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), and paleomagnetism (determining the magnetization of rock samples).

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 simplest example lies with
harmonic identification of a stable linear dynamical system:
the transfer-function *e.g.* the Cauchy formula.

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

Step 1 relates to 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, in particular
to count
critical points and to derive bounds, see Section . Step 2 raises
further issues in approximation theory regarding the rate of convergence and
the extent to which singularities of the
approximant (*i.e.* its poles) tend to singularities of the
approximated function; this is where logarithmic potential theory
becomes instrumental, see Section .

Applying a realization procedure to the result of step 2 yields an identification procedure from incomplete frequency data which was first demonstrated in to tune resonant microwave filters. Harmonic identification of nonlinear systems around a stable equilibrium can also be envisaged by combining the previous steps with exact linearization techniques from , see .

A similar path can be taken to approach design problems in the frequency domain, replacing the measured behavior by some desired behavior. However, describing achievable responses in terms of the design parameters is often cumbersome, and most constructive techniques rely on specific criteria adapted to the physics of the problem. This is especially true of filters, the design of which traditionally appeals to polynomial extremal problems , . Apics contributed to this area the use of Zolotarev-like problems for multi-band synthesis, although we presently favor interpolation techniques in which parameters arise in a more transparent manner, see Section .

The previous 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; this phenomenon is called
*balayage* . In the two steps approach
previously described,
we implicitly removed this indeterminacy by requiring in step 1
that the measure
be supported on the boundary (because we seek a function holomorphic
throughout the right half-space), and
by requiring in step 2
that the measure be discrete in the left half-plane (in fact: a sum of
point masses

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. This differs 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, Apics advocates the use of steps 1 and 2 above, along with some singularity analysis, to approach issues of nondestructive control in 2-D and 3-D , , . The team is currently engaged in the generalization to inverse source problems for the Laplace equation in 3-D, to be described further in Section . There, holomorphic functions are replaced by harmonic gradients; applications are to EEG/MEG and inverse magnetization problems in Geosciences, see Section .

The approximation-theoretic tools developed by Apics to handle issues mentioned so far are outlined in Section . In Section to come, we describe in more detail which problems are considered and which applications are targeted.

By standard properties of conjugate differentials, reconstructing Dirichlet-Neumann boundary conditions
for a function harmonic in a plane domain,
when these conditions are already known on a subset

Another application by the team deals with non-constant conductivity
over a doubly connected domain, the set *Tore Supra*
.
The 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 to free boundary problems of Bernoulli type,
using descent
algorithms based on shape-gradient for such approximation-theoretic
criteria, is an interesting prospect, still to be pursued.

The piece of work we just mentioned requires defining and studying Hardy spaces of the conjugate-Beltrami equation, which is an interesting topic by itself. For Sobolev-smooth coefficients of exponent greater than 2, they were investigated in , . The case of the critical exponent 2 is treated in , which apparently provides the first example of well-posedness for the Dirichlet problem in the non-strictly elliptic case: the conductivity may be unbounded or zero on sets of zero capacity and, accordingly, solutions need not be locally bounded.

The 3-D version of step 1 in Section is another
subject investigated by Apics: to recover a harmonic function
(up to an additive constant) in a ball or a half-space from partial knowledge of its
gradient on the boundary. This prototypical inverse problem
(*i.e.* inverse to the Cauchy problem for the Laplace equation)
often recurs in electromagnetism. At present, Apics is involved with
solving instances of this inverse problem arising
in two fields, namely medical imaging
*e.g.* for electroencephalography (EEG)
or magneto-encephalography (MEG), and
paleomagnetism (recovery of rocks magnetization)
, ,
see Section . In this connection, we collaborate with two groups of partners:
Athena Inria project-team,
CHU La Timone, and BESA company on the one hand,
Geosciences Lab. at MIT and Cerege CNRS Lab. on the other hand.
The question is considerably more difficult than its 2-D
counterpart, due mainly to the lack of multiplicative structure for harmonic
gradients. Still,
considerable progress has been made over the last years
using methods of harmonic analysis and operator theory.

The team is further concerned with 3-D generalizations and applications to
non-destructive control of step 2 in Section .
A typical problem is here to localize inhomogeneities or defaults such as
cracks, sources or occlusions in a planar or 3-dimensional object,
knowing thermal, electrical, or
magnetic measurements on the boundary.
These defaults can be expressed as a lack of harmonicity
of the solution to the associated Dirichlet-Neumann problem,
thereby posing an inverse potential problem in order to recover them.
In 2-D, finding an optimal discretization of the
potential in Sobolev norm amounts to solve a best rational approximation
problem, and the question arises as to how the location of the
singularities of the approximant (*i.e.* its poles)
reflects the location of the singularities of the potential
(*i.e.* the defaults we seek). This is a fairly deep issue
in approximation theory, to which Apics contributed convergence results
for certain classes of fields
expressed as Cauchy integrals over extremal contours for
the logarithmic potential
, .
Initial schemes to locate cracks or sources
*via* rational approximation on
planar domains were obtained this way , , . It is remarkable that finite inverse source problems
in 3-D balls, or more general algebraic surfaces,
can be approached using these 2-D techniques upon slicing the
domain into planar sections
, .
This bottom line generates a steady research activity
within Apics, and again applications are sought to medical imaging and
geosciences, see Sections ,
and .

