The team develops constructive, function-theoretic approaches to inverse problems arising in modeling and design, in particular for electro-magnetic systems as well as in the analysis of certain classes of signals.

Data typically consist of measurements or desired behaviors. The general thread is to approximate them by families of solutions to the equations governing the underlying system. This leads us to consider various interpolation and approximation problems in classes of rational and meromorphic functions, harmonic gradients, or solutions to more general elliptic partial differential equations (PDE), in connection with inverse potential problems. A recurring difficulty is to control the singularities of the approximants.

The mathematical tools pertain to complex and harmonic analysis, approximation theory, potential theory, system theory, differential topology, optimization and computer algebra. Targeted applications include:

identification and synthesis of analog microwave devices (filters, amplifiers),

non-destructive control from field measurements in medical engineering (source recovery in magneto/electro-encephalography), paleomagnetism (determining the magnetization of rock samples), and nuclear engineering (plasma shaping in tokamaks).

In each case, the endeavor is to develop algorithms resulting in dedicated software.

Within the extensive field of inverse problems, much of the research by APICS
deals with reconstructing solutions of classical elliptic PDEs from their
boundary behavior. Perhaps the most basic example of such a problem is
harmonic
identification of a stable linear dynamical system: the transfer-function *e.g.* by Cauchy formula.

Practice is not nearly as simple, for *i.e.* locating the

Step 1 makes connection with extremal
problems and analytic operator theory, see Section .
Step 2 involves optimization, and some Schur analysis
to parametrize transfer matrices of given Mc-Millan degree
when dealing with systems having several inputs and outputs,
see Section .
It also makes contact with the topology of rational functions, to count
critical points and to derive bounds, see Section . Moreover, this step raises
issues in approximation theory regarding the rate of convergence and whether
the singularities of the
approximant (*i.e.* its poles) converge to the singularities of the
approximated function; this is where logarithmic potential theory
becomes effective, see Section .

Iterating the previous steps coupled with a sensitivity analysis yields a tuning procedure which was first demonstrated in on resonant microwave filters.

Similar steps can be taken to approach design problems in the frequency domain, replacing measured behavior by desired behavior. However, describing achievable responses from the design parameters at hand is generally cumbersome, and most constructive techniques rely on rather specific criteria adapted to the physics of the problem. This is especially true of circuits and filters, whose design classically appeals to standard polynomial extremal problems and realization procedures from system theory , . APICS is active in this field, where we introduced the use of Zolotarev-like problems for microwave multi-band filter design. We currently favor interpolation techniques because of their transparency with respect to parameter use, see Section .

In another connection, the example of harmonic identification
quickly suggests a generalization
of itself. Indeed, on identifying *i.e.*, the field) on part of a hypersurface (a curve in 2-D)
encompassing the support of

Inverse potential problems are severely indeterminate because infinitely many measures within an open set produce the same field outside this set . In step 1 above we implicitly removed this indeterminacy by requiring that the measure be supported on the boundary (because we seek a function holomorphic throughout the right half space), and in step 2 by requiring, say, in case of rational approximation that the measure be discrete in the left half-plane. The same discreteness assumption prevails in 3-D inverse source problems. To recap, the gist of our approach is to approximate boundary data by (boundary traces of) fields arising from potentials of measures with specific support. Note this is different from standard approaches to inverse problems, where descent algorithms are applied to integration schemes of the direct problem; in such methods, it is the equation which gets approximated (in fact: discretized).

Along these lines, the team initiated the use of steps 1 and 2 above, along with singularity analysis, to approach issues of nondestructive control in 2 and 3-D , . We are currently engaged in two kinds of generalization, further described in Section . The first one deals with non-constant conductivities, where Cauchy-Riemann equations for holomorphic functions are replaced by conjugate Beltrami equations for pseudo-holomorphic functions; there we seek applications to inverse free boundary problems such as plasma confinement in the vessel of a tokamak. The other one lies with inverse source problems for Laplace's equation in 3-D, where holomorphic functions are replaced by harmonic gradients, developing applications to EEG/MEG and inverse magnetization problems in paleomagnetism, see Section .

The main approximation-theoretic tools developed by APICS to get to grips with issues mentioned so far are outlined in Section . In Section to come, we make more precise which problems are considered and for which applications.

This work has benefited from collaboration with Alexander Borichev (Aix-Marseille University).

Reconstructing Dirichlet-Neumann boundary conditions
for a function harmonic in a plane domain
when these are known on a strict subset

A recent application by the team deals with non-constant conductivity
over a doubly connected domain,

When the domain is regarded as separating the edge of a tokamak's vessel
from the plasma (rotational symmetry makes this a 2-D problem),
the procedure just described suits plasma control from magnetic confinement.
It was successfully applied in collaboration with CEA
(the French nuclear agency) and the University of Nice (JAD Lab.)
to data from *Tore Supra* . This procedure is fast because no numerical integration of
the underlying PDE is needed, as an explicit basis of solutions to the
conjugate Beltrami equation in terms of Bessel functions
was found in this case. Generalizing this approach in a more systematic
manner into descent
algorithms for boundary-value criteria
using the gradient of a shape is an interesting perspective.

Three-dimensional versions of step 1 in Section are also considered, namely to recover a harmonic function (up to a constant) in a ball or a half-space from partial knowledge of its gradient on the boundary. Such questions arise naturally in connection with neurosciences and medical imaging (electroencephalography, EEG) or in paleomagnetism (analysis of rocks magnetization) , , see Section . They are not yet as developed as the 2-D case where the power of complex analysis is at work, but considerable progress was made over the last years through methods of harmonic analysis and operator theory.

The team is also concerned with non-destructive control problems of localizing defaults such as cracks, sources or occlusions in a planar or 3-dimensional domain, from boundary data (which may correspond to thermal, electrical, or magnetic measurements). These defaults can be expressed as a lack of analyticity of the solution of the associated Dirichlet-Neumann problem and we approach them using techniques of best rational or meromorphic approximation on the boundary of the object , , see Sections and . In fact, the way singularities of the approximant relate to the singularities of the approximated function is an all-pervasive theme in approximation theory, and for appropriate classes of functions like those expressed as Cauchy integrals over certain extremal contours for the logarithmic potential, the location of the poles of a best rational approximant can be used as an estimator of the singularities of the approximated function (see Section ). This circle of ideas is driving step 2 in Section .

A genuine 3-dimensional theory of approximation by discrete potentials, though, is still in its infancy.

