The team develops constructive, function-theoretic approaches to inverse problems arising in modelling 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 behaviours. 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). 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 endeavour is to develop algorithms resulting in dedicated software.

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), Pau (EPI commune Inria Magique-3D), Provence (Marseille, CMI), Nice (Lab. JAD), with CEA-IRFM (Cadarache), CWI (the Netherlands), MIT (Boston, USA) Michigan State University (East-Lansing, USA), Vanderbilt University (Nashville 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).

The project is involved in the GDR-project AFHP (CNRS), in a EMS21-RTG NSF program (with MIT, Boston, and Vanderbilt University, Nashville, USA), in a LMS Grant with Leeds University (UK) and in a CSF program (with University of Cyprus).

Within the extensive field of inverse problems, much of the research by APICS
deals with reconstructing solutions of classical elliptic PDEs from their
boundary behaviour. 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

To determine a complete
model, that is, one which is defined
for every frequency, in a sufficiently flexible function
class (*e.g.* Hardy spaces). This ill-posed issue requires
regularization, for instance constraints on the behaviour at
non-measured frequencies.

To compute a reduced order model. This typically consists of rational approximation of the complete model obtained in step 1, or phase-shift thereof to account for delays. Derivation of the complete model is important to achieve stability of the reduced one.

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 output,
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 frequency domain, replacing measured behaviour by desired behaviour. 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 especailly 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 multiband 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 plasma confinement. 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 is done in collaboration with Alexander Borichev (Univ. Provence).

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* , see section
. 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 was found in this case.

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 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 much in the spirit of 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 modelled 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 as
a 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, and 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 a novel parametrization of stable transfer functions .

From this rational model, realizations meeting certain constraints imposed by the technology in use are computed (see section ). These constraints typically come from the nature and 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 topology. 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 ().

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, but further study is needed to make them 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 Tchebychev 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 to 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 ).

Later, investigations by the team 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 in such studies, 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 to the theory , . They are of fundamental use in 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 current studies by the team of synthesis and adaptation 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 multipoint Schur analysis in the team's investigation of spectral representation for certain non-stationary discrete stochastic processes , .

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 linearised 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
analyse 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 (Univ. Oregon at Eugene, USA).

To find an analytic function in

(

Here *a priori*
assumptions on
the behaviour 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/mgneto 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 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 undertook a
long-term 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, saddles...)
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
behaviour 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 become over years an increasingly significant part of the team's activity (see section ). For instance, convergence properties of multipoint Schur approximants, which are rational interpolants preserving contractivity of a function, were analysed 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 multiband 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 want to consider reads:
*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 people collaborate with us on this subject: Herbert Stahl (TFH Berlin), Maxim Yattselev (Univ. Oregon at Eugene, USA).

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

Generally speaking, the
behaviour 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
analysing the behaviour 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 ).

Sylvain Chevillard, joined team in November 2010. His coming
resulted in APICS hosting a research activity in certified computing,
centered around 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 Solya is not a long-term objective of APICS.

These domains are related to the problems described in sections and . They are handled using the techniques described in section .

This work is done in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena project-team.

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.
Actually, 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.

Numerical experiments gave very good results on simulated data and we are now proceeding with real experimental magneto-encephalographic data, see also sections and . The PhD thesis of A.-M. Nicu was concerned with these applications, see , in collaboration with the Athena team at Inria Sophia Antipolis, and neuroscience teams in partner-hospitals (hosp. Timone, Marseille).

Similar inverse potential problems appear naturally in magnetic reconstruction. A particular application, which is the object of a joint NSF project with Vanderbilt University and MIT, is to geophysics. There, the remanent magnetization of a rock is to be analysed to draw information on magnetic reversals and to reconstruct the rock history. Recently developed scanning magnetic microscopes measure the magnetic field down to very small scales in a “thin plate” geological sample at the Laboratory of planetary sciences at MIT, and the magnetization has to be recovered from the field measured on a plane located at small distance above the slab.

