2025‌Activity reportProject-TeamALPINES‌​‌

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.

7.1​​​‌ Latest software developments

7.2 Open data‌

Non applicable

8.1 Numerical shape​​​‌ and topology optimization of‌ regions supporting the boundary‌​‌ conditions of a physical​​ problem

With: Eric Bonnetier,​​​‌ Carlos Brito-Pacheco, Charles Dapogny‌ , Rafael Estevez.

The‌​‌ article 34 deals with​​ a particular class of​​​‌ shape and topology optimization‌ problems: the optimized design‌​‌ is a region $\mathrm{G}$ of the boundary $\mathrm{\partial ‌}\Omega$ of a given‌ domain $\Omega$, which‌​‌ supports a particular type​​ of boundary conditions in​​​‌ the state problem characterizing‌ the physical situation. In‌​‌ our analyses, we develop​​ adapted versions of the​​​‌ notions of shape and‌ topological derivatives, which are‌​‌ classically tailored to functions​​ of a “bulk” domain.​​​‌ This leads to two‌ complementary notions of derivatives‌​‌ for a quantity of​​ interest $J\left(\mathrm{G‌}\right)$ depending on a‌ region $G\subset \mathrm{\partial ‌‌}\Omega$ on the one​​ hand, we elaborate on​​​‌ the boundary variation method‌ of Hadamard for evaluating‌​‌ the sensitivity of $\mathrm{J}\left(G\right)$ with​​​‌ respect to “small” perturbations‌ of the boundary of‌​‌ $G$ within $\partial \mathrm{\Omega ‌}$. On the other​ hand, we use techniques​‌ from asymptotic analysis to​​ appraise the sensitivity of​​​‌ $J(G)$ with respect to the​‌ addition of a new​​ connected component to the​​​‌ region $G$ , shaped​ as a “small” surface​‌ disk. The calculation of​​ both types of derivatives​​​‌ raises original difficulties, which​ are closely related to​‌ the weakly singular behavior​​ of the solution to​​​‌ a boundary value problem​ at the points of​‌ $\partial \Omega$ where the​​ boundary conditions change types.​​​‌ These aspects are carefully​ detailed in a simple​‌ mathematical setting based on​​ the conductivity equation. We​​​‌ notably propose formal arguments​ to calculate our derivatives​‌ with a minimum amount​​ of technicality, and we​​​‌ show how they can​ be generalized to handle​‌ more intricate problems, arising​​ for instance in the​​​‌ physical contexts of acoustics​ and structural mechanics, respectively​‌ governed by the Helmholtz​​ equation and the linear​​​‌ elasticity system. In numerical​ applications, our derivatives are​‌ incorporated into a recent​​ algorithmic framework for tracking​​​‌ arbitrarily dramatic motions of​ a region G within​‌ a fixed ambient surface,​​ which combines the level​​​‌ set method with remeshing​ techniques to offer a​‌ clear, body-fitted discretization of​​ the evolving region. Finally,​​​‌ various 3d numerical examples​ are presented to illustrate​‌ the salient features of​​ our analysis.

8.2 Distributionally​​​‌ robust shape and topology​ optimization

With: Charles Dapogny​‌ , Julien Prando, Boris​​ Thibert.

In 24,​​​‌ we aim to introduce​ the paradigm of distributional​‌ robustness from the field​​ of convex optimization to​​​‌ tackle optimal design problems​ under uncertainty. We consider​‌ realistic situations where the​​ physical model, and thereby​​​‌ the cost function of​ the design to be​‌ minimized depend on uncertain​​ parameters. The probability distribution​​​‌ of the latter is​ itself known imperfectly, through​‌ a nominal law, reconstructed​​ from a few observed​​​‌ samples. The distributionally robust​ optimal design problem is​‌ an intricate bilevel program​​ which consists in minimizing​​​‌ the worst value of​ a statistical quantity of​‌ the cost function (typically,​​ its expectation) when the​​​‌ law of the uncertain​ parameters belongs to a​‌ certain "ambiguity set". We​​ address three classes of​​​‌ such problems: firstly, this​ ambiguity set is made​‌ of the probability laws​​ whose Wasserstein distance to​​​‌ the nominal law is​ less than a given​‌ threshold; secondly, the ambiguity​​ set is based on​​​‌ the first-and second-order moments​ of the actual and​‌ nominal probability laws. Eventually,​​ a statistical quantity of​​​‌ the cost other than​ its expectation is made​‌ robust with respect to​​ the law of the​​​‌ parameters, namely its conditional​ value at risk. Using​‌ techniques from convex duality,​​ we derive tractable, single-level​​​‌ reformulations of these problems,​ framed over augmented sets​‌ of variables. Our methods​​ are essentially agnostic of​​​‌ the optimal design framework;​ they are described in​‌ a unifying abstract framework,​​ before being applied to​​​‌ multiple situations in density-based​ topology optimization and in​‌ geometric shape optimization. Several​​ numerical examples are discussed​​​‌ in two and three​ space dimensions to appraise​‌ the features of the​​ proposed techniques.

