2023Activity reportProject-TeamSPHINX

3.1 Analysis, control, stabilization and optimization of heterogeneous systems

Fluid-Structure Interaction Systems are present in many physical problems and applications. Their study involves solving several challenging mathematical problems:

• Nonlinearity: One has to deal with a system of nonlinear PDE such as the Navier-Stokes or the Euler systems;
• Coupling: The corresponding equations couple two systems of different types and the methods associated with each system need to be suitably combined to successfully solve the full problem;
• Coordinates: The equations for the structure are classically written with Lagrangian coordinates whereas the equations for the fluid are written with Eulerian coordinates;
• Free boundary: The fluid domain is moving and its motion depends on the motion of the structure. The fluid domain is thus an unknown of the problem and one has to solve a free boundary problem.

In order to control such FSIS, one has first to analyze the corresponding system of PDE. The oldest works on FSIS go back to the pioneering contributions of Thomson, Tait and Kirchhoff in the 19th century and Lamb in the 20th century, who considered simplified models (potential fluid or Stokes system). The first mathematical studies in the case of a viscous incompressible fluid modeled by the Navier-Stokes system and a rigid body whose dynamics is modeled by Newton's laws appeared much later 105, 100, 79, and almost all mathematical results on such FSIS have been obtained in the last twenty years.

The most studied FSIS is the problem modeling a rigid body moving in a viscous incompressible fluid 61, 58, 99, 69, 73, 102, 104, 89, 71. Many other FSIS have been studied as well. Let us mention 91, 76, 72, 63, 49, 68, 50, 67 for different fluids. The case of deformable structures has also been considered, either for a fluid inside a moving structure (e.g., blood motion in arteries) or for a moving deformable structure immersed in a fluid (e.g., fish locomotion). The obtained coupled FSIS is a complex system and its study raises several difficulties. The main one comes from the fact that we gather two systems of different nature. Some studies have been performed for approximations of this system: 54, 49, 82, 62, 52). Without approximations, the only known results 59, 60 were obtained with very strong assumptions on the regularity of the initial data. Such assumptions are not satisfactory but seem inherent to this coupling between two systems of different natures. In order to study self-propelled motions of structures in a fluid, like fish locomotion, one can assume that the deformation of the structure is prescribed and known, whereas its displacement remains unknown 96. This permits to start the mathematical study of a challenging problem: understanding the locomotion mechanism of aquatic animals. This is related to control or stabilization problems for FSIS. Some first results in this direction were obtained in 77, 51, 93.

3.2 Inverse problems for heterogeneous systems

The area of inverse problems covers a large class of theoretical and practical issues which are important in many applications (see for instance the books of Isakov 78 or Kaltenbacher, Neubauer, and Scherzer 80). Roughly speaking, an inverse problem is a problem where one attempts to recover an unknown property of a given system from its response to an external probing signal. For systems described by evolution PDE, one can be interested in the reconstruction from partial measurements of the state (initial, final or current), the inputs (a source term, for instance) or the parameters of the model (a physical coefficient for example). For stationary or periodic problems (i.e., problems where the time dependency is given), one can be interested in determining from boundary data a local heterogeneity (shape of an obstacle, value of a physical coefficient describing the medium, etc.). Such inverse problems are known to be generally ill posed and their study raises the following questions:

• Uniqueness. The question here is to know whether the measurements uniquely determine the unknown quantity to be recovered. This theoretical issue is a preliminary step in the study of any inverse problem and can be a hard task.
• Stability. When uniqueness is ensured, the question of stability, which is closely related to sensitivity, deserves special attention. Stability estimates provide an upper bound for the parameter error given some uncertainty on the data. This issue is closely related to the so-called observability inequality in systems theory.
• Reconstruction. Inverse problems are usually ill-posed, one needs to develop specific reconstruction algorithms which are robust to noise, disturbances and discretization. A wide class of methods is based on optimization techniques.

We can split our research in inverse problems into two classes which both appear in FSIS and CWS:

1. Identification for evolution PDE.

   Driven by applications, the identification problem for systems of infinite dimension described by evolution PDE has seen in the last three decades a fast and significant growth. The unknown to be recovered can be the (initial/final) state (e.g., state estimation problems 44, 70, 75, 101 for the design of feedback controllers), an input (for instance source inverse problems 41, 53, 64) or a parameter of the system. These problems are generally ill-posed and many regularization approaches have been developed. Among the different methods used for identification, let us mention optimization techniques 57, specific one-dimensional techniques (like in 45) or observer-based methods as in 85.

