performances with Krylov algorithms.

8.10 A finite element toolbox for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

In 20, we present a finite element toolbox for the computation of Bogoliubov-de Gennes modes used to assess the linear stability of stationary solutions of the Gross-Pitaevskii (GP) equation. Applications concern one (single GP equation) or two-component (a system of coupled GP equations) Bose-Einstein condensates in one, two and three dimensions of space. An implementation using the free software FreeFem++ is distributed with this paper. For the computation of the GP stationary (complex or real) solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicated libraries for the associated eigenvalue problem. Mesh adaptivity is proved to considerably reduce the computational time for cases implying complex vortex states. Programs are validated through comparisons with known theoretical results for simple cases and 97 numerical results reported in the literature.

8.11 Mathematics and Finite Element Discretizations of Incompressible Navier-Stokes Flows

With : C. Bernardi, V. Girault, F. Hecht, P.-A. Raviart, B. Riviere.

This book in final preparation (2024) is a revised, updated, and augmented version of the out-of-print classic book “Finite Element Methods for Navier–Stokes Equations" by Girault and Raviart published by Springer in 1986 [GR]. The incompressible Navier–Stokes equations model the flow of incom- pressible Newtonian fluids and are used in many practical applications, including computational fluid dynamics. In addition to the basic theoretical analysis, this book presents a fairly exhaustive treatment of the up-to-date finite element discretizations of incompressible Navier–Stokes equa- tions and a variety of numerical algorithms used in their computer implementation. It covers the cases of standard and non-standard boundary conditions and their numerical discretizations via the finite element methods. Both conforming and non-conforming finite elements are examined in detail, as well as their stability or instability. The topic of time-dependent Navier–Stokes equa- tions, which was missing from [GR], is now presented in several chapters. In the same spirit as [GR], we have tried as much as possible to make this book self-contained and therefore we have either proved or recalled all the theoretical results required. This book can be used as at textbook for advanced graduate students.

8.12 Randomized Householder QR

With Laura Grigori, Edouard Timsit

In 8, we introduce a randomized Householder QR factorization (RHQR). This factorization can be used to obtain a well conditioned basis of a set of vectors and thus can be employed in a variety of applications. We discuss in particular the usage of this randomized Householder factorization in the Arnoldi process. Numerical experiments show that RHQR produces a well conditioned basis and an accurate factorization. We observe that for some cases, it can be more stable than Randomized Gram-Schmidt (RGS) in both single and double precision.

8.13 Randomized Implicitly Restarted Arnoldi method for the non-symmetric eigenvalue problem

With Jean-Guillaume de Damas, Laura Grigori

In  25 we introduce a randomized algorithm for solving the non-symmetric eigenvalue problem, referred to as randomized Implicitly Restarted Arnoldi (rIRA). This method relies on using a sketchorthogonal basis during the Arnoldi process while maintaining the Arnoldi relation and exploiting a restarting scheme to focus on a specific part of the spectrum. We analyze this method and show that it retains useful properties of the Implicitly Restarted Arnoldi (IRA) method, such as restarting without adding errors to the Ritz pairs and implicitly applying polynomial filtering. Experiments are presented to validate the numerical efficiency of the proposed randomized eigenvalue solver.

8.14 PDE-constrained optimization within FreeFEM

With Gontran Lance, and Emmanuel Trélat

The book  22 is aimed at students and researchers who want to learn how to efficiently solve constrained optimization problems involving partial differential equations (PDE) using the FreeFEM software. PDE-constrained optimization problems are frequently encountered in many academic and industrial contexts. Readers should have a basic knowledge of the analysis and numerical solution of partial differential equations using finite element methods, optimization, algorithms and numerical implementation.

9.1 Bilateral contracts with industry

Participants: F. Nataf, B. Antoine dit Urban.

• Contract with IFPEN and ONERA, January 2025 - December 2027, that funds half of the Phd thesis of Brieuc Antoine dit Urban on "Méthodes d’apprentissage pour les solveurs linéaires et préconditionneurs pour matrices creuses" . Supervisor Frédéric Nataf , T. Faney and E. Martin.