8.3 A​​ shape optimization approach for​​​‌ inferring sources of volcano‌ ground deformation

With: Théo‌​‌ Perrot, Freysteinn Sigmundsson, Charles​​ Dapogny .

One of​​​‌ the main goals of‌ volcano geodesy is to‌​‌ improve the understanding of​​ how an increase in​​​‌ pressure related to magma‌ accumulation causes ground deformation‌​‌ in order to evaluate​​ volcanic unrest. The inversion​​​‌ methods used for this‌ purpose rely on a‌​‌ parametrization of the shape​​ of the crustal volume​​​‌ in which pressure changes‌ due to magma inflow/outflow‌​‌ (the magma domain), to​​ search for the optimal​​​‌ parameters that minimize the‌ difference between model predicted‌​‌ and measured ground displacements.​​ However, these methods assume​​​‌ a predefined shape of‌ the magma domain, which‌​‌ limits their applicability.

@unpublishedperrot:hal-05373455,​​ In 40, we​​​‌ propose a new shape‌ optimization framework that can‌​‌ invert these sources without​​ such prior, formulating a​​​‌ reconstruction problem to infer‌ the complete shape of‌​‌ the magma domain. First,​​ we validate this approach​​​‌ using a synthetic test‌ case and then apply‌​‌ it to observations of​​ the Svartsengi volcanic system​​​‌ in Iceland.

8.4 Domain‌ decomposition preconditioners for efficient‌​‌ parallel simulations of single-phase​​ flow in three-dimensional fractured​​​‌ porous media with a‌ very large number of‌​‌ fractures

With: Pierre Jolivet,​​ Michel Kern, Frédéric Nataf​​​‌ , Géraldine Pichot, Daniel‌ Zegarra Vasquez.

In 30‌​‌, three-dimensional fractured-porous media​​ are classically described by​​​‌ the Discrete Fracture Matrix‌ (DFM) model, where the‌​‌ rock matrix remains three-dimensional,​​ while the fracture network​​​‌ is of co-dimension one.‌ This study deals with‌​‌ efficient solutions of the​​ linear system arising in​​​‌ single-phase flow in large‌ DFMs using a mixed-hybrid‌​‌ finite element method. It​​ demonstrates the effectiveness of​​​‌ two-level domain decomposition methods‌ based on the GenEO‌​‌ framework for preconditioning iterative​​ solvers applied to such​​​‌ flow problems. Due to‌ the mixed-dimensional nature of‌​‌ the geometry, a specific​​ partitioning strategy is required.​​​‌ The coarse spaces for‌ these preconditioners are built‌​‌ by locally solving spectral​​ problems. This paper presents​​​‌ simulations with very large‌ DFMs, with up to‌​‌ hundreds of thousands of​​ fractures, achieving a low​​​‌ iteration count and short‌ total computation time. Furthermore,‌​‌ this work confirms the​​ scalability of these preconditioners.​​​‌

8.5 A scalable Domain‌ Decomposition method for Saddle‌​‌ Point problems with GenEO​​ coarse spaces

With: Filippo​​​‌ Brunelli , Guillaume Delay,‌ Frédéric Nataf , Emile‌​‌ Parolin , Pierre-Henri Tournier​​ .