Conjectures can be raised on the behavior of optimal potential discretization in 3-D, but answering them is an ambitious program still in its infancy.

Through contacts with CNES (French space agency),
members of the team became involved in identification and tuning
of microwave electromagnetic filters used in space telecommunications,
see Section . The initial problem was
to recover, from band-limited frequency measurements, 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 comes into play, 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 provide a framework to transform this ill-posed issue into a series of regularized analytic and meromorphic approximation problems. More precisely, the procedure sketched in Section goes as follows:

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
a problem analogous to

A stable rational approximation of appropriate degree to the model obtained in the previous step is performed. For this, a descent method on the compact manifold of inner matrices of given size and degree is used, based on an original parametrization of stable transfer functions developed within the team , .

Realizations of this rational approximant are computed. To be useful, they must satisfy certain constraints imposed by the geometry of the device. These constraints typically come from the coupling topology of the equivalent electrical network used to model the filter. This network is composed of resonators, coupled according to some specific graph. This realization step can be recast, under appropriate compatibility conditions , as solving a zero-dimensional multivariate polynomial system. To tackle this problem in practice, we use Gröbner basis techniques and continuation methods which team up in the Dedale-HF software (see Section ).

Let us mention that extensions of classical coupling matrix theory to frequency-dependent (reactive) couplings have been carried-out in recent years for wide-band design applications.

Apics also investigates issues pertaining to
design rather than identification.
Given the topology of the filter,
a basic problem in this connection is to find the optimal response
subject to specifications
that bear on rejection, transmission and group delay of the
scattering parameters.
Generalizing the classical approach based on Chebyshev polynomials
for single band
filters, we recast the problem of multi-band response synthesis
as a generalization of the 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 were implemented in the software
easy-FF (see easy-FF). Currently, the team is engaged
in the synthesis of more complex microwave devices
like multiplexers and routers, which connect several
filters through wave guides.
Schur analysis plays an important role here, because
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.
The so-called Schur parameters of a function
may be viewed as Taylor coefficients for the hyperbolic metric of the disk, and
the fact that Schur functions are contractions for that metric lies at the
root of Schur's test.
Generalizations thereof turn out to be efficient to parametrize
solutions to contractive interpolation problems .
Dwelling on this, Apics contributed
differential parametrizations (atlases of charts) of lossless
matrix functions , , which
are fundamental to our rational approximation
software RARL2 (see Section ).
Schur analysis is also instrumental to approach de-embedding issues,
and provides one with considerable
insight into the so-called matching problem. The latter consists in
maximizing the power a multiport can pass to a given load, and for
reasons of efficiency it
is all-pervasive in microwave and electric network design, *e.g.* of
antennas, multiplexers, wifi cards and more. It can be viewed as a
rational approximation problem in the hyperbolic metric, and the team
presently gets to grips with this hot topic using multipoint
contractive interpolation in
the framework of the (defense funded) ANR COCORAM,
see Sections and .

In recent years,
our attention was driven by CNES and UPV (Bilbao)
to questions about stability of high-frequency amplifiers,
see Section .
Contrary to previously discussed devices, these are *active* components.
The response of an amplifier can be linearized around a
set of primary current and voltages,
and then admittances of the corresponding electrical network
can be computed at various frequencies, using the so-called harmonic
balance method.
The initial goal is to check for stability of the linearized model,
so as to ascertain existence of a well-defined working state.
The network is composed of lumped electrical elements namely
inductors, capacitors, negative *and* positive reactors,
transmission lines, and controlled current sources.
Our research so far focuses on describing the algebraic structure
of admittance functions, so as to set up a function-theoretic framework
where the two-steps approach outlined in Section
can be put to work. The main discovery so far is that
the unstable part of each partial transfer function is rational and can
be computed by analytic projection,
see Section .

To find an analytic function

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

To fix terminology, we refer to *bounded extremal problem*.
As shown in , ,
,
the solution to this convex
infinite-dimensional optimization problem can be obtained
when

(

The case

Various modifications of

The analog of Problem *seek the inner
boundary*, knowing it is a level curve of the solution.
In this case, the Lagrange parameter indicates
how to deform the inner contour in order to improve
data fitting.
Similar topics are discussed in Section for more general equations than the Laplacian, namely
isotropic conductivity equations of the form

Though originally considered in dimension 2,
Problem

When

On the ball, the analog
of Problem

When *Hardy-Hodge* decomposition,
allowing us to express a *i.e.* those generating no field
in the upper half space) .

Just like solving problem

Problem

Companion to problem

Note that

The techniques set forth in this section are used to solve
step 2 in Section and instrumental to
approach inverse boundary value problems
for the Poisson equation

We put

A natural generalization of problem

(

Only for

The case where *stable* rational
approximant to *not* be unique.