   In the last few years, we have developed observers to solve initial data inverse problems for a class of linear systems of infinite dimension. Let us recall that observers, or Luenberger observers  84, have been introduced in automatic control theory to estimate the state of a dynamical system of finite dimension from the knowledge of an output (for more references, see for instance 90 or 103). Using observers, we have proposed in 92, 74 an iterative algorithm to reconstruct initial data from partial measurements for some evolution equations. We are deepening our activities in this direction by considering more general operators or more general sources and the reconstruction of coefficients for the wave equation. In connection with this problem, we study the stability in the determination of these coefficients. To achieve this, we use geometrical optics, which is a classical albeit powerful tool to obtain quantitative stability estimates on some inverse problems with a geometrical background, see for instance  47, 46.

2. Geometric inverse problems.

   We investigate some geometric inverse problems that appear naturally in many applications, like medical imaging and non-destructive testing. A typical problem we have in mind is the following: given a domain $\Omega$ containing an (unknown) local heterogeneity $\omega$, we consider the boundary value problem of the form

   $$\left\{\begin{array}{cc}Lu=0,\hfill & (\Omega \setminus \omega )\hfill \\ u=f,\hfill & \left(\partial \Omega \right)\hfill \\ Bu=0,\hfill & \left(\partial \omega \right)\hfill \end{array}\right.$$

   where $L$ is a given partial differential operator describing the physical phenomenon under consideration (typically a second order differential operator), $B$ the (possibly unknown) operator describing the boundary condition on the boundary of the heterogeneity and $f$ the exterior source used to probe the medium. The question is then to recover the shape of $\omega$ and/or the boundary operator $B$ from some measurement on the outer boundary $\partial \Omega$. This setting includes in particular inverse scattering problems in acoustics and electromagnetics (in this case $\Omega$ is the whole space and the data are far field measurements) and the inverse problem of detecting solids moving in a fluid. It also includes, with slight modifications, more general situations of incomplete data (i.e., measurements on part of the outer boundary) or penetrable inhomogeneities. Our approach to tackle this type of problems is based on the derivation of a series expansion of the input-to-output map of the problem (typically the Dirichlet-to-Neumann map of the problem for the Calderón problem) in terms of the size of the obstacle.

3.3 Numerical analysis and simulation of heterogeneous systems

Within the team, we have developed in the last few years numerical codes for the simulation of FSIS and CWS. We plan to continue our efforts in this direction.

• In the case of FSIS, our main objective is to provide computational tools for the scientific community, essentially to solve academic problems.
• In the case of CWS, our main objective is to build tools general enough to handle industrial problems. Our strong collaboration with Christophe Geuzaine's team in Liège (Belgium) makes this objective credible, through the combination of DDM (Domain Decomposition Methods) and parallel computing.

Below, we explain in detail the corresponding scientific program.

• Simulation of FSIS: In order to simulate fluid-structure systems, one has to deal with the fact that the fluid domain is moving and that the two systems for the fluid and for the structure are strongly coupled. To overcome this free boundary problem, three main families of methods are usually applied to numerically compute in an efficient way the solutions of the fluid-structure interaction systems. The first method consists in suitably displacing the mesh of the fluid domain in order to follow the displacement and the deformation of the structure. A classical method based on this idea is the A.L.E. (Arbitrary Lagrangian Eulerian) method: with such a procedure, it is possible to keep a good precision at the interface between the fluid and the structure. However, such methods are difficult to apply for large displacements (typically the motion of rigid bodies). The second family of methods consists in using a fixed mesh for both the fluid and the structure and to simultaneously compute the velocity field of the fluid with the displacement velocity of the structure. The presence of the structure is taken into account through the numerical scheme. Finally, the third class of methods consists in transforming the set of PDEs governing the flow into a system of integral equations set on the boundary of the immersed structure. The members of SPHINX have already worked on these three families of numerical methods for FSIS systems with rigid bodies (see e.g., 97, 81, 98, 94, 95, 86).
• Simulation of CWS: Solving acoustic or electromagnetic scattering problems can become a tremendously hard task in some specific situations. In the high frequency regime (i.e., for small wavelength), acoustic (Helmholtz's equation) or electromagnetic (Maxwell's equations) scattering problems are known to be difficult to solve while being crucial for industrial applications (e.g., in aeronautics and aerospace engineering). Our particularity is to develop new numerical methods based on the hybridization of standard numerical techniques (like algebraic preconditioners, etc.) with approaches borrowed from asymptotic microlocal analysis. Most particularly, we contribute to building hybrid algebraic/analytical preconditioners and quasi-optimal Domain Decomposition Methods (DDM) 48, 65, 66 for highly indefinite linear systems. Corresponding three-dimensional solvers (like for example GetDDM) will be developed and tested on realistic configurations (e.g., submarines, complete or parts of an aircraft, etc.) provided by industrial partners (Thales, Airbus). Another situation where scattering problems can be hard to solve is the one of dense multiple (acoustic, electromagnetic or elastic) scattering media. Computing waves in such media requires us to take into account not only the interactions between the incident wave and the scatterers, but also the effects of the interactions between the scatterers themselves. When the number of scatterers is very large (and possibly at high frequency 42, 43), specific deterministic or stochastic numerical methods and algorithms are needed. We introduce new optimized numerical methods for solving such complex configurations. Many applications are related to this problem, such as osteoporosis diagnosis where quantitative ultrasound is a recent and promising technique to detect a risk of fracture. Therefore, numerical simulation of wave propagation in multiple scattering elastic media in the high frequency regime is a very useful tool for this purpose.