In 22,​​​‌ we present an adaptive‌ domain decomposition (DD) preconditioning‌​‌ technique for the solution​​ of saddle point problems​​​‌ with a 2x2 blocks‌ structure. This work utilises‌​‌ the GenEO theory for​​ symmetric positive definite (SPD)​​​‌ problems (Spillane et al.,‌ 2014) to tackle the‌​‌ solution of saddle point​​ linear systems with iterative​​​‌ methods. The latters are‌ preferred over direct methods‌​‌ due to the typically​​ large size of these​​​‌ problems. Under the same‌ assumptions of the GenEO‌​‌ method we can develop​​ a robust and scalable​​​‌ preconditioning technique for the‌ saddle point linear system.‌​‌ We present a preconditioner​​ for the primal-primal block​​​‌ and one for the‌ Schur complement matrix, both‌​‌ based on the DD​​​‌ paradigm. Then, these ingredients​ are used to precondition​‌ the 2x2 block saddle​​ point matrix. Numerical tests​​​‌ are performed on a​ benchmark systems of partial​‌ differential equations (PDEs) discretized​​ with finite elements, for​​​‌ which the assumptions are​ easy to verify. The​‌ results emphasize preconditioner's robustness​​ with respect to the​​​‌ specific domain's partitioning.

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

With: Niall Bootland, Victorita​‌ Dolean, Frédéric Nataf ,​​ Pierre-Henri Tournier .

In​​​‌ 17, 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.

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

With: Frédéric​​ Nataf , Emile Parolin​​​‌ .

In 39,​ 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.

8.8 Coarse spaces​​ using extended generalized eigenproblems​​​‌ for heterogeneous Helmholtz problems​

With: Emile Parolin ,​‌ Frédéric Nataf , Aminata​​ Diarra .

An abstract​​​‌ construction of coarse spaces​ for non-Hermitian problems and​‌ non-Hermitian domain decomposition preconditioners​​ based on extended generalized​​​‌ eigenproblems was proposed in​ § 8.7 and analyzed​‌ on the matrix formulation.​​ Building upon this work,​​​‌ in 31, we​ consider instead the specific​‌ case of heterogeneous Helmholtz​​ problems, and the derivation​​​‌ and analysis is performed​ at the continuous level.​‌

Additional numerical investigations and​​ a computational cheaper alternative​​​‌ construction of the coarse​ space have been obtained​‌ during the Master thesis​​ of Aminata Diarra .​​​‌

8.9 Numerical comparison of​ coarse spaces for heterogeneous​‌ Helmholtz problems

With: Victorita​​ Dolean, Mark Fry, Matthias​​​‌ Langer, Emile Parolin ,​ Pierre-Henri Tournier .

Solving​‌ time-harmonic wave propagation problems​​ in the frequency domain​​​‌ within heterogeneous media poses​ significant mathematical and computational​‌ challenges, particularly in the​​ high-frequency regime. Among the​​​‌ available numerical approaches, domain​ decomposition methods can achieve​‌ near-constant time-to-solution as the​​ wavenumber increases, though often​​ at the expense of​​​‌ a computationally intensive coarse‌ correction step. In 25‌​‌, we focus on​​ identifying the best algorithms​​​‌ and numerical strategies for‌ benchmark problems modelled by‌​‌ the Helmholtz equation. Specifically,​​ we examine and compare​​​‌ several coarse spaces which‌ are part of different‌​‌ families, e.g. GenEO (Generalised​​ Eigenvalue Overlap) type coarse​​​‌ spaces and harmonic coarse‌ spaces (including the one‌​‌ in §8.8),​​ that underpin two-level domain​​​‌ decomposition methods. By leveraging‌ spectral information and multiscale‌​‌ approaches, we aim to​​ provide a comprehensive overview​​​‌ of the strengths and‌ weaknesses of these methods.‌​‌ Numerical experiments demonstrate that​​ the effectiveness of these​​​‌ coarse spaces depends on‌ the specific problem and‌​‌ numerical configuration, highlighting the​​ trade-offs between computational cost,​​​‌ robustness, and practical applicability.‌

8.10 Modal analysis of‌​‌ a domain decomposition method​​ for Maxwell's equations in​​​‌ a waveguide

With: Victorita‌ Dolean, Antoine Tonnoir, Pierre-Henri‌​‌ Tournier .