The former Miaou project (predecessor of Apics) designed a dedicated
steepest-descent algorithm
for the case *local minimum* is
guaranteed; until now it seems to be the only procedure meeting this
property. This gradient algorithm proceeds
recursively with respect to *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
(local minima, saddle points...), in which
tools from differential topology and
operator theory team up with classical interpolation theory
, .
Based on this work,
uniqueness or asymptotic uniqueness of the approximant
was proved for certain classes of functions like
transfer functions of relaxation
systems (*i.e.*
Markov functions) and more
generally Cauchy integrals over hyperbolic geodesic arcs .
These are the only results of this kind. Research by Apics on this topic
remained dormant for a while by reasons of opportunity,
but revisiting the work in higher dimension is still
a worthy endeavor. Meanwhile,
an analog to AAK theory
was carried out for

A common
feature to the above-mentioned problems
is that critical point equations
yield non-Hermitian orthogonality relations for the denominator
of the approximant. This stresses connections with interpolation,
which is a standard way to build approximants,
and in many respects best or near-best rational approximation
may be regarded as a clever manner to pick interpolation points.
This was exploited in , ,
and is used in an essential manner to assess the
behavior of poles of best approximants to functions with branched
singularities,
which is of particular interest for inverse source problems
(*cf.* Sections and ).

In higher dimensions, the analog of Problem

Besides,
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 (*i.e.* a Schur function). In particular, Schur interpolation
lately received renewed attention
from the team, in connection with matching problems.
There, interpolation data are subject to
a well-known compatibility condition (positive definiteness of the so-called
Pick matrix), and the main difficulty is to put interpolation
points on the boundary of

Matrix-valued approximation is necessary to handle systems with several
inputs and outputs but it generates additional difficulties
as compared to scalar-valued approximation,
both theoretically and algorithmically. In the matrix case,
the McMillan degree (*i.e.* the degree of a minimal realization in
the System-Theoretic sense) generalizes the usual notion of degree
for rational functions.

The basic problem that we consider now goes as follows:
*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 derived in
and mentioned in
Section
generalizes to
the matrix-valued situation . The
first difficulty here is to parametrize
inner matrices (*i.e.* matrix-valued functions
analytic in the unit disk and unitary on the unit circle) of
given McMillan degree degree

Difficulties relative to multiple local minima of course 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 degree

Let us stress that RARL2 seems the only algorithm handling rational approximation in the matrix case that demonstrably converges to a local minimum while meeting stability constraints on the approximant. It is still a working pin of many developments by Apics on frequency optimization and design.

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

Generally speaking in approximation theory, assessing the
behavior of poles of rational approximants is essential
to obtain error rates as the degree goes large, and to tackle
constructive issues like
uniqueness. However, as explained in Section ,
Apics considers this issue foremost as a means
to extract information on
singularities of the solution to a
Dirichlet-Neumann problem.
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
initial condition for more precise but much heavier
numerical optimizations which often do not even converge
unless properly initialized.
As regards crack detection or source recovery, this approach
boils down to
analyzing the behavior of best meromorphic
approximants of a function with branch points, which is the prototype of
a polar singular set.
For piecewise analytic cracks, or in the case of sources, we were able to
prove (, , ),
that the poles of the
approximants accumulate, when the degree goes large,
to some extremal cut of minimum weighted
logarithmic capacity connecting
the singular points of the crack, or the sources
.
Moreover, the asymptotic density
of the poles turns out to be the Green equilibrium distribution
on this cut in

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

It is remarkable that inverse source problems inside a sphere or an ellipsoid in 3-D can be approached with such 2-D techniques, as applied to planar sections, see Section . The technique is implemented in the software FindSources3D, see Section .

In addition to the above-mentioned research activities, Apics develops and maintains a number of long-term software tools that either implement and illustrate effectiveness of the algorithms theoretically developed by the team or serve as tools to help further research by team members. We present briefly the most important of them.

http://

Dedale-HF is a software dedicated to solve exhaustively the coupling matrix synthesis problem in reasonable time for the filtering community. Given a 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.

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. corresponding to the user-specified filter characteristics. The reference files are computed off-line using Gröbner basis techniques or numerical techniques based on the exploration of a monodromy group. The use of such continuation techniques, combined with an efficient implementation of the integrator, drastically reduces the computational time.

Dedale-HF has been licensed to, and is currently used by TAS-España.

This work is conducted in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena EPI.

FindSources3D is a Matlab software program dedicated to the resolution of inverse source problems in electroencephalography (EEG) (see http://

The program is currently being tested by BESA company (Munich). Our purpose is to distribute FindSources3D to teams in partner-hospitals (like la Timone, Marseille). It has a CeCILL license.