Mathematically
speaking, EEG and magnetization inverse problems both amount to recover the (3-D valued) quantity

outside the volume

The team is also getting engaged in problems with variable conductivity governed by a 2-D conjugate-Beltrami equation, see , , . The application we have in mind is to plasma confinement for thermonuclear fusion in a Tokamak, more precisely with the extrapolation of magnetic data on the boundary of the chamber from the outer boundary of the plasma, which is a level curve for the poloidal flux solving the original div-grad equation. Solving this inverse problem of Bernoulli type is of importance to determine the appropriate boundary conditions to be applied to the chamber in order to shape the plasma . These issues are the topics of the PhD theses of S. Chaabi and D. Ponomarev , and of a joint collaboration with the Laboratoire J.-A. Dieudonné at the Univ. of Nice-SA (and the Inria team Castor), and the CMI-LATP at the Univ. of Aix-Marseille I (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 modelled 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 linearising 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 Tchebychev 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 filter's responses under modulus constraints is due to this ability to "forget" about latter spectral equation, and express all design constraints on the filtering functions

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 ). An appropriate Moebius 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 modelled 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/personnel/Fabien.Seyfert/Presto_web_page/presto_pres.html. 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 (see section ). 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 2011, to TAS-Espagna to tune filter, with topologies with multiple solutions. The usage of Dedale-HF is here considered together with Presto-HF.

Status: A stable version is maintained.

This software has been developed by Vincent Lunot (Taiwan Univ.) during his Ph.d. 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 EPI, and with Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris).

FindSources3D is a software dedicated to source
recovery for the inverse EEG problem, in 3-layer spherical settings, from pointwise
data (see
http://

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

This software is developed in collaboration with Christoph Lauter (LIP6) and Mioara Joldeş (Uppsala University, Sweden).

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.

Amongst 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://

The works presented here are done in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena EPI, with Doug Hardin and Edward Saff from Vanderbilt University (Nashville, USA), and with Abderrazek Karoui (Univ. Bizerte, Tunisie) and Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris).

This section in dedicated to 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 a common problem to both which has been recently achieved, see ,, .These are sums of (distributional) derivatives of Sobolev functions vanishing on the boundary.

In 3-D, functional or clinical active regions in the cortex are often represented by pointwise 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. A milestone in a long-haul research on the behaviour of poles of best rational approximants of fixed degree to functions with branch points has been reached this year , which shows that the technique carries over to finitely many sources (see section ). In this connection, a dedicated software “FindSources3D” (see section ) has been developed, in collaboration with the team Athena , .

Further, it appears that in the rational approximation step of these schemes, *multiple* poles possess a nice behaviour with respect to the branched singularities. This is due to the very basic 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 these multiple poles have such strong accumulation
properties, which remains an intriguing observation.

Issues of robust interpolation on the sphere from incomplete pointwise data are also under study in order to improve numerical accuracy of our reconstruction schemes. Spherical harmonics, Slepian bases and related special functions are of special interest (thesis of A.-M. Nicu , ), while other techniques should be considered as well.

Also, magnetic data from MEG (magneto-encephalography) will soon become available, which should enhance the accuracy of source recovery algorithms.

It turns out that discretization issues in geophysics can also be approached by these approximation techniques. Namely, in geodesy or for GPS computations, one may need to get a best 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 a physicist colleague (IGN, LAREG, geodesy). Related geometrical issues (finding out the geoid, level surface of the gravitational potential) are worthy of consideration as well.

Magnetic sources localization from observations of the field away from the
support of the magnetization is an issue under investigation in a joint effort
with
the Math. department of Vanderbilt University and the Earth Sciences
department at MIT. The goal is to recover the 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).

The magnetization operator is the Riesz potential of the divergence of the magnetization. The problem of recovering a thin plate magnetization distribution from measurements of the field in a plane above the sample lead us to an analysis of the kernel of this operator, which we characterized in various function and distribution spaces (arbitrary compactly supported distributions or derivatives of bounded functions). For this purpose, we introduced a generalization of the Hodge decomposition in terms of Riesz transforms and showed that a thin plate magnetization is “silent” (i.e. in the kernel) if the normal component is zero and the tangential component is divergence free. In particular, we show that a unidirectional non-trivial magnetization with compact support cannot be silent. The same is true for bidirectional magnetizations if at least one of the directions is nontangential. We also proved that any magnetization is equivalent to a unidirectional. We did introduce notions of being silent from above and silent from below, which are in general distinct. These results have been reported in a paper to appear .

We currently work on Fourier based inversion techniques for unidirectional magnetizations, and Figures , , and show an example of reconstruction. A joint paper with our collaborators from VU and MIT is being written on this topic.

For more general magnetizations, the severe ill-posedness of reconstruction challenges discrete Fourier methods, one of the main problems being the truncation of the observations outside the range of the SQUID measurements. We look forward to develop extrapolation techniques in the spirit of step 1 in section .