Time-harmonic wave​​ propagation problems, especially those​​​‌ governed by Maxwell's equations,‌ pose significant computational challenges‌​‌ due to the non-self-adjoint​​ nature of the operators​​​‌ and the large, non-Hermitian‌ linear systems resulting from‌​‌ discretization. Domain decomposition methods,​​ particularly one-level Schwarz methods,​​​‌ offer a promising framework‌ to tackle these challenges,‌​‌ with recent advancements showing​​ the potential for weak​​​‌ scalability under certain conditions.‌ In 26, we‌​‌ analyze the weak scalability​​ of one-level Schwarz methods​​​‌ for Maxwell's equations in‌ strip-wise domain decompositions, focusing‌​‌ on waveguides with general​​ cross sections and different​​​‌ types of transmission conditions‌ such as impedance or‌​‌ perfectly matched layers (PMLs).​​ By combining techniques from​​​‌ the limiting spectrum analysis‌ of Toeplitz matrices and‌​‌ the modal decomposition of​​ Maxwell's solutions, we provide​​​‌ a novel theoretical framework‌ that extends previous work‌​‌ to more complex geometries​​ and transmission conditions. Numerical​​​‌ experiments confirm that the‌ limiting spectrum effectively predicts‌​‌ practical behavior even with​​ a modest number of​​​‌ subdomains. Furthermore, we demonstrate‌ that the one-level Schwarz‌​‌ method can achieve robustness​​ with respect to the​​​‌ wave number under specific‌ domain decomposition parameters, offering‌​‌ new insights into its​​ applicability for large-scale electromagnetic​​​‌ wave problems.

8.11 A‌ Robust Two-Level Schwarz Preconditioner‌​‌ For Sparse Matrices

With:​​ Hussam Al Daas, Pierre​​​‌ Jolivet, Frédéric Nataf ,‌ Pierre-Henri Tournier .

In‌​‌ 16, 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.

8.12 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 37 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 36, 35​​, 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%.​​​‌

8.13 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.

8.14​​ Reduced models for highly​​​‌ oscillating differential equations

With:‌ Sever Hirstoaga

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.

The aim in‌ this context is to‌​‌ derive reduced models of​​ first-order 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​​​‌ (see 38). In‌ 27 we extend this‌​‌ 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 obtain first-order equations‌​‌ approximating stiff differential equations​​ in this general case.​​​‌ We develop numerical simulations‌ to support the approximation‌​‌ results and the efficiency​​ in terms of computational​​​‌ times. More precisely, we‌ demonstrate on several numerical‌​‌ examples that the first-order​​ two-scale solution (i) is​​​‌ much less costly than‌ the reference solution of‌​‌ the initial model and​​ (ii) is much more​​​‌ accurate than the zero-order‌ two-scale solution.

In view‌​‌ of these encouraging results,​​ we continue working on​​​‌ the derivation of first-order‌ models by homogenization in‌​‌ time, in the case​​ of partial differential equations​​​‌ as Vlasov-Poisson, by expressing‌ its solution using characteristics.‌​‌ The implementation in this​​ framework have yielded very​​​‌ good numerical results, emphasizing‌ the efficiency of the‌​‌ first-order reduced model.The numerical​​ framework is the particle-in-cell​​​‌ approach, widely used for‌ kinetic simulations. Developed in‌​‌ 3D+3V (six dimensions plus​​ time), this code is​​​‌ intended to be used‌ in a parallel frame‌​‌ with distributed and shared​​ memory paradigms, in order​​​‌ to achieve high performance‌ computing. This is ongoing‌​‌ work.

8.15 Stable Trefftz​​ approximation of Helmholtz solutions​​​‌ using evanescent plane waves‌

With: Nicola Galante ,‌​‌ Andrea Moiola, Emile Parolin​​ .

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 18​​​‌ 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.​‌

8.16 A Trefftz Continuous​​ Galerkin method for Helmholtz​​​‌ problems

With: Nicola Galante​ , Bruno Després, Emile​‌ Parolin .

In 28​​, we introduce a​​​‌ novel Trefftz Continuous Galerkin​ (TCG) method for 2D​‌ Helmholtz problems based on​​ evanescent plane waves (EPWs).​​​‌ We construct a new​ globally-conforming discrete space, departing​‌ from standard discontinuous Trefftz​​ formulations, and investigate its​​​‌ approximation properties, providing wavenumber-explicit​ best-approximation error estimates. The​‌ mesh is defined by​​ intersecting the domain with​​​‌ a Cartesian grid, and​ the basis functions are​‌ continuous in the whole​​ computational domain, compactly supported,​​​‌ and can be expressed​ as simple linear combinations​‌ of EPWs within each​​ element. This ensures they​​​‌ remain local solutions to​ the Helmholtz equation and​‌ allows the system matrix​​ to be assembled in​​​‌ closed form for polygonal​ domains. The discrete space​‌ provides stable approximations with​​ bounded coefficients and spectral​​​‌ accuracy for analytic Helmholtz​ solutions. The approximation error​‌ is proved to decay​​ exponentially both at a​​​‌ fixed frequency, with respect​ to the discretization parameters,​‌ and along suitable sequences​​ of increasing wavenumbers, with​​​‌ the number of degrees​ of freedom scaling linearly​‌ with the frequency. Numerical​​ results confirm these theoretical​​​‌ estimates for the full​ Galerkin error.