Presto-HF is a toolbox dedicated to low-pass parameter identification for microwave filters https://

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. Constrained realizations are computed using the Dedale-HF 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 assumption: far off the pass-band, one can reasonably expect a good approximation of the rational components of S11 and S22 by the first few terms of their Taylor expansion at infinity, a small degree polynomial in 1/s. Using this idea, a sequence of quadratic convex optimization problems are solved, in order to obtain appropriate compensations. In order to check the previous assumption, one has to measure the filter on a larger band, typically three times the pass band.

This toolbox has been licensed to, and is currently used by Thales Alenia Space in Toulouse and Madrid, Thales airborne systems and Flextronics (two licenses). XLIM (University of Limoges) is a heavy user of Presto-HF among the academic filtering community and some free license agreements have been granted to the microwave department of the University of Erlangen (Germany) and the Royal Military College (Kingston, Canada).

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

RARL2 computes a stable rational L2-approximation of specified order to a given L2-stable (L2 on the unit circle, analytic in the complement of the unit disk) matrix-valued function. This can be 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 the first or the second case, and leans on frequency data identification in the third. For 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.

RARL2 performs the rational approximation step in the software tools PRESTO-HF (see Section ) and FindSources3D (see Section ). It is distributed under a particular license, allowing unlimited usage for academic research purposes. It was released to the universities of Delft and Maastricht (the Netherlands), Cork (Ireland), Brussels (Belgium), Macao (China) and BITS-Pilani Hyderabad Campus (India).

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

**Preliminary remark: ** The coming of Sylvain Chevillard in the team in 2010 resulted in Apics hosting a research activity in certified computing, centered on the software *Sollya*. 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.

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. As well, it 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://

Application 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 as described in Section .

This work is conducted in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena EPI.

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, see . Indeed, the latter involves propagating the
initial conditions through several layers of different conductivities,
from the boundary shell
down to the center of the domain where the
singularities (*i.e.* the sources) lie.
Once propagated
to the innermost sphere, it turns out that traces of the
boundary data on 2-D cross sections coincide
with analytic functions with branched singularities
in the slicing plane
. The singularities are
related to the actual location of the sources, namely their moduli
reach in turn a
maximum when the plane contains one of the sources. Hence we are
back to the 2-D framework of Section ,
and recovering these singularities
can be performed *via* best rational approximation.
The goal is to produce a fast and sufficiently accurate
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 obtained with our software FindSources3D give very good results on simulated data and we are now engaged in the process of handling real experimental data (see 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 appearing in Section , occur 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, which led us to form the Inria Associate Team “IMPINGE” (Inverse Magnetization Problems IN GEosciences) together with MIT and Vanderbilt University. A recent collaboration with Cerege (CNRS, Aix-en-Provence), in the framework of the ANR-project MagLune, completes this picture, see Section .

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 sub-millimeter 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

This is joint work with Stéphane Bila (XLIM, Limoges).

One of the best training grounds for function-theoretic applications by the team is the identification and design of physical systems whose performance is assessed frequency-wise. 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 an essentially discrete set of wave vectors is selected. In the considered range of frequency, the electrical field in each cavity can be 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 to the Helmholtz equations is given by a second order differential equation. Thus, one obtains an electrical model of the filter as a sequence of electrically-coupled resonant circuits, each circuit being modeled by two resonators, one per mode, the resonance frequency of which represents the frequency of a mode, and whose resistance accounts for electric losses (surface currents) in the cavities.

This way, the filter can be seen as a quadripole, with two ports, when
plugged onto a resistor at one end and fed with some potential at the other end.
One is now
interested in the power which is transmitted and reflected. This leads
one to define 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 no longer
have real
coefficients) but whose degree is divided by 2 (8 in the example).

In short, the strategy for identification 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 confirmation of the linearity assumption on 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, which are 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 proposed, in order to derive guaranteed optimal multi-band filter responses subject to modulus constraints . This generalizes the classical single band design techniques based on Chebyshev polynomials and elliptic functions. The approach relies on the fact that the modulus of the scattering parameter

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, owes much to the possibility of
forgetting about Feldtkeller's equation and express all design constraints
in terms of the filtering function. This no longer the case when
considering the synthesis

Through contacts with CNES (Toulouse) and UPV (Bilbao),
Apics got further involved
in the design of amplifiers which, unlike filters, are active devices.
A prominent issue here is stability. A twenty years back, it was not
possible to simulate unstable responses, and only after building a device
could one detect instability. The advent of so-called *harmonic balance*
techniques, which compute steady state responses of linear elements in
the frequency domain and look for a periodic state in the time domain of
a network connecting these linear elements *via*
static non-linearities made it possible to compute the harmonic response
of a (possibly nonlinear and unstable) device .
This has had tremendous impact on
design, and there is a growing demand for software analyzers.

There are two types of stability involved. The first is stability of a fixed
point around which the linearized transfer function
accounts for small signal amplification. The second is stability of a
limit cycle which is reached when the input signal is no longer small
and truly nonlinear amplification is attained
(*e.g.* because of saturation).
Work by the team so far has been concerned with the first type of stability,
and emphasis is put on defining and extracting the “unstable part” of the response, see Section .