Through initial contacts with CNES, the French space agency,
the team came to work on identification-for-tuning
of microwave electromagnetic filters used in space telecommunications
(see Section ). The problem was
to recover, from band-limited frequency measurements, the physical
parameters of the device under examination.
The latter consists of interconnected dual-mode resonant cavities with
negligible loss, hence its scattering matrix is modeled by a

This is where system theory enters the scene, through the
so-called *realization* process mapping
a rational transfer function in the frequency domain
to a state-space representation of the underlying system
of linear differential equations in the time domain.
Specifically, realizing the scattering matrix
allows one to construct
a virtual electrical network, equivalent to the filter,
the parameters of which mediate in between the frequency response
and the
geometric characteristics of the cavities (*i.e.* the tuning parameters).

Hardy spaces, in particular the Hilbert space

infer from the pointwise boundary data in the bandwidth
a stable transfer function (*i.e.* one which is holomorphic
in the right half-plane), that may be infinite dimensional
(numerically: of high degree). This is done by solving in the Hardy space

From this stable model, a rational stable approximation of appropriate degree is computed. For this a descent method is used on the relatively compact manifold of inner matrices of given size and degree, using an original parametrization of stable transfer functions developed by the team .

From this rational model, realizations meeting certain constraints imposed by the technology in use are computed. These constraints typically come from the nature and coupling topology of the equivalent electrical network used to model the filter. This network is composed of resonators, coupled to each other by some specific coupling graph. Performing this realization step for given coupling topology can be recast, under appropriate compatibility conditions , as the problem of solving a zero-dimensional multivariate polynomial system. To tackle this problem in practice, we use Groebner basis techniques as well as continuation methods as implemented in the Dedale-HF software (see Section ).

Let us also mention that extensions of classical coupling matrix theory to frequency-dependent (reactive) couplings have lately been carried-out for wide-band design applications, although further study is needed to make them computationally effective.

Subsequently APICS started investigating issues pertaining to filter design rather than identification. Given the topology of the filter, a basic problem is to find the optimal response with respect to amplitude specifications in frequency domain bearing on rejection, transmission and group delay of scattering parameters. Generalizing the approach based on Chebyshev polynomials for single band filters, we recast the problem of multi-band response synthesis in terms of a generalization of classical Zolotarev min-max problem for rational functions . Thanks to quasi-convexity, the latter can be solved efficiently using iterative methods relying on linear programming. These are implemented in the software easy-FF (see Section ).

Investigations by the team have extended to design and
identification of more complex microwave devices,
like multiplexers and routers, which connect several
filters through wave guides.
Schur analysis plays an important role here, which is no surprise
since scattering matrices of passive systems are of Schur type
(*i.e.* contractive in the stability region).
The theory originates with the work of I. Schur ,
who devised a recursive test to
check for contractivity of a holomorphic function in the disk.
Generalizations thereof turned out to be very efficient to parametrize
solutions to contractive interpolation problems subject to
a well-known compatibility condition (positive definiteness of the so-called
Pick matrix) .
Schur analysis became quite popular
in electrical engineering, as the Schur recursion precisely describes how
to chain two-port circuits.

Dwelling on this, members of the team contributed to differential parametrizations (atlases of charts) of lossless matrix functions , . These are fundamental to our rational approximation software RARL2 (see Section ). Schur analysis is also instrumental to approach de-embedding issues considered in Section , and provides further background to synthesis and matching problems for multiplexers. At the heart of the latter lies a variant of contractive interpolation with degree constraint introduced in .

We also mention the role played by multi-point Schur analysis in the team's investigation of spectral representation for certain non-stationary discrete stochastic processes , .

More recently, in collaboration with UPV (Bilbao),
our attention was driven by CNES,
to questions of stability relative to high-frequency amplifiers,
see Section .
Contrary to previously mentioned devices, these are *active* components.
The amplifier can be linearized at a functioning point
and admittances of the corresponding electrical network
can be computed at various frequencies, using the so-called harmonic
balance method.
The goal is to check for stability of this linearized model.
The latter is composed of lumped electrical elements namely
inductors, capacitors, negative *and* positive reactors,
transmission lines, and commanded current sources.
Research so far focused on determining the algebraic structure
of admittance functions, and setting up a function-theoretic framework to
analyze them. In particular, much effort was put on realistic assumptions
under which a stable/unstable decomposition can be claimed in

The following people are collaborating with us on these topics: Bernard Hanzon (Univ. Cork, Ireland), Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris), Jonathan Partington (Univ. Leeds, UK), Ralf Peeters (Univ. Maastricht, NL), Edward Saff (Vanderbilt University, Nashville, USA), Herbert Stahl (TFH Berlin), Maxim Yattselev (Purdue Univ. at Indianapolis, USA).

To find an analytic function in

Here *a priori*
assumptions on
the behavior of the model off

To fix terminology we refer to *bounded extremal problem*.
As shown in , ,
, for

(

The case

Various modifications of

The analog of Problem *seek the inner
boundary*, knowing it is a level curve of the flux.
Then, the Lagrange parameter indicates
which deformation should be applied on the inner contour in order to improve
data fitting.

This is discussed in Sections and
for more general equations than the Laplacian, namely
isotropic conductivity equations of the form

Though originally considered in dimension 2,
Problem

When

When

Problem

Such problems arise in connection with source recovery in electro/magneto encephalography and paleomagnetism, as discussed in Sections and .

The techniques explained in this section are used to solve
step 2 in Section *via* conformal mapping
and subsequently are instrumental to
approach inverse boundary value problems
for Poisson equation

Let as before

A natural generalization of Problem

(

Only for

The case *complement* of *stable* rational
approximant to *not* be unique.

The former Miaou project (predecessor of APICS) has designed an
adapted steepest-descent algorithm
for the case *local minimum* is
guaranteed; until now it seems to be the only procedure meeting this
property. Roughly speaking, it is a gradient algorithm that proceeds
recursively with respect to the order *critical points* of lower degree
(as is done by the RARL2 software, Section ).

In order to establish global convergence results, APICS has undertaken a
deeper study of the number and nature of critical points, in which
tools from differential topology and
operator theory team up with classical approximation theory.
The main discovery is that
the nature of the critical points
(*e.g.*, local minima, saddle points...)
depends on the decrease of the interpolation
error to *i.e.*
Markov functions) and more
generally Cauchy integrals over hyperbolic geodesic arcs
and certain entire functions .