4.1 Robotic swimmers

Some companies aim at building biomimetic robots that can swim in an aquarium, as toys but also for medical purposes. An objective of SPHINX is to model and to analyze several models of these robotic swimmers. For the moment, we focus on the motion of a nanorobot. In that case, the size of the swimmers leads us to neglect the inertia forces and to only consider the viscosity effects. Such nanorobots could be used for medical purposes to deliver some medicine or perform small surgical operations. In order to get a better understanding of such robotic swimmers, we have obtained control results via shape changes and we have developed simulation tools (see 55, 56, 86, 83). Among all the important issues, we aim to consider the following ones:

1. Solve the control problem by limiting the set of admissible deformations.
2. Find the “best” location of the actuators, in the sense of being the closest to the exact optimal control.

The main tools for this investigation are the 3D codes that we have developed for simulating the fish in a viscous incompressible fluid (SUSHI3D) or in an inviscid incompressible fluid (SOLEIL).

4.2 Aeronautics

We will develop robust and efficient solvers for problems arising in aeronautics (or aerospace) like electromagnetic compatibility and acoustic problems related to noise reduction in an aircraft. Our interest for these issues is motivated by our close contact with companies like Airbus or “Thales Systèmes Aéroportés”. We will propose new applications needed by these partners and assist them in integrating these new scientific developments in their home-made solvers. In particular, in collaboration with C. Geuzaine (Université de Liège), we are building a freely available parallel solver based on Domain Decomposition Methods that can handle complex engineering simulations, in terms of geometry, discretization methods as well as physics problems, see here.

5.1 Awards

• Yannick Privat was awarded the Blaise Pascal Prize of the French Academy of Sciences in 2023.

6.1 Analysis, control, stabilization and optimization of heterogeneous systems

Participants: Rémi Buffe, Imene Djebour, Alessandro Duca, Ludovick Gagnon, Julien Lequeurre, Karim Ramdani, Jean-François Scheid, Takéo Takahashi, Julie Valein, Christophe Zhang.

Analysis

In 38, we extend the theory of single and double layer potentials (well documented for functions with $H_{loc}^1$ regularity) to locally square integrable functions. Having in mind numerical simulations for which functions are usually defined on a polygonal mesh, we wish this theory to cover the cases of non-smooth domains (i.e., with Lipschitz continuous or polygonal boundaries).

In 11, we consider a wave equation with a structural damping coupled with an undamped wave equation located at its boundary. We prove that, due to the coupling, the full system is parabolic. In order to show that the underlying operator generates an analytical semigroup, we study in particular the effect of the damping of the "interior" wave equation on the "boundary" wave equation and show that it generates a structural damping.

In 25, we study the weak uniqueness and the regularity of the weak solutions of a fluid-structure interaction system. More precisely, we consider the motion of a rigid ball in a viscous incompressible fluid and we assume that the fluid-rigid body system fills the entire space ${ℝ}^3$. We prove that the corresponding weak solutions that additionally satisfy a classical Prodi-Serrin condition, including a critical one, are unique. We also show that the weak solutions are regular under the Prodi-Serrin conditions, with a smallness condition in the critical case.