8.17 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.

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

In‌​‌ 41, 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.

8.19​​ 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.

8.20​​​‌ 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.

8.21​ Randomized Krylov-Schur eigensolver with​‌ deflation

With: Jean-Guillaume De​​ Damas , Laura Grigori.​​​‌

In 23, we​ introduce a novel algorithm​‌ to solve large-scale eigenvalue​​ problems and seek a​​​‌ small set of eigenpairs.​ The method, called randomized​‌ Krylov-Schur (rKS), has a​​ simple implementation and benefits​​​‌ from fast and efficient​ operations in low-dimensional spaces,​‌ such as sketch-orthogonalization processes​​ and stable reordering of​​​‌ Schur factorizations. It also​ includes a practical deflation​‌ technique for converged eigenpairs,​​ enabling the computation of​​​‌ the eigenspace associated with​ a given part of​‌ the spectrum. Numerical experiments​​ are provided to demonstrate​​​‌ the scalability and accuracy​ of the method.

8.22​‌ PDE-constrained optimization within FreeFEM​​

With Gontran Lance, and​​​‌ Emmanuel Trélat

The book​  20 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.

Participants: C. Dapogny​‌.

• Contract with Schneider​​ Electric, April 2025 -​​​‌ April 2028, that funds​ the Phd thesis of​‌ Stéphane Gaydier on "Shape​​ and topology optimization of​​​‌ current sensors and actuators"​ . Supervisors Charles Dapogny​‌ , Olivier Chadebec, Carlos​​ Valdivieso.

9.2 Bilateral training​​​‌ with industry

We organized​ a three-day training session​‌ on the finite element​​ toolbox FreeFEM followed by​​​‌ a specialized training on​ problems from the company​‌ Alice & Bob at​​ a pace of 2​​​‌ hours per week online​ for two months.

10.1​​ European initiatives

10.2​‌ National initiatives

11.1 Promoting scientific activities​​​‌

11.2​​ Teaching - Supervision -​​​‌ Juries - Educational and‌ pedagogical outreach

• Master 2:‌​‌ Frédéric Nataf , Course​​ on Domain Decomposition Methods,​​​‌ Sorbonne University, 22h.
• Master‌ 2: Emile Parolin ,‌​‌ High Performance Computing, Sorbonne​​ Université, 15h.
• Master 1:​​​‌ Emile Parolin , Numerical‌ simulation and approximation of‌​‌ solutions to elliptic PDEs,​​ Sorbonne Université, 20h.
• Licence​​​‌ 3: Sever Hirstoaga Méthodes‌ numériques, cours et TP,‌​‌ Polytech Sorbonne, EISE 3,​​ 37 hetd.
• Master 2:​​​‌ Sever Hirstoaga , Cours‌ sur les méthodes numériques‌​‌ pour les EDO, Sorbonne​​ université, 15h.
• Pierre-Henri Tournier​​​‌ , Numerical simulation in‌ physics using the finite‌​‌ element method, PSL advanced​​​‌ training, Mines Paris -​ PSL, 12h.