This work has been performed in collaboration with Yannick Fischer from the Magique3D EPI (Inria Bordeaux, Pau).

In collaboration with the
CMI-LATP (University Aix-Marseille I), the team considers 2-D diffusion processes with variable conductivity.
In particular its complexified version, the so-called *conjugate* or
*real Beltrami
equation*,
was investigated.
In the case of a smooth domain, and for Lipschitz
conductivity, we analyzed the Dirichlet problem
for solutions in Sobolev and then in Hardy classes .

Their traces merely lie in

We generalized the construction to finitely connected Dini-smooth
domains and

In the transversal section of a tokamak (which is a disk if the vessel is idealized into a torus), the so-called poloidal flux is subject to some conductivity equation outside the plasma volume for some simple explicit smooth conductivity function, while the boundary of the plasma (in the Tore Supra tokamak) is a level line of this flux . Related magnetic measurements are available on the chamber, which furnish incomplete boundary data from which one wants to recover the inner (plasma) boundary. This free boundary problem (of Bernoulli type) can be handled through the solutions of a family of bounded extremal problems in generalized Hardy classes of solutions to real Beltrami equations, in the annular framework .

In the particular case at hand, the conductivity is

The PhD work of S. Chaabi is devoted to further aspects of Dirichlet
problems for the
conjugate Beltrami equation. On the one hand, a
method based on Foka's
approach to boundary value problems, which uses Lax pairs and
solves for a Riemann-Hilbert problem, has been devised to
compute in semi explicit form solutions to Dirichlet and Neumann problems for
the conductivity equation satisfied by the poloidal flux.
Also, for more general conductivities, namely bounded below and
lying in

Finally, note that the conductivity equation can be expressed like a static Schrödinger equation, for smooth enough conductivity coefficients. This provides a link with the following results recently set up by D. Ponomarev, who recently join the team for his PhD. 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 such setting, Helmholtz equation is approximated by a linear Schrödinger equation with one of spatial coordinates being an evolutionary variable. Explicit comparison of the solutions in the whole half-space allows to establish global justification of the Schrodinger model for sufficiently smooth pulses . This phenomenon can also be described by a nonstationary model that relies on the spatial nonlinear Schrödinger (NLS) equation with the time-dependent refractive index. A toy problem is 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. The NLS approximation is derived from a 2-D quasi-linear wave equation, for small time intervals and smooth initial data. Numerical simulations illustrate the approximation result in the 1-D case.

This work has been done in collaboration with Smain Amari (Royal Military College, Kingston, Canada), Jean Charles Faugère (SALSA EPI, Inria Rocquencourt), Giuseppe Macchiarella (Politecnico di Milano, Milan, Italy), Uwe Rosenberg (Design and Project Engineering, Osterholz-Scharmbeck, Germany) and Matteo Oldoni (Politecnico di Milano, Milan, Italy).

We continued our work on the circuit realizations of filters' responses with mixed type (inductive or capacitive) coupling elements and constrained topologies . For inline circuits, methods based on sequential extractions of electrical elements are best suited due to their computational simplicity. On the other hand, for circuits with no inline topology ,such methods are inefficient while algebraic methods (based on a Groebner basis) can be used, but at high computational cost. In order to tackle large order circuits, our approach is to decompose them into connected inline sections, which can be directly realized by extraction techniques, and into complex sections, where algebraic methods are needed for realization. In order to do this, we started studying the synthesis of filter responses by means of circuits with reactive non-resonating nodes (dangling resonators) . Links of this topic with Potapov's factorization of J-inner functions are currently being investigated.

In this connection, sensitivity analysis of the electrical response of a filter with respect to the electrical parameters of the underlying circuit has been published in collaboration with the University of Cartagena and ESA . We essentially proved that the total electrical sensitivity of a filters' response does not depend on the coupling topology of the underlying circuit: the latter however controls the distribution of this sensitivity within each resonator.

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 (Politecnico di Milano, 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.
We conjecture that existence and uniqueness of the solution still holds
in our case, where interpolation takes place on the boundary,
provided

Let

This work is pursued in collaboration with Thales Alenia Space, Politecnico di Milano, Xlim and CNES in particular within the contract CNES-Inria 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 2011, 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.