In 34, we consider a fluid-structure interaction system coupling a viscous fluid governed by the compressible Navier-Stokes equations and a rigid body immersed in the fluid and modeled by Newton's law. In this work, we consider the Navier slip boundary conditions. Our aim is to show the local existence and uniqueness of the strong solution to the corresponding problem. The main step of this work is that we use a Lagrangian change of variables in order to handle the transport equation and to reduce the problem in the initial domain. However, the specificity here is that the Lagrangian coordinates do not coincide with the Eulerian coordinates at the boundaries since we consider slip boundary conditions. Therefore, it brings some extra nonlinear terms in the boundary conditions. The strategy is based on the study of the linearized system with nonhomogeneous boundary conditions and on the Banach fixed-point theorem.

Control

Controlling coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Our team has a strong expertise to tackle this kind of problems in the context of fluid-structure interaction systems. More precisely, we obtained the following results.

In 12, we consider the controllability of an abstract parabolic system by using switching controls. More precisely, we show that under general hypotheses, if a parabolic system is null controllable for any positive time with $N$ controls, then it is also null controllable with the property that at each time, only one of these controls is active. The main difference with previous results in the literature is that we can handle the case where the main operator of the system is not self-adjoint. We give several examples to illustrate our result: coupled heat equations with terms of orders 0 and 1, the Oseen system or the Boussinesq system.

In 14, we prove the null-controllability of a fluid-structure system coupling the Navier-Stokes equation for the fluid and a plate equation at the boundary. The control acts on arbitrarily small subsets of the fluid domain and in a small subset of the vibrating boundary. By proving a proper observability inequality, we obtain the local controllability for the nonlinear system. The proof relies on microlocal argument to handle the pressure terms.

In 32, we consider the controllability of a fluid-structure interaction system, where the fluid is modeled by the Navier-Stokes system and where the structure is a damped beam located on a part of its boundary. The motion of the fluid is bi-dimensional whereas the deformation of the structure is one-dimensional and we use periodic boundary conditions in the horizontal direction. Our result is the local null-controllability of this free-boundary system by using only one scalar control acting on an arbitrary small part of the fluid domain. This improves a previous result obtained by the authors where three scalar controls were needed to achieve the local null-controllability. In order to show the result, we prove the final-state observability of a linear Stokes-beam interaction system in a cylindrical domain. This is done by using a Fourier decomposition, proving Carleman inequalities for the corresponding system for the low-frequency solutions and in the case where the observation domain is a horizontal strip. Then we conclude this observability result by using a Lebeau-Robbiano strategy for the heat equation and a uniform exponential decay for the high-frequency solutions. Then, the result on the nonlinear system can be obtained by a change of variables and a fixed-point argument.

In 17, we study the local null-controllability of a modified Navier-Stokes system which includes nonlocal spatial terms. We generalize a previous work where the nonlocal spatial term is given by the linearization of a Ladyzhenskaya model for a viscous incompressible fluid. Here the nonlocal spatial term is more complicated and we consider a control with one vanishing component. The proof of the result is based on a Carleman estimate where the main difficulty consists of handling the nonlocal spatial terms. One of the key points is a particular decomposition of the solution of the adjoint system that allows us to overcome regularity issues. With a similar approach, we also show the existence of insensitizing controls for the same system.

In 18, we show the boundary controllability to stationary states of the Stefan problem with two phases and in one dimension in the space variable. For an initial condition that is a stationary state and for a time of control large enough, we also obtain the controllability to stationary states together with the sign constraints associated with the problem. Our method is based on the flatness approach that consists of writing the solution and the controls through two outputs and their derivatives. We construct these outputs as Gevrey functions of order σ so that their solution and controls are also in a Gevrey class.

In 19, we study a one-dimensional cross diffusion system with a free boundary modeling physical vapor deposition. Using the flatness approach, we show several results of boundary controllability for this system in spaces of Gevrey class functions. One of the main difficulties consists in the physical constraints on the state and the control. More precisely, the state corresponds to volume fractions of the $n+1$ chemical species and to the thickness of the film produced in the process, whereas the controls are the fluxes of the chemical species. We obtain the local controllability in the case where we apply $n+1$ nonnegative controls and a controllability result for large time in the case where we apply $n$ controls without any sign constraints. We also show in this last case that the controllability may fail for small times. We illustrate these results with some numerical simulations.