An analog to AAK theory
has been carried out for

A common
feature to all these problems
is that critical point equations
express non-Hermitian orthogonality relations for the denominator
of the approximant. This makes connection with interpolation theory
, and
is used in an essential manner to assess the
behavior of the poles of the approximants to functions with branchpoint-type
singularities,
which is of particular interest for inverse source problems
(*cf.* Sections and ).

In higher dimensions, the analog of Problem

Certain constrained rational approximation problems, of special interest in identification and design of passive systems, arise when putting additional requirements on the approximant, for instance that it should be smaller than 1 in modulus. Such questions have attracted significant attention of members of the team (see Section ). For instance, convergence properties of multi-point Schur approximants, which are rational interpolants preserving contractivity of a function, were analyzed in . Such approximants are useful in prediction theory of stochastic processes, but since they interpolate inside the domain of holomorphy they are of limited use in frequency design.

In another connection, the generalization to several arcs of classical Zolotarev problems is an achievement by the team which is useful for multi-band synthesis . Still, though the modulus of the response is the first concern in filter design, variation of the phase must nevertheless remain under control to avoid unacceptable distortion of the signal. This specific but important issue has less structure and was approached using constrained optimization; a dedicated code has been developed under contract with the CNES (see Section ).

Matrix-valued approximation is necessary for handling systems with several
inputs and outputs, and it generates substantial additional difficulties
with respect to scalar approximation,
theoretically as well as algorithmically. In the matrix case,
the McMillan degree (*i.e.* the degree of a minimal realization in
the System-Theoretic sense) generalizes the degree.

The problem we consider is now:
*let $\mathcal{F}\in {\left({H}^{2}\right)}^{m\times l}$ and $n$ an
integer; find a rational matrix of size $m\times l$ without
poles in the unit disk and of McMillan degree at most $n$ which is nearest possible
to $\mathcal{F}$ in ${\left({H}^{2}\right)}^{m\times l}$.*
Here the

The scalar approximation algorithm , mentioned in Section ,
generalizes to
the matrix-valued situation . The
first difficulty here consists in the parametrization
of transfer matrices of given
McMillan degree *i.e.* matrix-valued functions
that are analytic in the unit disk and unitary on the circle) of degree

Difficulties relative to multiple local minima naturally arise in the matrix-valued case as well, and deriving criteria that guarantee uniqueness is even more difficult than in the scalar case. The case of rational functions of sought degree or small perturbations thereof (the consistency problem) was solved in . The case of matrix-valued Markov functions, the first example beyond rational functions, was treated in .

Let us stress that the algorithms mentioned above are first to handle rational approximation in the matrix case in a way that converges to local minima, while meeting stability constraints on the approximant.

The following persons did collaborate with us on this subject: Herbert Stahl (TFH Berlin), Maxim Yattselev (Purdue Univ. at Indianapolis, USA).

We refer here to the behavior of poles of best
meromorphic approximants, in the

Generally speaking, the
behavior of poles is particularly important in meromorphic approximation
to obtain error rates as the degree goes large and to tackle
constructive issues like
uniqueness. As explained in Section ,
we consider this issue in connection with
approximation of the solution to a
Dirichlet-Neumann problem, so as to extract information on the
singularities. The general theme is thus *how do the singularities
of the approximant reflect those of the approximated function?*
This approach to inverse problem for the 2-D Laplacian turns out
to be attractive when singularities
are zero- or one-dimensional (see Section ). It can be used
as a computationally cheap
initialization of more precise but heavier
numerical optimizations.

As regards crack detection or source recovery, the approach in
question boils
down to
analyzing the behavior of best meromorphic
approximants of a function with branch points.
For piecewise analytic cracks, or in the case of sources, we were able to
prove (, , ),
that the poles of the
approximants accumulate on some extremal contour of minimum weighted energy
linkings the singular points of the crack, or the sources
.
Moreover, the asymptotic density
of the poles turns out to be the Green equilibrium distribution
of this contour in

The case of two-dimensional singularities is still an outstanding open problem.

It is interesting that inverse source problems inside a sphere or an ellipsoid in 3-D can be attacked with the above 2-D techniques, as applied to planar sections (see Section ). This is at work in the software FindSources3D, see Section .

Sylvain Chevillard, joined team in November 2010. His coming
resulted in APICS hosting a research activity in certified computing,
centered on the software *Sollya* of which S. Chevillard is a
co-author, see Section . On the one hand, Sollya is an
Inria software which still requires some tuning to a growing community of
users. On the other hand, approximation-theoretic methods
at work in Sollya are potentially useful for certified solutions to
constrained analytic problems described in Section .
However, developing Sollya is not a long-term objective of APICS.

These domains are naturally linked to the problems described in Sections and . By and large, they split into a systems-and-circuits part and an inverse-source-and-boundary-problems part, united under a common umbrella of function-theoretic techniques described in Section .

This work is done in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena Project-Team, and Jean-Paul Marmorat (Centre de mathématiques appliquées - CMA, École des Mines de Paris).

Solving overdetermined Cauchy problems for the Laplace equation on a
spherical layer (in 3-D) in order to extrapolate
incomplete data (see Section ) is
a necessary
ingredient of the team's approach to inverse source problems, in particular
for applications to EEG since the latter involves propagating the
initial conditions through several layers of different conductivities,
from the boundary
down to the center of the domain where the
singularities (*i.e.* the sources) lie.
Then, once propagated
to the innermost sphere, it turns out that that traces of the
boundary data on 2-D cross sections (disks) coincide
with analytic functions in the slicing plane,
that has branched singularities inside the disk . These
singularities are
related to the actual location of the sources (namely, they reach in turn a
maximum in modulus when the plane contains one of the sources). Hence, we are
back to the 2-D framework of Section
where approximately recovering these singularities
can be performed using best rational approximation.
The goal is to produce a fast but already good enough
initial guess on the number
and location of the sources in order to run heavier
descent algorithms on the direct problem, which are more precise but
computationally costly, and often
fail to converge if not properly initialized.

Numerical experiments give very good results on simulated data and we are now engaged in the process of handling real experimental magneto-encephalographic data, see also Sections and , in collaboration with the Athena team at Inria Sophia Antipolis, neuroscience teams in partner-hospitals (la Timone, Marseille), and the BESA company (Munich).

Generally speaking, inverse potential problems similar to the one in Section appear naturally in connection with systems governed by Maxwell's equation in the quasi-static approximation regime. In particular, they arise in magnetic reconstruction issues. A specific application is to geophysics, whose study led us to form an Inria Associate Team (“IMPINGE” for Inverse Magnetization Problems IN GEosciences) together with MIT and Vanderbilt University.