In 2012, we focused on the kind of instabilities that these ideal systems can exhibit. We showed that a circuit can be unstable, although it has 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. This pathological example is unrealistic, though, because it assumes 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. Under this simple assumption, we proved that the class of transfer functions of realistic circuits is much smaller than in previous situation. In fact, 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. An article is currently being written on the subject.

This work is performed in collaboration with Jonathan Partington (Univ. Leeds, UK).

Continuing effort is being paid by the team to
carry over the solution to bounded extremal problems of section
to various settings.
We mentioned already in section
the extension to 2-D diffusion equations with variable
conductivity for the determination of free boundaries in
plasma control and the development of
a generalized Hardy class theory.
We also investigate the ordinary Laplacian in

Still, questions about the behaviour of solutions to the
standard bounded extremal problems

In another connection, weighted composition operators on Lebesgue, Sobolev, and Hardy spaces appear in changes of variables while expressing conformal equivalence of plane domains. A universality property related to the existence of invariant subspaces for these important classes of operators has been established in . Additional density properties also allow one to handle some of their dynamical aspects (like cyclicity).

This work has been done in collaboration with Bernard Hanzon and Conor Sexton from Univ. Cork.

The problem is to fit a probability density function on a large set of financial data. The model class is the set of non-negative EPT (Exponential-Polynomials-Trigonometric) functions which provides a useful framework for probabilistic calculation as illustrated in the link
http://

This work has been done in collaboration with Herbert Stahl (TFH Berlin) and Maxim Yattselev (Univ. Oregon at Eugene, USA).

We completed and published this year the proof of an important result in
approximation theory, namely the counting measure of
poles of best

This result warrants source recovery techniques used in section .

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. In the case of 4 branchpoints, expressing the computation of Padé approximants in terms of the solution to a Riemann-Hilbert problem on the Riemann surface of the function, we proved that the pole behaviour generically shows deterministic chaos .

The overall and long-term goal is to enhance the quality of numerical computations. The progress made during year 2012 is the following:

Publication of a work about the implementation of functions erf and erfc in multiple precision and with correct rounding . It corresponds to a work initially begun in the Arénaire team and finished in the Caramel team. The goal of this work is to show on a representative example the different steps of the rigorous implementation of a function in multiple precision arithmetic (choice of a series approximating the function, choice of the truncation rank and working precision used for the computation, roundoff analysis, etc.). The steps are described in such a way that they can easily be reproduced by someone who would like to implement another function. Moreover, it is showed that the process is very regular, which suggests that it (or at least large parts of it) could be automated.

In the same field of multiple precision arithmetic, and with Marc Mezzarobba (Aric team), we proposed an algorithm for the efficient evaluation of the Airy

Finally, 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.

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.

**LMS grant**, support of collaborative research with Leeds Univ., U.K., School of Mathematics (no. 41130, 2012).

**PHC Utique CMCU**, cooperation France-Tunisia (no. 10G 1503, led by Univ. Orléans, MAPMO).

**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.”

Smain Amari (RMC Ontario).

Bernard Hanzon (Univ. Cork, External Collaborator).

Tahar Moumni (Univ. Bizerte, Tunisia).

Jonathan R. Partington (Univ. Leeds, U.K., External Collaborator).

Vladimir Peller (Michigan state Univ. at East Lansing)

Yves Rolain (Vrije Universiteit Brussels).

Nikos Stylianopoulos (Univ. of Cyprus).

Shubham KUMAR (from May 2012 until Sep 2012)

Subject: Mathematical methods for multiplexers study

Institution: IIT Delhi (India)

Dmitry Ponomarev (from Jun 2012 until Aug 2012)

Subject: Constrained optimization with prescribed values on the disk

Pre-doctoral trainee

Rahul PRAKASH (from May 2012 until Sep 2012)

Subject: Mathematical methods for multiplexers study

Institution: IIT Delhi (India)

Xuan Zhang (from May 2012 until Sep 2012)

Subject: Groebner basis methods for multiplexers study

Institution: Polytech'Nice

Jie Zhou (from May 2012 until Aug 2012)

Subject: A Hardy-Hodge Decomposition on the 2D Sphere

Institution: Ecole des Mines de Nancy

The following people are external collaborators of the team:

Smain Amari [RMC (Royal Military College), Kingston, Canada, since October].

Ben Hanzon [Univ. Cork, Ireland, since October].

Mohamed Jaoua [French Univ. of Egypt].