Recent developments allow to use Dedale-HF in combination with Presto-HF and in replacement of the former software RGC. A circuit optimizer has also been added to handle specific coupling topologies, the admissible set of which is not known in terms of a simple polynomial description.

A new (Matlab) version of the software that automatically performs the estimation of the quantity of sources is being built (see Section ). It uses an alignment criterion in addition to other clustering tests for the selection. Also, the team benefit from an “Action de Développement Technologique” (ADT Inria) BOLIS, 2014-2016, and of the young engineer N. Schnitzler at half-part of the time. The aim is to get from FindSources3D a modular, ergonomic, accessible and interactive platform, providing a convenient graphical interface and a tool that can be easily distributed and used, for medical imaging (EEG, MEG, EIT) or other applications (like inverse source problems in planetary sciences, see Section ). Modularity is now granted, though still in progress (using the tools dtk, Qt, still with compiled Matlab libraries; translation in C++ will be continued). The related version of the software now offers a detailed and nice visualization of the data and tuning parameters, of the processing steps and of the computed results (using VTK).

This section is concerned with inverse problems for 3-D Poisson-Laplace equations, among which source recovery issues. Though the geometrical settings differ in Sections and , the characterization of silent sources (those giving rise to a vanishing field) is one common problem to both which has been resolved in the magnetization setup .

This work is conducted in collaboration with Jean-Paul Marmorat and Nicolas Schnitzler, together with Maureen Clerc and Théo Papadopoulo from the Athena EPI.

In 3-D, functional or clinical active regions in the cortex are often modeled by pointwise sources that have to be localized from measurements taken by electrodes on the scalp of an electrical potential satisfying a Laplace equation (EEG, electroencephalography). In the works on the behavior of poles in best rational approximants of fixed degree to functions with branch points, 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 finitely many sources (see Section ).

In this connection, a dedicated software FindSources3D (see Section )
is being developed, in collaboration with the team Athena and the CMA. We continued this year algorithmic developments, prompted by
a fruitful collaboration with the firm BESA,
namely automatic detection of the number of sources (which was left to the user until recently).
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 3, at the moment).
Further, a modular and ergonomic platform version of the software is under development.

In connection with these and other brain exploration modalities like electrical impedance tomography (EIT), we are now studying conductivity estimation problems. This is the topic of the PhD research work of C. Papageorgakis (co-advised with the Athena project-team and BESA GmbH). In layered models, it concerns the estimation of the conductivity of the skull (intermediate layer). Indeed, the skull was assumed until now to have a given isotropic constant conductivity, whose value can differ from one individual to another. A preliminary issue in this direction is: can we uniquely recover and estimate a single-valued skull conductivity from one EEG recording? This has been established in the spherical setting when the sources are known, see . Situations where sources are only partially known and the geometry is more realistic than a sphere are currently under study. When the sources are unknown, we should look for more data (additional clinical and/or functional EEG, EIT, ...) that could be incorporated in order to recover both the sources locations and the skull conductivity. Furthermore, while the skull essentially consists of hard bone part that may be assumed to have constant electrical conductivity, it also contains spongy bone compartments. These two distinct parts of the skull possess quite different conductivities. The influence of that second value on the overall model is now being studied .

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, Michael Northington 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 has been the fundamental
issue under investigation by IMPINGE. The goal was 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 (superconducting quantum interference
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). Note however that EEG data consist
of values of the normal current and of the associated potential, while in the present setting only values of the normal magnetic field are measured.

Over the previous years, we mainly focused on developing techniques to recover magnetizations with rather sparse support. To this end, we set up a heuristic procedure to recover sparse magnetizations, based on iterative truncation of the support of the recovered magnetization. In this heuristics, magnetizations were represented by dipoles placed at the points of a regular rectangular

The procedure turned out to be poor when trying to recover the magnetization itself, due to the severe ill-posedness of the problem and the unexpected existence of magnetizations that produce almost no field at the height where measurements are performed, although the corresponding magnetic distributions strongly differ from truly silent distributions. Nevertheless, whenever the support could be significantly shrunk while keeping the error small (*i.e.*, explaining the data satisfactorily), estimates of the net moment so far,
based on the dipolar model obtained by inversion, have been good.

This suggests that recovering the net moment and recovering the magnetization are rather different problems, the first one being less ill-posed than the second. Although the information provided by the net moment of the sample seems to be much weaker than knowing the full magnetic distribution, its importance has been emphasized by the geophysicists at MIT for at least three reasons:

It yields important geological information on the sample in particular to estimate the magnitude of the ambient magnetic field at the time the rock was formed.

It can be estimated independently to some extent, using a magnetometer, thereby allowing one to cross-validate the approach.

From a computation point of view, knowledge of the net moment should lead to numerically stable reconstruction of an equivalent unidirectional magnetization. The support of the latter would provide us with valuable information to test for unidirectionality of the true magnetization, which is an important question to physicists in connection with rocks history and formation.