To set up the context, recall that the Earth's geomagnetic field is generated by convection of the liquid metallic core (geodynamo) and that rocks become magnetized by the ambient field as they are formed or after subsequent alteration. Their remanent magnetization provides records of past variations of the geodynamo, which is used to study important processes in Earth sciences like motion of tectonic plates and geomagnetic reversals. Rocks from Mars, the Moon, and asteroids also contain remanent magnetization which indicates the past presence of core dynamos. Magnetization in meteorites may even record fields produced by the young sun and the protoplanetary disk which may have played a key role in solar system formation.

For a long time, paleomagnetic techniques were only capable of analyzing bulk samples and compute their net magnetic moment. The development of SQUID microscopes has recently extended the spatial resolution to submillimeter scales, raising new physical and algorithmic challenges. This associate team aims at tackling them, experimenting with the SQUID microscope set up in the Paleomagnetism Laboratory of the department of Earth, Atmospheric and Planetary Sciences at MIT. Typically, pieces of rock are sanded down to a thin slab, and the magnetization has to be recovered from the field measured on a parallel plane at small distance above the slab.

Mathematically
speaking, both inverse source problems for EEG from Section and inverse magnetization problems described presently
amount to recover the (3-D valued) quantity

outside the volume

The team has engaged in the study of
problems with variable conductivity
*cf.* in particular,
the PhD thesis of S. Chaabi , ,
jointly supervised with the CMI-LATP at the Aix-Marseille University.
(see Section ).

This work is done in collaboration with Stéphane Bila (XLIM, Limoges) and Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris).

One of the best training grounds for the research of the team in function theory is the identification and design of physical systems for which the linearity assumption works well in the considered range of frequency, and whose specifications are made in the frequency domain. This is the case of electromagnetic resonant systems which are of common use in telecommunications.

In space telecommunications (satellite transmissions), constraints specific to on-board technology lead to the use of filters with resonant cavities in the microwave range. These filters serve multiplexing purposes (before or after amplification), and consist of a sequence of cylindrical hollow bodies, magnetically coupled by irises (orthogonal double slits). The electromagnetic wave that traverses the cavities satisfies the Maxwell equations, forcing the tangent electrical field along the body of the cavity to be zero. A deeper study of the Helmholtz equation states that essentially only a discrete set of wave vectors is selected. In the considered range of frequency, the electrical field in each cavity can be seen as being decomposed along two orthogonal modes, perpendicular to the axis of the cavity (other modes are far off in the frequency domain, and their influence can be neglected).

Near the resonance frequency, a good approximation of the Maxwell equations is given by the solution of a second order differential equation. One obtains thus an electrical model for our filter as a sequence of electrically-coupled resonant circuits, and each circuit will be modeled by two resonators, one per mode, whose resonance frequency represents the frequency of a mode, and whose resistance represent the electric losses (current on the surface).

In this way, the filter can be seen as a quadripole, with two ports, when
plugged on a resistor at one end and fed with some potential at the other end.
We are
then interested in the power which is transmitted and reflected. This leads to
defining a
scattering matrix

In fact, resonance is not studied via the electrical model,
but via a low-pass
equivalent circuit obtained upon linearizing near the central frequency, which is no
longer
conjugate symmetric (*i.e.* the underlying system may not have real
coefficients) but whose degree is divided by 2 (8 in the example).

In short, the identification strategy is as follows:

measuring the scattering matrix of the filter near the optimal frequency over twice the pass band (which is 80MHz in the example).

Solving bounded extremal problems for the transmission and the reflection (the modulus of he response being respectively close to 0 and 1 outside the interval measurement, cf. Section ). This provides us with a scattering matrix of order roughly 1/4 of the number of data points.

Approximating this scattering matrix by a rational transfer-function of fixed degree (8 in this example) via the Endymion or RARL2 software (cf. Section ).

A realization of the transfer function is thus obtained, and some additional symmetry constraints are imposed.

Finally one builds a realization of the approximant and looks for a change of variables that eliminates non-physical couplings. This is obtained by using algebraic-solvers and continuation algorithms on the group of orthogonal complex matrices (symmetry forces this type of transformation).

The final approximation is of high quality. This can be interpreted as
a validation of the linearity hypothesis for the system:
the relative

The above considerations are valid for a large class of filters. These developments have also been used for the design of non-symmetric filters, useful for the synthesis of repeating devices.

The team also investigates problems relative to the design of optimal responses for microwave devices. The resolution of a quasi-convex Zolotarev problems was for example proposed, in order to derive guaranteed optimal multi-band filter's responses subject to modulus constraints . This generalizes the classical single band design techniques based on Chebyshev polynomials and elliptic functions. These techniques rely on the fact that the modulus of the scattering parameters of a filters, say

The filtering function appears to be the ratio of two polynomials

The relative simplicity of the derivation of a filter's response under
modulus constraints is due to the possibility of
"forgetting" about Feldtkeller's equation, and express all design constraints
in terms of the filtering function

Status: Currently under development. A stable version is maintained.

This software is developed in collaboration with Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris).

RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for
rational approximation (see Section )
http://

The software RARL2 computes, from a given matrix-valued function in *stable and of prescribed McMillan degree*
(see Section ). It was initially developed in the context of linear (discrete-time) system theory and makes an heavy use of the classical concepts in this field. The matrix-valued function to be approximated can be viewed as the transfer function of a multivariable discrete-time stable system. RARL2 takes as input either:

its internal realization,

its first

discretized (uniformly distributed) values on the circle. In this case, a least-square criterion is used instead of
the

It thus performs model reduction in case 1) and 2) and frequency data identification in case 3). In the case of band-limited frequency data, it could be necessary to infer the behavior of the system outside the bandwidth before performing rational approximation (see Section ). An appropriate Möbius transformation allows to use the software for continuous-time systems as well.

The method is a steepest-descent algorithm. A parametrization of MIMO systems is used, which ensures that the stability constraint on the approximant is met. The implementation, in Matlab, is based on state-space representations.

The number of local minima can be rather high so that the choice of an initial point for the optimization can play a crucial role. Two methods can be used: 1) An initialization with a best Hankel approximant. 2) An iterative research strategy on the degree of the local minima, similar in principle to that of RARL2, increases the chance of obtaining the absolute minimum by generating, in a structured manner, several initial conditions.