Jean-Paul Marmorat [Centre de mathématiques appliquées (CMA), École des Mines de Paris].

Jonathan Partington [Univ. Leeds, UK].

Edward Saff [Vanderbilt University, Nashville, USA].

L. Baratchart, S. Chevillard and J. Leblond gave communications at the Workshop on Inverse Magnetization Problems, Nashville, USA (Apr.).

L. Baratchart and J. Leblond gave communications at PICOF, Conference Problèmes Inverses, Contrôle et Optimisation de Formes, Palaiseau, France (Apr.).

L. Baratchart gave invited talks at the Workshop on Potential Theory and Applications, Szeged, Hungary (June), and at SIGMA 2012, CIRM-Luminy (Nov). He gave a talk at the Conférence en l'honneur de Gauthier Sallet, Saint Louis du Sénégal (Dec.). He was a colloquim speaker at the State University of New York, Albany, USA (October) and at the University of Oregon, USA (Oct.).

S. Chevillard gave a talk at the ERNSI 2012 conference in Maastricht (Netherlands). He reviewed an article for the Journal of Symbolic Computation.

J. Leblond was invited to give a talk at the following conferences: Conference Control & Inverse Problems for PDE (CIPPDE), Santiago, Chili (Jan.), Workshop Control of Fluid-Structure Systems & Inverse Problems, Toulouse, France (Jun.), International Conference on Constructive Complex Approximation, Lille, France (Jun.), Joint Congress of the French & Vietnamese Math. Soc. (VMS-SMF), Hué, Vietnam (Aug.), Congress on Numerical MEthods & MOdelisation (MEMO), Tunis, Tunisie (Dec.). She also gave communications at the seminars of the School of Mathematics, Univ. Leeds, U.K. (Feb.), of the Institut de Mathématiques de Bordeaux (IMB, Univ. Bordeaux, Mar.), of the Department of Math. & Geosciences, Univ. Trieste, Italy (Oct.), and at the 2nd Nice Physical Day (“Journées de la Physique de Nice”), Nice (Dec.).

M. Olivi was co-organizer (with B. Hanzon and R. Peeters) of an invited session “model reduction/approximation” at the 16th IFAC Symposium on System Identification, Brussels, July 2012.

D. Ponomarev presented a poster at the 2nd PhD Event in Fusion Science and Engineering, Pont-a-Mousson (Oct.).

E. Pozzi gave several communications at seminars at Univ. of Besançon, Grenoble (Jan.), Bordeaux, Orléans (Mar.), Marseille, Lille (Apr.)

F. Seyfert was invited to give a talk at the European Microwave Week 2012, Workshop on Advances of N-port networks for Space Application, Amsterdam, Netherlands.

Licence: E. Pozzi, Numerical algorithmics, 26h ETD (from Sep.), L3, Computer Sciences, Polytech'Nice, Univ. Nice Sopia Antipolis, France.

PhD: A.-M. Nicu, Approximation et représentation des fonctions sur la sphère. Applications aux problèmes inverses de la géodésie et l'imagerie médicale . Univ. Nice Sophia Antipolis, ED STIC, Feb. 2012 (advisor: J. Leblond).

PhD in progress: S. Chaabi, Boundary value problems for pseudo-holomorphic functions, since Nov. 2008 (advisors: L. Baratchart and A. Borichev).

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

J. Leblond (advisor) and M. Olivi (examinator) were members of the PhD jury of A.-M. Nicu (Univ. Nice Sophia Antipolis, Feb.) .

J. Leblond was a member (reviewer) of the PhD jury of N. Chaulet (Ecole Polytechnique, Nov.).

L. Baratchart was the head of the PhD jury of Matteo Santacesaria (Ecole Polytechnique, Dec.).

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

J. Leblond and E. Pozzi are members of this committee.

E. Pozzi participated to the “Filles et mathématiques” day, Avignon, Nov.

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 Univ. Provence (Aix-Marseille).

S. Chevillard is representative at the « comité de centre » and at the « comité des projets » (Research Center Inria-Sophia).

J. Leblond is an elected member of the “Conseil Scientifique” of Inria. Together with C. Calvet from Human Resources, she is in charge of the mission “Conseil et soutien aux chercheurs” within the Research Centre, and she participated to the working group BEAT (“Bien Être Au Travail”).

M. Olivi is a member of the CSD (Comité de Suivi Doctoral) of the Research Center. She is responsible for scientific mediation.

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