This year, we addressed the problem of directly recovering the net moment, without recourse to full inversion. Indeed, the latter is rather inefficient as it requires using a cluster and even then, for some samples, days of evaluation in order to obtain only a coarse estimate of the net moment. This research effort led us to investigate three different and complementary approaches.

First, we improved over Fourier based techniques previously designed
by reformulating the problem with the help of the Kelvin transform. This gave us an asymptotic expansion of the net moment involving, at the first order, the integrals

In parallel, and based on the results obtained with Fourier transform, we investigated a second approach, consisting in directly computing asymptotic expansions of the above integrals, on several domains (namely, the 2-D balls of radius

where

Finally, a third more ambitious approach has been investigated. As an attempt to generalize the previous expansions, our initial question was: given measurement of

Still in the course of D. Ponomarev's PhD research, the study of a 2D spectral problem for the truncated Poisson operator in planar geometry has been pursued.
It is a simplified formulation of the relation between the magnetization and the magnetic potential (of which the magnetic field is the gradient) and is expected to produce an efficient representation basis (the eigenfunctions of
the magnetization-to-field operator).
This is a long-standing problem. Noteworthy properties of solutions have been obtained through connections with other spectral problems and
asymptotic reductions for large and small values of the main parameters (distance

The year 2015 was the last of our “équipe associée” Impinge with the MIT and Vanderbilt University. The final report is available on the web page of the associate team

The team APICS is a partner of the ANR project MagLune concerning Lunar magnetism, associated to the Geophysics and Planetology Department of Cerege, CNRS, Aix-en-Provence (see Section ). Measurements of the remanent magnetic field of the Moon let geoscientists think that the Moon used to have a magnetic dynamo for some time, but the exact process that triggered and fed this dynamo is not yet understood, much less why it stopped. In particular, the Moon is too small to have a convecting dynamo like the Earth has. The overall goal of the project is to devise models to explain how this dynamo phenomenon was possible on the Moon.

To this end, the geophysicists from Cerege will go to NASA to perform some measurements on samples brought back from the Moon by Apollo missions. The samples are kept inside bags with a protective atmosphere, and geophysicists are not allowed to open the bags, nor to take out the samples from NASA facilities. Therefore, measurements must be performed with some rudimentary instrument and our colleagues from Cerege designed a specific magnetometer. This device allows them to obtain measurements of the components of the magnetic field produced by the sample, at some discrete set of points located on disks belonging to three cylinders (see Figure ).

This collaboration started this year and some preparatory work was necessary
fix conventions used by our colleagues from Cerege in order to handle their measurements. During his Master 2 internship, Konstantinos Mavreas has developed a method based on rational approximation, using the same ideas as those underlying the FindSources3D tool (see Sections and ), for
the case where the field produced by the sample can be well explained by a single magnetic dipole, whose position and moment are unknown. See his report

This is collaborative work with Stéphane Bila (Xlim, Limoges, France), Yohann Sence (Xlim, Limoges, France), Thierry Monediere (Xlim, Limoges, France), Francois Torrès (Xlim, Limoges, France).

Filter synthesis is usually performed under the hypothesis that both ports of the filter are loaded on a constant resistive load (usually 50 Ohm). In complex systems, filters are however cascaded with other devices, and end up being loaded, at least at one port, on a non purely resistive frequency varying load. This is for example the case when synthesizing a multiplexer: each filter is here loaded at one of its ports on a common junction. Thus, the load varies with frequency by construction, and is not purely resistive either. Likewise, in an emitter-receiver, the antenna is followed by a filter. Whereas the antenna can usually be regarded as a resistive load at some frequencies, this is far from being true on the whole pass-band. A mismatch between the antenna and the filter, however, causes irremediable power losses, both in emission and transmission. Our goal is therefore to develop a method for filter synthesis that allows us to match varying loads on specific frequency bands, while enforcing some rejection properties away from the pass-band.

The matching problem of minimizing

When the degree

where

which accounts for the losslessness of the filter. The frequencies

The previous interpolation procedure provides us with a matching/rejecting filtering characteristics at a discrete set of frequencies. This can serve as a
starting point for heavier optimization procedures where the matching and rejection specifications are expressed uniformly over the bandwidth. Although the practical results thus obtained have shown to be quite convincing, we have no proof of their global optimality. This led us to seek alternative approaches able to asses, at least in simple cases, global optimality of the derived response. Following the approach of Fano and Youla, we considered the problem of a designing a

This work was conducted in collaboration with Yves Rolain (VUB, Brussels, Belgium)

Coupling topologies that admit multiple realizations may lead to ambiguous de-embedding tuning procedures where distinct coupled resonator circuits are identified from the same measurements. This is for example the case of the well-known coupling topologies in triplets, quadruplets and extended boxed. If no additional measurements are performed on the DUT (device under tuning), the different solutions to the coupling matrix synthesis problem are undistinguishable, as they yield similar scattering responses. We therefore studied specific tuning strategies to discriminate among them. The later uses a sequence of measurements of the DUT, obtained after varying some discriminating tuning parameters of the filter and testing for coherence of the extracted circuits. This work was presented by Matthias Caenepeel at IMS 2015 in Phoenix and at the EuMC 2015 in Paris . In a similar vein Matthias is currently developing techniques taking advantage of the differential information provided by EM solvers in order to compute the Jacobian matrix of the identified coupling matrix(ces) with respect to the geometrical parameters of the filter.