RARL2 performs the rational approximation step in our applications to filter identification (see Section ) as well as sources or cracks recovery (see Section ). It was released to the universities of Delft, Maastricht, Cork and Brussels. The parametrization embodied in RARL2 was also used for a multi-objective control synthesis problem provided by ESTEC-ESA, The Netherlands. An extension of the software to the case of triple poles approximants is now available. It provides satisfactory results in the source recovery problem and it is used by FindSources3D (see Section ).

Status: A stable version is maintained.

This software is developed in collaboration with Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris).

The identification of filters modeled by an electrical
circuit that was developed by the team (see Section )
led us to compute the electrical parameters of the underlying
filter. This means finding a particular realization

Status: Currently under development. A stable version is maintained.

PRESTO-HF: a toolbox dedicated to lowpass parameter identification for microwave filters http://www-sop.inria.fr/apics/Presto-HF. In order to allow the industrial transfer of our methods, a Matlab-based toolbox has been developed, dedicated to the problem of identification of low-pass microwave filter parameters. It allows one to run the following algorithmic steps, either individually or in a single shot:

determination of delay components caused by the access devices (automatic reference plane adjustment),

automatic determination of an analytic completion, bounded in modulus for each channel,

rational approximation of fixed McMillan degree,

determination of a constrained realization.

For the matrix-valued rational approximation step, Presto-HF relies on RARL2 (see Section ), a rational approximation engine developed within the team. Constrained realizations are computed by the RGC software. As a toolbox, Presto-HF has a modular structure, which allows one for example to include some building blocks in an already existing software.

The delay compensation algorithm is based on the following strong assumption:
far off the passband, one can reasonably expect a good approximation of the
rational components of

This toolbox is currently used by Thales Alenia Space in Toulouse, Thales airborn systems and a license agreement has been recently negotiated with TAS-Espagna. XLIM (University of Limoges) is a heavy user of Presto-HF among the academic filtering community and some free license agreements are currently being considered with the microwave department of the University of Erlangen (Germany) and the Royal Military College (Kingston, Canada).

Status: Currently under development. A stable version is maintained.

Dedale-HF is a software dedicated to solve exhaustively the coupling matrix synthesis problem in reasonable time for the users of the filtering community. For a given coupling topology, the coupling matrix synthesis problem (C.M. problem for short) consists in finding all possible electromagnetic coupling values between resonators that yield a realization of given filter characteristics. Solving the latter problem is crucial during the design step of a filter in order to derive its physical dimensions as well as during the tuning process where coupling values need to be extracted from frequency measurements (see Figure ).

Dedale-HF consists in two parts: a database of coupling topologies as well as
a dedicated predictor-corrector code. Roughly speaking each reference file of
the database contains, for a given coupling topology, the complete solution
to the C.M. problem associated to particular filtering characteristics. The
latter is then used as a starting point for a predictor-corrector integration
method that computes the solution to the C.M. problem of the user,
*i.e.* the one corresponding to user-specified filter characteristics. The
reference files are computed off-line using Groebner basis techniques or
numerical techniques based on the exploration of a monodromy group. The use of
such a continuation technique combined with an efficient implementation of the
integrator produces a drastic reduction, by a factor of 20, of the computational time.

Access to the database and integrator code is done via the web on http://www-sop.inria.fr/apics/Dedale/WebPages. The software is free of charge for academic research purposes: a registration is however needed in order to access full functionality. Up to now 90 users have registered world wide (mainly: Europe, U.S.A, Canada and China) and 4000 reference files have been downloaded.

A license of this software has been sold end of 2011, to TAS-Espagna, in order for it to tune filters with topologies having multiple solutions. The use of Dedale-HF is here coupled with that of Presto-HF.

Status: A stable version is maintained.

This software has been developed by Vincent Lunot (Taiwan Univ.) during his PhD. He still continues to maintain it.

EasyFF is a software dedicated to the computation of complex, and in particular multi-band, filtering functions. The software takes as input, specifications on the modulus of the scattering matrix (transmission and rejection), the filter's order and the number of transmission zeros. The output is an "optimal" filtering characteristic in the sense that it is the solution of an associated min-max Zolotarev problem. Computations are based on a Remez-type algorithm (if transmission zeros are fixed) or on linear programming techniques if transmission zeros are part of the optimization .

Status: Currently under development. A stable version is maintained.

This software is developed in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena Project-Team, and with Jean-Paul Marmorat (Centre de mathématiques appliquées - CMA, École des Mines de Paris).

FindSources3D

Status: Currently under development. A stable version is maintained.

This software is developed in collaboration with Christoph Lauter (LIP6) and Mioara Joldeş (LAAS).

Sollya is an interactive tool where the developers of mathematical floating-point libraries (libm) can experiment before actually developing code. The environment is safe with respect to floating-point errors, *i.e.* the user precisely knows when rounding errors or approximation errors happen, and rigorous bounds are always provided for these errors.

Among other features, it offers a fast Remez algorithm for computing polynomial approximations of real functions and also an algorithm for finding good polynomial approximants with floating-point coefficients to any real function. It also provides algorithms for the certification of numerical codes, such as Taylor Models, interval arithmetic or certified supremum norms.

It is available as a free software under the CeCILL-C license at http://

This section is concerned with inverse problems for 3-D Poisson-Laplace equations. Though the geometrical settings differ in the 2 sections below, the characterization of silent sources (that give rise to a vanishing potential at measurement points) is one of the common problems to both which has been recently achieved in the magnetization setup, see .

This work is conducted in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena Project-Team, and with Jean-Paul Marmorat (Centre de mathématiques appliquées - CMA, École des Mines de Paris).

In 3-D, functional or clinical active regions in the cortex are often modeled by point-wise sources that have to be localized from measurements on the scalp of a potential satisfying a Laplace equation (EEG, electroencephalography). In the work it was shown how to proceed via best rational approximation on a sequence of 2-D disks cut along the inner sphere, for the case where there are at most 2 sources. Last year, a milestone was reached in the research on the behavior of poles in best rational approximants of fixed degree to functions with branch points , to the effect that the technique carries over to finitely many sources (see Section ).

In this connection, a dedicated software “FindSources3D”
is being developed, in collaboration with the team Athena and the CMA. We took on this year algorithmic developments, prompted by
recent and promising contacts with the firm BESA (see Section ),
namely automatic detection of the number of sources
(which is left to the user at the moment) and simultaneous processing
of data from several time instants.
It appears that in the rational approximation step,
*multiple* poles possess a nice behavior with respect to branched
singularities. This is due to the very physical assumptions on the model
(for EEG data, one should consider *triple* poles). Though numerically
observed in , there is no mathematical
justification so far why multiple poles generate such strong accumulation
of the poles of the approximants. This intriguing property, however,
is definitely helping source recovery.
It is used in order to automatically estimate the “most plausible”
number of sources (numerically: up to 2, at the moment).