In 33, we study the boundary controllability of $2\times 2$ system of heat equations by using a flatness approach. According to the relation between the diffusion coefficients of the heat equation, it is known that the system can be not null controllable or null controllable for any $T>T_0$ where $T_0\in [0,\infty ]$. Here we recover this result in the case that $T_0\in [0,\infty )$ by using the flatness method, and we obtain an explicit formula for the control and for the corresponding solutions. In particular, the state and the control have Gevrey regularity in time and in space.

In 28, we deal with the controllability properties of a system of $m$ coupled Stokes systems or $m$ coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control an $m\times N$ state (corresponding to the velocities of the fluids) by $N-1$ distributed scalar controls. The proof of the controllability of the coupled Stokes system is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point.

In 27, we consider the controllability of a class of systems of $n$ Stokes equations, coupled through terms of order zero and controlled by $m$ distributed controls. Our main result states that such a system is null controllable if and only if a Kalman type condition is satisfied. This generalizes the case of finite-dimensional systems and the case of systems of coupled linear heat equations. The proof of the main result relies on the use of the Kalman operator introduced and on a Carleman estimate for a cascade type system of Stokes equations. Using a fixed-point argument, we also obtain that if the Kalman condition is verified, then the corresponding system of Navier-Stokes equations is locally null controllable.

In 40, we consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the "leader" control is the null-controllability property whereas the objective of "follower" control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are led to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.

In 31, we ensure an observability inequality, also known as spectral inequality, within spaces spanned by the first eigenfunctions for a family of one-dimensional degenerate operators ${\partial }_x{x}^{\alpha }{\partial }_x$. We give a bound for the blow-up of the constant when the frequency cut goes to infinity, which is known to be optimal for the Laplace operator. A new application to observability for degenerate parabolic equations is given. Finally, the associated degenerate parabolic equation is known to be not null controllable when $\alpha \ge 2$. we prove a bound for the blow-up of the constants of the spectral inequality when $\alpha \to 2^{-}$. The proof relies on a combination of the moment method and Carleman estimates.

Finally, in 39, we consider the internal control of linear parabolic equations through on-off shape controls with a prescribed maximal measure. They establish small-time approximate controllability towards all possible final states allowed by the comparison principle with non-negative controls and manage to build controls with constant amplitude.

Stabilization

Stabilization of infinite dimensional systems governed by PDEs is a challenging problem. In our team, we have investigated this issue for different kinds of systems (fluid systems and wave systems) using different techniques.

In 35, we consider the stabilization of a class of linear evolution systems $z^{\text{'}}=Az+Bv$ under the observation $y=Cz$ by means of a finite dimensional control $v$. The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the infinite dimensional case, the operator $A$ is supposed to generate an analytical semigroup with compact resolvent and the operators $B$ and $C$ are unbounded operators whereas, in the finite dimensional case, $A$ is assumed to be a self-adjoint operator with compact resolvent, $B$ and $C$ are supposed to be bounded operators. In both cases, we show that if $(A,B)$ and $(A,C)$ verify the Fattorini-Hautus Criterion, then we can construct an observer-based control $v$ of finite dimension (greater or equal to the largest geometric multiplicity of the unstable eigenvalues of $A$) such that the evolution problem is exponentially stable. As an application, we study the stabilization of the $N$-dimensional convection-diffusion system with Dirichlet boundary control and an internal observation.

In 26, we consider the Korteweg-de Vries equation with time-dependent delay on the boundary or internal feedbacks. Under some assumptions on the time-dependent delay, on the weights of the feedbacks, and on the length of the spatial domain, we prove the exponential stability results, using appropriate Lyapunov functionals. We finish with some numerical simulations that illustrate the stability results and the influence of the delay on the decay rate.

In 23, we show the stabilization by a finite number of controllers of a fluid-structure interaction system where the fluid is modeled by the Navier-Stokes system into a periodical canal and where the structure is an elastic wall localized on top of the fluid domain. The elastic deformation of the structure follows a damped beam equation. We also assume that the fluid can slip on its boundaries and we model this by using the Navier slip boundary conditions. Our result states the local exponential stabilization around a stationary state of strong solutions by using dynamical controllers in order to handle the compatibility conditions at initial time. The proof is based on a change of variables to write the fluid-structure interaction system in a fixed domain and on the stabilization of the linearization of the corresponding system around the stationary state. One of the main difficulties consists in handling the nonlinear terms coming from the change of variables in the boundary conditions.