We studied this year the asymptotic behavior of the orthonormal polynomials

with

outside the convex hull of

This generalizes considerably known asymptotics on
analytic domains with Hölder smooth non vanishing weights .
The proof rests on some Hardy space theory, conformal mapping and
*Orthogonal and Multiple Orthogonal Polynomials*, August 9-14 2015, Oaxaca
(Mexico). An article is being written to report on this result.

This is joint work with M. Yattselev (IUPUI).

We studied best rational approximants in the *sup* norm to an analytic
function

The proof rests on a blend of AAK-theory and potential theory.

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

This contract (reference CNES: RS14/TG-0001-019) involving CNES, University of Bilbao (UPV/EHU) and Inria aims at setting up a methodology for testing the stability of amplifying devices. The work at Inria is concerned with the design of frequency optimization techniques to identify the unstable part of the linearized response and analyze the linear periodic components.

This is a research agreement between Inria (Apics and Athena teams) and the German company BESA

Flextronics, active in the manufacturing of communication devices all over the world, bought two sets of licenses for Presto-HF and Dedale-HF. Deployment of our tools in their production facilities for wireless communication units is being studied.

Contract Provence Alpes Côte d'Azur (PACA) Region - Inria, BDO (no. 2014-05764) funding the research grant of C. Papageorgakis, see Sections , .

The ANR (Astrid) project COCORAM (Co-design et co-intégration de réseaux d’antennes actives multi-bandes pour systèmes de radionavigation par satellite) started January 2014. We are associated with three other teams from XLIM (Limoges University), geared respectively towards filters, antennas and amplifiers design. The core idea of the project is to realize dual band reception an emission chains by co-conceiving the antenna, the filters, and the amplifier. We are specifically in charge of the theoretical design of the filters, matching the impedance of a bi-polarized dual band antenna. This represent a perfect training ground to test, apply and adapt our work on matching problems (see Section ).

The ANR project MagLune (Magnétisme de la Lune) has been approved July 2014. It involves the Cerege (Centre de Recherche et d’Enseignement de Géosciences de l’Environnement, joint laboratory between Université Aix-Marseille, CNRS and IRD), the IPGP (Institut de Physique du Globe de Paris) and ISTerre (Institut des Sciences de la Terre). Associated with Cerege are Inria (Apics team) and Irphe (Institut de Recherche sur les Phénomènes Hors Équilibre, joint laboratory between Université Aix-Marseille, CNRS and École Centrale de Marseille). The goal of this project (led by geologists) is to understand the past magnetic activity of the Moon, especially to answer the question whether it had a dynamo in the past and which mechanisms were at work to generate it. Apics participates in the project by providing mathematical tools and algorithms to recover the remanent magnetization of rock samples from the moon on the basis of measurements of the magnetic field it generates. The techniques described in Section are instrumental for this purpose.

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

System identification deals with the derivation, estimation and validation of mathematical models of dynamical phenomena from experimental data.

Title: Inverse Magnetization Problems IN GEosciences.

International Partner (Institution - Laboratory - Researcher):

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

Start year: 2013

See also: 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 of Mathematicians at Vanderbilt Univ.

**MIT-France seed funding** is a competitive collaborative research
program ran
by the Massachusetts Institute of Technology (Cambridge, Ma, USA). Together with
E. Lima and B. Weiss from the Earth and Planetary Sciences dept. at MIT,
Apics obtained two-years support from the above-mentioned program to run a project entitled:
“Development of Ultra-high Sensitivity Magnetometry for Analyzing Ancient Rock Magnetism”

**NSF Grant** L. Baratchart, S. Chevillard and J. Leblond are
external investigators in the NSF Grant 2015-2018,
"Collaborative Research: Computational
methods for ultra-high sensitivity magnetometry of geological samples"
led by E.B. Saff (Vanderbilt Univ.) and B. Weiss.
(MIT).

Andrea Gombani (IEIIT-CNR, Padova, Italy, February 16-27).

Michael Northington (Vanderbilt University, Nashville, Tennessee, USA, July 21-30).

Vladimir Peller (Michigan State Univ., East Lansing, USA, September 2-30).

Eduardo Lima (MIT, Boston, Massachusetts, USA, September 6-12).

Isabella Sanders (MIT, Boston, Massachusetts, USA, September 6-12).

Konstantinos Mavreas, Master 2 Computational Biology - UNSA (5 months), Dipole localization in Moon rocks from sparse magnetic data.

L. Baratchart was a visiting scientist at Indiana University-Purdue University at Indianapolis (IUPUI), November 2015.

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), BESA company (Munich), Flextronics.