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 $𝐇(‌\mathrm{𝐜𝐮𝐫𝐥})$ problems in​ general non-convex three-dimensional geometries​‌.November 2023HAL​​
• 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 Meeting​‌Portland, Oregon, United States​​IEEESeptember 2023,​​​‌ 1879-1880HALDOIback​ to text
• 3 inproceedings​‌S.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, Italy​​​‌March 2023, 5​HALback 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 Problems3911​​2023, 115003HAL​​​‌DOIback to text​
• 6 bookV.Victorita​‌ Dolean, P.Pierre​​ Jolivet and F.Frédéric​​​‌ Nataf. An Introduction​ to Domain Decomposition Methods:​‌ algorithms, theory and parallel​​ implementation.SIAM2015​​​‌back 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 Physics​475February 2023,​‌ 111864HALDOIback​​ to text
• 8 misc​​​‌L.Laura Grigori and​ E.Edouard Timsit.​‌ Randomized Householder QR.​​July 2023HALback​​​‌ to text
• 9 misc​F.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 2023HAL​back to text
• 10​‌ articleF.F. Hecht​​. New development in​​​‌ FreeFem++.J. Numer.​ Math.203-42012​‌, 251--265
• 11 article​​F.Frédéric Nataf and​​​‌ P.-H. H.Pierre-Henri H​ Tournier. A GenEO​‌ Domain Decomposition method for​​ Saddle Point problems.​​​‌Comptes Rendus. Mécanique351​S1March 2023,​‌ 1-18HALDOIback​​ to text
• 12 article​​​‌E.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 Analysis57​​6October 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 2023​​​‌HALback 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 Communications294‌​‌January 2024, 108948​​HALDOI
• 15 article​​​‌N.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-y​​​‌DOI

12.2 Publications of‌ the year

12.3 Cited publications

• 32​ articleF.F Assous​‌, M.M Kray​​, F.F Nataf​​​‌ and E.E Turkel​. Time-reversed absorbing condition:​‌ application to inverse problems​​.Inverse Problems27​​​‌62011, 065003​URL: http://stacks.iop.org/0266-5611/27/i=6/a=065003back to​‌ text
• 33 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
• 34 articleE.​​​‌Eric Bonnetier, C.​Carlos Brito-Pacheco, C.​‌Charles Dapogny and R.​​Rafael Estevez. Numerical​​​‌ shape and topology optimization​ of regions supporting the​‌ boundary conditions of a​​ physical problem.ESAIM:​​​‌ Control, Optimisation and Calculus​ of Variations31August​‌ 2025, 75HAL​​DOIback to text​​​‌
• 35 inproceedingsS.Sahar​ Borzooei, P.-H.Pierre-Henri​‌ Tournier, V.Victorita​​ Dolean and C.Claire​​​‌ Migliaccio. A SVM-based​ approach for detecting tendon​‌ Injury.AP-S/URSI 2024​​ - International Symposium on​​​‌ Antennas and Propagation and​ ITNC-USNC-URSI Radio Sciencehttps://2024.apsursi.org/​‌Florence, ItalyJuly 2024​​, pp. 1511-1512HAL​​​‌DOIback to text​
• 36 articleS.Sahar​‌ Borzooei, P.-H.Pierre-Henri​​ Tournier, V.Victorita​​ Dolean and C.Claire​​​‌ Migliaccio. Microwave Digital‌ Twin Prototype for Shoulder‌​‌ Injury Detection.Sensors​​2420October 2024​​​‌, 1-17HALDOI‌back to text
• 37‌​‌ articleS.Sahar Borzooei​​, P.-H.Pierre-Henri Tournier​​​‌, V.Victorita Dolean‌, C.Christian Pichot‌​‌, N.Nadine Joachimowicz​​, H.Helene Roussel​​​‌ and C.Claire Migliaccio‌. Numerical Modeling for‌​‌ Shoulder Injury Detection Using​​ Microwave Imaging.IEEE​​​‌ Journal of Electromagnetics, RF‌ and Microwaves in Medicine‌​‌ and BiologyJune 2024​​, 1-8HALDOI​​​‌back to text
• 38‌ articleS. A.Sever‌​‌ Adrian Hirstoaga. A​​ first-order reduced model for​​​‌ a highly oscillating differential‌ equation with application in‌​‌ Penning traps.SIAM​​ Journal on Scientific Computing​​​‌2024, S225-S245HAL‌DOIback to text‌​‌
• 39 unpublishedF.Frédéric​​ Nataf and E.Emile​​​‌ Parolin. Coarse spaces‌ for non-symmetric two-level preconditioners‌​‌ based on local generalized​​ eigenproblems.April 2024​​​‌, working paper or‌ preprintHALback to‌​‌ text
• 40 unpublishedT.​​Théo Perrot, F.​​​‌Freysteinn Sigmundsson and C.‌Charles Dapogny. A‌​‌ shape optimization approach for​​ inferring sources of volcano​​​‌ ground deformation.November‌ 2025, working paper‌​‌ or preprintHALback​​ to text
• 41 article​​​‌G.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​‌back to text
• 42​​ 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