In 36, we prove the rapid stabilization of the linearized water waves equation with the Fredholm backstepping method. This result is achieved by overcoming an important theoretical threshold imposed by the classical methodology, namely, the quadratically close criterion. Indeed, the spatial operator of the linearized water waves exhibits an insufficient growth of the eigenvalues and the quadratically close criterion is not true in this case. We introduce the duality compactness method for general skew-adjoint operators to circumvent this difficulty. In turn, we prove the existence of a Fredholm backstepping transformation for a wide range of equations, opening the path to an abstract framework for this widely used method.

Optimization

In 16, an optimal shape problem for a general functional depending on the solution of a bidimensional Fluid-Structure Interaction problem (FSI) is studied. The system is composed of a coupling stationary Stokes-Elasticity sub-system for modeling the deformation of an elastic structure immersed in a viscous fluid. The differentiability with respect to the reference domain of the elastic structure is proved under shape perturbations with diffeomorphisms. The shape-derivative is then calculated with the use of the velocity method. This derivative involves the material derivatives of the solution of this fluid–structure interaction problem. The adjoint method is then used to obtain a simplified expression for the shape derivative. The main difficulty for studying the shape sensitivity of this FSI problem lies in the coupling between the Stokes problem written in a Eulerian frame and the linear elasticity problem written in a Lagrangian form.

6.2 Direct and inverse problems for heterogeneous systems

Participants: Alessandro Duca, Anthony Gerber-Roth, Alexandre Munnier, Karim Ramdani, Jean-François Scheid.

Direct problems

Negative materials are artificially structured composite materials (also known as metamaterials), whose dielectric permittivity and magnetic permeability are simultaneously negative in some frequency ranges. K. Ramdani continued his collaboration with R. Bunoiu on the homogenization of composite materials involving both positive and negative materials. Due to the sign-changing coefficients in the equations, classical homogenization theory fails, since it is based on uniform energy estimates which are known only for positive (more precisely constant sign) coefficients.

In 15, in collaboration with C. Timofte, K. Ramdani and R. Bunoiu investigate the homogenization of a diffusion-type problem, for sign-changing conductivities in the case of imperfect interface conditions is considered, by allowing flux jumps across their oscillating interface. The main difficulties of this study are due to the sign-changing coefficients and the appearance of an unsigned surface integral term in the variational formulation. A proof by contradiction (nonstandard in this context) and $T$-coercivity techniques are used in order to cope with these difficulties.

Inverse problems

Supervised by A. Munnier and K. Ramdani, the PhD of Anthony Gerber-Roth is devoted to the investigation of some geometric inverse problems, and can be seen as a continuation of the work initiated by the two supervisors in 88 and 87. In these papers, the authors addressed a particular case of Calderòn's inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e., heterogeneities of infinitely high conductivities). The first contribution of Anthony Gerber-Roth was to apply the method proposed in 87 to tackle a two-dimensional inverse gravimetric problem. The strong connection with the important notion of quadrature domains in this context has been highlighted. An efficient reconstruction algorithm has been proposed (and rigorously justified in some cases) for this geometric inverse problem. These results are detailed in 5.

In 29, we are interested in an inverse problem set on a tree shaped network where each edge behaves according to the wave equation with potential, external nodes have Dirichlet boundary conditions and internal nodes follow the Kirchoff law. The main goal is the reconstruction of the potential everywhere on the network, from the Neumann boundary measurements at all but one external vertices. Leveraging from the Lipschitz stability of this inverse problem, we aim to provide an efficient reconstruction algorithm based on the use of a specific global Carleman estimate. The proof of the main tool and of the convergence of the algorithm are provided; along with a detailed description of the numerical illustrations given at the end of the article.

In 37, we address the classical inverse problem of recovering the position and shape of obstacles immersed in a planar Stokes flow using boundary measurements. We prove that this problem can be transformed into a shape-from-moments problem to which ad hoc reconstruction methods can be applied. The effectiveness of this approach is confirmed by numerical tests that show significant improvements over those available in the literature to date.

6.3 Numerical analysis and simulation of heterogeneous systems

Participants: Yannick Privat, Jean-François Scheid.