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), Inria Saclay (Lab. Poems), Cerege-CNRS (Aix-en-Provence), 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), Indiana University-Purdue University at Indianapolis, 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), 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 the ANR (Défis de tous les savoirs program) project MagLune (with Cerege, IPGP, ISTerre, Irphe), in a MIT-France collaborative seed funding, in the Associate Inria Team IMPINGE (with MIT, Boston), and in a NSF grant (with Vanderbilt University and MIT).

L. Baratchart gave a talk at the International Instrumentation and Measurement Technology Conference (I2MTC IEEE), May 11-14 2015, Pisa (Italy), at the AMS-EMS-SPM meeting, June 10-13 2015, Porto (Portugal), at the 10-th ISAAC congress (International Society for Analysis, its Applications and Computation), August 3-8 2015, Macau (China); he was a colloquium speaker at IUPUI, November 2015.

M. Caenepeel gave a talk at the IMS 2015 in Phoenix and at the EuMC 2015 in Paris .

S. Chevillard gave a talk at the 27th IFIP TC7 Conference on System Modelling and Optimization, Sophia Antipolis (July 2015).

J. Leblond presented a communication at the seminar “Mathématiques pour l'Analyse des Données” (MAD), Nice, May 21.

C. Papageorgakis presented posters at the 1st International Conference on Mathematical Neuroscience (ICMNS, Juan les Pins, Jun.) and at the International Conference on Basic and Clinical Multimodal Imaging (BACI, Utrecht, the Netherlands, September) , .

D. Ponomarev gave talks at the seminar “Modèles et Algorithmes Déterministes” of Lab. Jean Kuntzmann, Univ. J. Fourier, Grenoble (May 28) and at the 27th IFIP TC7 Conference on System Modelling and Optimization, Sophia Antipolis (July 2015).

F. Seyfert gave a talk at the European Microwave Week 2015 in the workshop dedicated to "Recent Advances in the Synthesis of Microwave Filters and Multiplexers", Paris, France

L. Baratchart and J. Leblond organized a special session on “Inverse Elliptic Problems” at the 27th IFIP TC7 Conference on System Modelling and Optimization, Sophia Antipolis (July 2015).

K. Mavreas and C. Papageorgakis are the PhD students in charge of the PhD students Seminar within the Research Center (since September).

L. Baratchart was on the Program Committee of the 17th IFAC Symposium on System Identification (SYSID 2015), Beijing, China, October 19-21, 2015.

F. Seyfert was a member of the technical committee of the conference "International Workshop on Microwave Filters" in Toulouse, France, March 23-25 2015, http://

L. Baratchart is a member of the Editorial Boards of the journals *Constructive Methods and Function Theory* and *Complex Analysis and Operator Theory*.

L. Baratchart was a reviewer for several journals including,
*SIAM Journal on Analysis, Inverse Problems, Journal of Approximation Theory, Annales de l'Institut Fourier*.

S. Chevillard was a reviewer for the journal *ACM Transactions on Mathematical Software*.

J. Leblond was a reviewer for the journals *Applied Mathematical Modelling, International Journal of Computer Mathematics*.

F. Seyfert was a reviewer of the journal *IEEE Microwave Transaction on Theory and Techniques*

L. Baratchart was an invited speaker at the Workshop on Blaschke Products and Function Theory, July 29-31 2015, Hong-Kong (China), and at the conference Orthogonal and Multiple Orthogonal Polynomials, August 9-14 2015, Oaxaca (Mexico). He was a plenary speaker at the “Journées du Gdr Analyse Fonctionnelle, Harmonique et Probabilités”, November 30-December 4 2015, Luminy (France).

J. Leblond was invited to give a communication at the conference “Harmonic Analysis, Function Theory, Operator Theory and Applications” (in honor of J. Esterle), Bordeaux, Jun. 1-4.

F. Seyfert gave a mini course on "Advanced Filter Synthesis" at the "International Workshop on Microwave Filters" in Toulouse, France, March 23-25 2015

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” and of the “Commission Administrative Paritaire” of Inria. She is in charge of the mission “Conseil et soutien aux chercheurs” within the Research Center. She is also a member of the “Conseil Académique” of the Univ. Côte d'Azur (UCA).

**Colles**: S. Chevillard is giving “Colles” at Centre International de Valbonne (CIV) (2 hours per week).

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, The development of models for the design of RF/microwave filters, since Feb. 2013 (advisors: Y. Rolain, M. Olivi, F. Seyfert).

PhD in progress: C. Papageorgakis, Conductivity model estimation, since Oct. 2014 (advisors: J. Leblond, M. Clerc, B. Lanfer).

PhD in progress: K. Mavreas, Inverse source problems in planetary sciences: dipole localization in Moon rocks from sparse magnetic data, since Oct. 2015 (advisors: S. Chevillard, J. Leblond).

L. Baratchart was a reviewer for the “Mémoire d'habilitation à diriger des Recherches” of Rachid Zarouf, Univ. Aix-Marseille, December 2015.

J. Leblond was a member of the “jury d'admission du concours CR” of Inria (Jun.).

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