In connection with the work related to inverse magnetization issues (see Section ), the characterization of silent sources for EEG has been carried out . These are sums of (distributional) derivatives of Sobolev functions vanishing on the boundary.

In a near future, magnetic data from MEG (magneto-encephalography) will become available along with EEG data; indeed, it is now possible to use simultaneously corresponding measurement devices, in order to measure both electrical and magnetic fields. This should enhance the accuracy of our source recovery algorithms.

Let us mention that discretization issues in geophysics can also be approached by such techniques. Namely, in geodesy or for GPS computations, one is led to seek a discrete approximation of the gravitational potential on the Earth's surface, from partial data collected there. This is the topic of a beginning collaboration with physicist colleagues (IGN, LAREG, geodesy). Related geometrical issues (finding out the geoid, level surface of the gravitational potential) are worthy of consideration as well.

This work is carried out in the framework of the “équipe associée Inria” IMPINGE, comprising Eduardo Andrade Lima and Benjamin Weiss from the Earth Sciences department at MIT (Boston, USA) and Douglas Hardin and Edward Saff from the Mathematics department at Vanderbilt University (Nashville, USA),

Localizing magnetic sources from measurements of the magnetic field
away from the support of the magnetization is the fundamental
issue under investigation by IMPINGE The goal is to determine
magnetic properties of rock
samples (*e.g.* meteorites or stalactites) from fine field measurements
close to the sample that
can nowadays be obtained using SQUIDs (supraconducting coil devices).
Currently, rock samples are cut into thin slabs and the magnetization
distribution is considered to lie in a plane, which makes for a
somewhat less indeterminate framework than EEG as regards inverse problems
because “less” magnetizations can produce the same field
(for the slab has no inner volume).

The magnetization operator
is the Riesz potential of the divergence of the magnetization,
see ().
Last year, the problem of recovering a thin plate magnetization distribution
from measurements of the field in a plane above the sample
led us to an analysis of the kernel of this operator, which we
characterized in various functional and distributional spaces .
Using a generalization of the Hodge decomposition, we were able
to describe all magnetizations equivalent to a given one.
Here, equivalent means that the magnetizations generate the same field from
above and from below if, say, the slab is horizontal. When magnetizations have bounded support, which is the
case for rock samples, we proved that magnetizations equivalent from above
are also equivalent from below, but this is no longer true for unbounded
supports. In fact, even for unidirectional magnetizations,
uniqueness of a magnetization generating a given field depends on
the boundedness of the support, as we proved that *any* magnetization
is equivalent from above to a unidirectional one (with infinite support in
general). This
helps explaining why methods in the Fourier domain (which essentially loose
track of the support information) do encounter problems. It also
shows that
information on the support must be used in a crucial way
to solve the problem.

This year, we produced a fast inversion scheme for magnetic field maps of unidirectional planar geological magnetization with discrete support located on a regular grid, based on discrete Fourier transform . Figures , , and show an example of reconstruction. As the just mentioned article shows, the Fourier approach is computationally attractive but undergoes aliasing phenomena that tend to offset its efficiency. In particular, estimating the total moment of the magnetization sample seems to require data extrapolation techniques which are to take place in the space domain. This is why we have started to study regularization schemes based on truncation of the support in connection with singular values analysis of the discretized problem.

In a joint effort by all members of IMPINGE, we set up a heuristics to recover dipolar magnetizations, using a discrete least square criterion. At the moment, it is solved by a singular value decomposition procedure of the magnetization-to-field operator, along with a regularization technique based on truncation of the support. Preliminary experiments on synthetic data give quite accurate results to recover the net moment of a sample, see
the preliminary document http://

This shows that the technique we use to reduce the support, which is based on
thresholding contributions of dipoles to the observations, is capable
of eliminating some nearly silent dipole distributions which flaw the singular
value analysis.
In order to better understand the geometric nature of such distributions,
and thus affirm theoretical bases to the above mentioned heuristics,
we raised the question of determining an eigenbasis for the
positive self adjoint
operator mapping a

This is not such an easy problem and currently, in the framework of the PhD thesis of D. Ponomarev, we investigate a simplified two-dimensional analog, defined via convolution of a function on a segment with the Poisson kernel of the upper half-plane and then restriction to a parallel segment in that half-plane. Surprisingly perhaps, this issue was apparently not considered in spite of its natural character and the fact that it makes contact with classical spectral theory. Specifically, it amounts to spectral representation of certain compressed Toeplitz operators with exponential-of-modulus symbols. Beyond the bibliographical research needed to understand the status of this question, only preliminary results have been attained so far.

This work was the occasion of collaborations with Alexander Borichev (Aix-Marseille University), Jonathan Partington (Univ. Leeds, UK), and Emmanuel Russ (Univ. Grenoble, IJF).

As we mentioned in Section
2-D diffusion equations of the form

The study of such Hardy spaces for Lipschitz

In 2013, completing a study begun last year in the framework of
the PhD of S. Chaabi, we established similar
results in the case where

The PhD work of S. Chaabi (defended December 2) contains further work on the Weinstein equation and certain generalizations thereof. This equation results from 2-D projection of Laplace's equation in the presence of rotation symmetry in 3-D. In particular, it is the equation governing the free boundary problem of plasma confinement in the plane section of a tokamak. A method dwelling on Fokas's approach to elliptic boundary value problems has been developed which uses Lax pairs and solves for a Riemann-Hilbert problem on a Riemann surface. It was used to devise semi-explicit forms of solutions to Dirichlet and Neumann problems for the conductivity equation satisfied by the poloidal flux.

In another connection, the conductivity equation can also be regarded as a static Schrödinger equation for smooth coefficients. In particular, a description of laser beam propagation in photopolymers can be crudely approximated by a stationary two-dimensional model of wave propagation in a medium with negligible change of refractive index. In this setting, Helmholtz equation is approximated by a linear Schrödinger equation with one spatial coordinate as evolutionary variable. This phenomenon can be described by a non-stationary model that relies on a spatial nonlinear Schrödinger (NLS) equation with time-dependent refractive index. A model problem has been considered in , when the rate of change of refractive index is proportional to the squared amplitude of the electric field and the spatial domain is a plane.