10.1 European initiatives

10.2 National initiatives

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

12.1 Major publications

• 1 miscN.Niall Bootland, V.Victorita Dolean, F.Frédéric Nataf and P.-H.Pierre-Henri Tournier. A robust and adaptive GenEO-type domain decomposition preconditioner for $𝐇\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$ problems in general non-convex three-dimensional geometries.November 2023HALback to text
• 2 inproceedingsS.Sahar Borzooei, C.Claire Migliaccio, V.Victorita Dolean, P.-H.Pierre-Henri Tournier and C.Christian Pichot. High-Performance Numerical Modeling for Detection of Rotator Cuff Tear.IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science MeetingPortland, Oregon, United StatesIEEESeptember 2023, 1879-1880HALDOIback to text
• 3 inproceedingsS.Sahar Borzooei, P.-H.Pierre-Henri Tournier, C.Claire Migliaccio, V.Victorita Dolean and C.Christian Pichot. Microwave tomographic imaging of shoulder injury.Conférence EUCAP 2023 - 17th European Conference on Antennas and PropagationFlorence, ItalyMarch 2023, 5HALback to text
• 4 miscH. A.Hussam Al Daas, P.Pierre Jolivet, F.Frédéric Nataf and P.-H.Pierre-Henri Tournier. A Robust Two-Level Schwarz Preconditioner For Sparse Matrices.January 2024HAL
• 5 articleM.Marion Darbas, J.Juliette Leblond, J.-P.Jean-Paul Marmorat and P.-H.Pierre-Henri Tournier. Numerical resolution of the inverse source problem for EEG using the quasi-reversibility method.Inverse Problems39112023, 115003HALDOIback to text
• 6 bookV.Victorita Dolean, P.Pierre Jolivet and F.Frédéric Nataf. An Introduction to Domain Decomposition Methods: algorithms, theory and parallel implementation.SIAM2015back to text
• 7 articleF.François Golse, F.Frédéric Hecht, O.Olivier Pironneau, D.Didier Smets and P.-H.Pierre-Henri Tournier. Radiative Transfer For Variable 3D Atmospheres.Journal of Computational Physics475February 2023, 111864HALDOIback to text
• 8 miscL.Laura Grigori and E.Edouard Timsit. Randomized Householder QR.July 2023HALback to text
• 9 miscF.Frédéric Hecht, S.-M.Sidi-Mahmoud Kaber, L.Lucas Perrin, A.Alain Plagne and J.Julien Salomon. Parallel approximation of the exponential of Hermitian matrices.June 2023HALback to text
• 10 articleF.F. Hecht. New development in FreeFem++.J. Numer. Math.203-42012, 251--265
• 11 articleF.Frédéric Nataf and P.-H. H.Pierre-Henri H Tournier. A GenEO Domain Decomposition method for Saddle Point problems.Comptes Rendus. Mécanique351S1March 2023, 1-18HALDOIback to text
• 12 articleE.Emile Parolin, D.Daan Huybrechs and A.Andrea Moiola. Stable approximation of Helmholtz solutions in the disk by evanescent plane waves.ESAIM: Mathematical Modelling and Numerical Analysis576October 2023, 3499-3536In press. HALDOIback to text
• 13 miscO.Olivier Pironneau and P.-H.Pierre-Henri Tournier. Reflective Conditions for Radiative Transfer in Integral Form with Compressed H-Matrices.March 2023HALback to text
• 14 articleG.Georges Sadaka, V.Victor Kalt, I.Ionut Danaila and F.Frédéric Hecht. A finite element toolbox for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates.Computer Physics Communications294January 2024, 108948HALDOI
• 15 articleN.Nicole Spillane, V.Victorita Dolean, P.Patrice Hauret, F.Frédéric Nataf, C.Clemens Pechstein and R.Robert Scheichl. Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps.Numer. Math.12642014, 741--770URL: http://dx.doi.org/10.1007/s00211-013-0576-yDOI

12.2 Publications of the year

12.3 Cited publications

• 29 articleF.F Assous, M.M Kray, F.F Nataf and E.E Turkel. Time-reversed absorbing condition: application to inverse problems.Inverse Problems2762011, 065003URL: http://stacks.iop.org/0266-5611/27/i=6/a=065003back to text
• 30 articleF.Franck Assous and F.Frédéric Nataf. Full-waveform redatuming via a TRAC approach: a first step towards target oriented inverse problem.Journal of Computational Physics4402021, 110377back to text
• 31 unpublishedJ.Josselin Garnier, L.Laure Giovangigli, Q.Quentin Goepfert and P.Pierre Millien. Scattered wavefield in the stochastic homogenization regime.September 2023, working paper or preprintHALback to text
• 32 articleP.-H.P.-H. Tournier, I.I. Aliferis, M.M. Bonazzoli, M.M. de Buhan, M.M. Darbas, V.V. Dolean, F.F. Hecht, P.P. Jolivet, I. E.I. El Kanfoud, C.C. Migliaccio, F.F. Nataf, C.Ch. Pichot and S.S. Semenov. Microwave tomographic imaging of cerebrovascular accidents by using high-performance computing.Parallel Computing852019, 88-97URL: https://www.sciencedirect.com/science/article/pii/S0167819118301042DOIback to text
• 33 articleP.-H.Pierre-Henri Tournier, M.Marcella Bonazzoli, V.Victorita Dolean, F.Francesca Rapetti, F.Frederic Hecht, F.Frederic Nataf, I.Iannis Aliferis, I.Ibtissam El Kanfoud, C.Claire Migliaccio, M.Maya de Buhan, M.Marion Darbas, S.Serguei Semenov and C.Christian Pichot. Numerical Modeling and High-Speed Parallel Computing: New Perspectives on Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring.IEEE Antennas and Propagation Magazine5952017, 98-110DOIback to text
• 34 articleP.-H.Pierre-Henri Tournier, P.Pierre Jolivet, V.Victorita Dolean, H. S.Hossein S. Aghamiry, S.Stéphane Operto and S.Sebastian Riffo. 3D finite-difference and finite-element frequency-domain wave simulation with multilevel optimized additive Schwarz domain-decomposition preconditioner: A tool for full-waveform inversion of sparse node data sets.GEOPHYSICS8752022, T381-T402DOIback to text