The work in 13 is devoted to the modeling and numerical simulations of a one-dimensional model for localized corrosion phenomena. Localized corrosion involves the dissolution of metal in an aqueous solution of a number of chemical species together with their mass transport by diffusion and migration, and their reactions in solution. From a mathematical point of view, this problem can be identified as a Stefan problem involving a convection-reaction-diffusion system of PDEs with a moving boundary between the aqueous solution and the metal. The unknowns of this system are the concentrations of the chemical species, the electric potential and the position of the free boundary. The dissolution law steering the evolution of the free boundary is given by the nonlinear Butler-Volmer formula. In this work, the mathematical procedure for solving this strongly coupled differential equations system and the numerical development for simulations are presented. A finite-difference ALE (Abritrary Lagrangian Eulerian) scheme is used for the numerical computation of the solutions of this free boundary problem, leading to a nonlinear discrete system which is then solved using a Newton procedure. The numerical simulations obtained are in good agreement with experimental results.

7.1 International initiatives

7.2 National initiatives

8.1 Promoting scientific activities

8.2 Teaching - Supervision - Juries

8.3 Popularization

9.1 Major publications

• 1 articleL.Loredana Bălilescu, J.Jorge San Martín and T.Takéo Takahashi. Fluid-structure interaction system with Coulomb's law.SIAM Journal on Mathematical Analysis2017HAL
• 2 articleR.Renata Bunoiu, L.Lucas Chesnel, K.Karim Ramdani and M.Mahran Rihani. Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients.Annales de la Faculté des Sciences de Toulouse. Mathématiques.2020HAL
• 3 articleN.Nicolas Burq, D.David Dos Santos Ferreira and K.Katya Krupchyk. From semiclassical Strichartz estimates to uniform $L^p$ resolvent estimates on compact manifolds.Int. Math. Res. Not. IMRN162018, 5178--5218URL: https://doi.org/10.1093/imrn/rnx042DOI
• 4 articleL.Ludovick Gagnon. Lagrangian controllability of the 1-dimensional Korteweg--de Vries equation.SIAM J. Control Optim.5462016, 3152--3173URL: https://doi.org/10.1137/140964783DOI
• 5 articleA.Anthony Gerber-Roth, A.Alexandre Munnier and K.Karim Ramdani. A reconstruction method for the inverse gravimetric problem.SMAI Journal of Computational Mathematics92023, 197-225HALback to text
• 6 articleO.Olivier Glass, A.Alexandre Munnier and F.Franck Sueur. Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid.Inventiones Mathematicae21412018, 171-287HALDOI
• 7 articleC.Céline Grandmont, M.Matthieu Hillairet and J.Julien Lequeurre. Existence of local strong solutions to fluid-beam and fluid-rod interaction systems.Annales de l'Institut Henri Poincaré (C) Non Linear Analysis364July 2019, 1105-1149HALDOI
• 8 articleA.Alexandre Munnier and K.Karim Ramdani. Calderón cavities inverse problem as a shape-from-moments problem.Quarterly of Applied Mathematics762018, 407-435HAL
• 9 articleK.Karim Ramdani, J.Julie Valein and J.-C.Jean-Claude Vivalda. Adaptive observer for age-structured population with spatial diffusion.North-Western European Journal of Mathematics42018, 39-58HAL
• 10 articleJ.-F.Jean-François Scheid and J.Jan Sokolowski. Shape optimization for a fluid-elasticity system.Pure Appl. Funct. Anal.312018, 193--217

9.2 Publications of the year

9.3 Cited publications

• 41 articleC.Carlos Alves, A. L.Ana Leonor Silvestre, T.T. Takahashi and M.Marius Tucsnak. Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation.SIAM J. Control Optim.4832009, 1632-1659back to text
• 42 incollectionX.Xavier Antoine, C.Christophe Geuzaine and K.Karim Ramdani. Computational Methods for Multiple Scattering at High Frequency with Applications to Periodic Structures Calculations.Wave Propagation in Periodic MediaProgress in Computational Physics, Vol. 1Bentham2010, 73-107back to text
• 43 articleX.Xavier Antoine, K.Karim Ramdani and B.Bertrand Thierry. Wide Frequency Band Numerical Approaches for Multiple Scattering Problems by Disks.Journal of Algorithms & Computational Technologies622012, 241--259back to text
• 44 articleD.D. Auroux and J.J. Blum. A nudging-based data assimilation method : the Back and Forth Nudging (BFN) algorithm.Nonlin. Proc. Geophys.15305-3192008back to text
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