We have also studied composition operators on generalized Hardy spaces in the framework of . In the work submitted for publication, we provide necessary and/or sufficient conditions on the composition map, depending on the geometry of the domains, ensuring that these operators are bounded, invertible, isometric or compact.

Several questions about the behavior of solutions to the
bounded extremal problem

This work has been done in collaboration with Stéphane Bila (XLIM, Limoges, France), Hussein Ezzedin (XLIM, Limoges, France), Damien Pacaud (Thales Alenia Space, Toulouse, France), Giuseppe Macchiarella (Politecnico di Milano, Milan, Italy), and Matteo Oldoni (Siae Microelettronica, Milan, Italy).

We focused our research on multiplexer with a star topology. These are
comprised of a central

where

This problem can be seen as an extended Nevanlinna-Pick interpolation problem, which was considered in when the interpolation
frequencies lie in the *open* left half-plane.
Last year we conjectured the existence and uniqueness of a solution,
which were eventually proved true this year when

This research lies at heart of our collaboration with CNES on multiplexer synthesis and the core of the starting ANR project COCORAM on co-integration of filters and antennas (see Section ).

Let

It was therefore natural to consider following de-embedding problem. Given

the

the coupling geometry of their circuital realization is known,

what can be said about the filter's responses ? It was shown that under
the above hypotheses, in particular with no a priori knowledge of

This work is pursued in collaboration with Thales Alenia Space, Siae Microelettronica, XLIM and CNES in particular under contract with
CNES on compact

This work is conducted in collaboration with Jean-Baptiste Pomet from the McTao team. It is a continuation of a collaboration with CNES and the University of Bilbao.The goal is to help developing amplifiers, in particular to detect instability at an early stage of the design.

Currently, electrical engineers from the University of Bilbao, under contract with CNES (the French Space Agency), use heuristics to diagnose instability before the circuit is physically implemented. We intend to set up a rigorously founded algorithm, based on properties of transfer functions of such amplifiers which belong to particular classes of analytic functions.

In non-degenerate cases, non-linear electrical components can be replaced by their first order approximation when studying stability to small perturbations. Using this approximation, diodes appear as perfect negative resistors and transistors as perfect current sources controlled by the voltages at certain points of the circuit.

In previous years, we had proved that the class of transfer functions which can be realized with such ideal components and standard passive components (resistors, selfs, capacitors and transmission lines) is rather large since it contains all rational functions in the variable and in the exponentials thereof. This makes possible to design circuits that are unstable, although they have no pole in the right half-plane. This remains true even if a high resistor is put in parallel of the circuit, which is rather unusual. These pathological examples are unrealistic, though, because they assume that non-linear elements continue to provide gain even at very high frequencies. In practice, small capacitive and inductive effects (negligible at moderate frequencies) make these components passive for very high frequencies.

In 2013, we showed that under this simple assumption that there are small inductive and capacitive effects in active components, the class of transfer functions of realistic circuits is much smaller than in previous situation. Our main result is that a realistic circuit is unstable if and only if it has poles in the right half-plane. Moreover, there can only be finitely many of them. Besides this result, we also generalized our description of the class of transfer functions achievable with ideal components, to include the case of transmission lines with loss. An article is currently being written on this subject.

This work has been done in collaboration with Herbert Stahl (Beuth-Hochsch.), Maxim Yattselev (Purdue Univ. at Indianapolis, USA), Tao Qian (Univ. Macao).

We published last year an important result in
approximation theory, namely the counting measure of
poles of best

We also studied partial realizations, or equivalently Padé approximants to transfer functions with branchpoints. Identification techniques based on partial realizations of a stable infinite-dimensional transfer function are known to often provide unstable models, but the question as to whether this is due to noise or to intrinsic instability was not clear. This year, we published a paper showing that, in the case of 4 branchpoints, the pole behavior generically has deterministic chaos to it .

We also considered the issue of lower bounds in rational approximation.
Prompted by renewed interest for linearizing techniques
such as vector fitting in the identification community, we studied linearized
errors in light of the topological approach
in , to find that, when
properly normalized, they give rise to lower bounds in

The overall and long-term goal is to enhance the quality of numerical computations. The progress made during year 2013 is the following:

Publication of a work with Marc Mezzarobba (who was with Aric project-team at that time, and who is now with LIP6) about the efficient evaluation of the Airy

A more general endeavor is to develop a tool that helps developers of libms in their task. This is performed by the software Sollya

Contract (reference Inria: 7066, CNES: 127 197/00)
involving CNES, XLIM and Inria, focuses on the development
of synthesis procedures for

Contract (reference CNES: RS10/TG-0001-019) involving CNES, University of Bilbao (UPV/EHU) and Inria whose objective is to set up a methodology for testing the stability of amplifying devices. The work at Inria concerns the design of frequency optimization techniques to identify the linearized response and analyze the linear periodic components.

The ANR (Astrid) project COCORAM (Co-design et co-intégration de réseaux d’antennes actives multi-bandes pour systèmes de radionavigation par satellite) has been accepted and will officially start January 2014. We are associated in this project with three other teams from XLIM (Limoges University), specialized respectively on filters, antennas and amplifiers. The core idea of the project is to work on the co-integration of various microwave devices in the context of GPS satellite systems and in particular for us to work on matching problems (see Section ).

APICS is part of the European Research Network on System Identification (ERNSI) since 1992.

Subject: System identification concerns the construction, estimation and validation of mathematical models of dynamical physical or engineering phenomena from experimental data.

Title: Inverse Magnetization Problems IN GEosciences.

Inria principal investigator: Laurent Baratchart

International Partner (Institution - Laboratory - Researcher):

MIT - Department of Earth, Atmospheric and Planetary Sciences (United States) - Benjamin Weiss

Duration: 2013 - 2015

See details at : http://

The purpose of the associate team IMPINGE is to develop efficient algorithms to recover the magnetization distribution of rock slabs from measurements of the magnetic field above the slab using a SQUID microscope (developed at MIT). The US team also involves a group at Vanderbilt Univ.

**NSF CMG** collaborative research grant DMS/0934630,
“Imaging magnetization distributions in geological samples”, with Vanderbilt University and the MIT (USA).

**Cyprus NF grant**
“Orthogonal polynomials in the complex plane: distribution of zeros, strong asymptotics and shape reconstruction”.

**PHC Utique CMCU** (led by Fédération Denis Poisson, Univ. Orléans), “Harmonic analysis and applications”.

As mentioned in
Sections and , a cooperation with the German firm BESA

Douglas Hardin (Vanderbilt University, Nashville, USA, Jun 2013)

Matteo Oldoni (Siae Microelettronica, Milano, Italy, Nov 2013)

Vladimir Peller (Michigan University, East Lansing, from May until Jun 2013)

Yannick Privat (CNRS, Univ. P. et M. Curie, Paris, Dec 2013).

Tao Qian (University of Macau, Taipa, China, Jul 2013)

Edward Saff (Vanderbilt University, Nashville, USA, from May until Jun 2013)

Michael Stessin (New York state University at Albany, USA, Jun 2013)

Nikos Stylianopoulos (Univ. of Cyprus).

Ian Sloan (University of New South Wales, Sydney, Australia, Jun. 2013).

Maxim Yattselev (Indiana University–Purdue University, Indianapolis, USA, Mar 2013)

K. Bashtova, Master 2 Mathmods - UNSA (6 months), Inverse source problems for elecromagnetic fields, with physical applications.

Collaboration under contract with Thales Alenia Space (Toulouse, Cannes, and Paris), CNES (Toulouse), XLIM (Limoges), University of Bilbao (Universidad del País Vasco / Euskal Herriko Unibertsitatea, Spain).

Regular contacts with research groups at UST (Villeneuve d'Asq), Universities of Bordeaux-I (Talence), Orléans (MAPMO), Aix-Marseille (CMI-LATP), Nice Sophia Antipolis (Lab. JAD), Grenoble (IJF and LJK), Paris 6 (P. et M. Curie, Lab. JLL), Paris Diderot (LAREG-IGN), CWI (the Netherlands), MIT (Boston, USA), Vanderbilt University (Nashville USA), Steklov Institute (Moscow), Michigan State University (East-Lansing, USA), Texas A&M University (College Station USA), State University of New-York (Albany, USA), University of Oregon (Eugene, USA), Politecnico di Milano (Milan, Italy), University of Trieste (Italy), RMC (Kingston, Canada), University of Leeds (UK), of Maastricht (The Netherlands), of Cork (Ireland), Vrije Universiteit Brussel (Belgium), TU-Wien (Austria), TFH-Berlin (Germany), ENIT (Tunis), KTH (Stockholm), University of Cyprus (Nicosia, Cyprus), University of Macau (Macau, China), BESA company (Munich), SIAE Microelettronica (Milano).

The project is involved in the GDR-project AFHP (CNRS), in the ANR (Astrid program) project COCORAM (with XLIM, Limoges, and DGA), in a EMS21-RTG NSF program (with MIT, Boston, and Vanderbilt University, Nashville, USA), in the Associate Inria Team IMPINGE (with MIT, Boston), and in a CSF program (with University of Cyprus).

F. Seyfert was invited to give a talk at the department "Optimization and System Theory" of KTH University (Stockholm, Sweden), on "Generalized Nevanlinna-Pick interpolation on the boundary"

J. Leblond and D. Ponomarev gave communications at the *11 $th$ International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2013)*, Tunis, June (http://

M. Olivi gave a talk at the SSSC 2013 conference in Grenoble (France) and presented a poster at the ERNSI 2013 conference in Nancy (France).

E. Pozzi gave communications at
the *Spring School in Functional and Harmonic analysis and Operator theory*, Lens (may), at the *Workshop in Operator Theory, Harmonic and Complex Analysis*, Lille (may),
and at the
seminar of the Lab. J.-A. Dieudonné, Université Nice-Sophia Antipolis.

L. Baratchart was an invited speaker at the workshop "Inverse Problems and Nonlinear Equations", May, Palaiseau. He was an invited speaker at the workshop "Frames and Bases in Banach Spaces of Holomorphic Functions”, October, Bordeaux. He was an invited speaker at the conference in honor of A.A. Gonchar, November, Steklov Institute, Moscow. He was a speaker at CMFT 2013 (Shantou, China). He was an invited speaker at the seminar in Guangzhou University (China), the University of Bordeaux (LMB)and of Grenoble (Laboratoire J. Kuntzmann). He was a visitor at the University of Macao, MIT, and the University of Cyprus.

Master, PhD: J. Leblond, Inverse source problems, 3h, Franco-German summer school for inverse problems and PDE, Univ. Brême, All.

PhD: S. Chaabi, Analyse complexe et problèmes de Dirichlet dans le plan : équation de Weinstein et autres conductivités non-bornées, defended Dec.2d, 2013 (advisors: L. Baratchart, A. Borichev).

PhD in progress: D. Ponomarev, Inverse problems for planar conductivity and Schrödinger PDEs, since Nov. 2012 (advisors: J. Leblond, L. Baratchart).

PhD in progress: M. Caenepeel, A hierarchical framework for design oriented modeling, since Feb. 2013 (advisors: Y. Rolain, M. Olivi, F. Seyfert).

L. Baratchart was a referee of the PhD. manuscript of J. Vayssettes (Univ. Poitiers).

J. Leblond was a member of the hiring committee for a professor in applied mathematics, Univ. Lorraine, and of the PhD jury of A. Blandinières (Univ. Lyon), R. Tytgat (Aix-Marseille University), and A. Abdelmoula (Univ. Rennes, reviewer).

M. Olivi was a member (reviewer) of the HdR jury of Sylvie Icart (Université de Nice-Sophia Antipolis, March).

J. Leblond is a member of the Committee MASTIC. She gave a communication within the “Café-in” of the Research Center (Sept.).

M. Olivi is co-president with I. Castellani of the Committee MASTIC (Commission d'Animation et de Médiation Scientifique) https://

E. Pozzi was a member of the Committee MASTIC.

S. Chevillard published a popularization blog post on the website “Mathém

L. Baratchart is a member of the Editorial Boards of *Constructive Methods and Function Theory* and *Complex Analysis and Operator Theory*.
He is Inria's representative at the “conseil scientifique” of the Aix-Marseille University.

S. Chevillard is representative at the “comité de centre” and at the “comité des projets” (Research Center Inria-Sophia). He was a member of the work-group “Books” whose assignment was to propose different scenarios regarding the future of the books currently stored at the library of the research center.

J. Leblond is an elected member of the “Conseil Scientifique” of Inria. She is one of the two researchers in charge of the mission “Conseil et soutien aux chercheurs” within the Research Center. She is a member of the “Comité de Suivi National PRPS - QVT” (Prévention des Risques Psycho-Sociaux et la Qualité de Vie au Travail).

M. Olivi is responsible for scientific mediation and co-president of the committee MASTIC.

F. Seyfert is a member of CUMIR at Inria Sophia-Antipolis-Méditerrannée.