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 Factas
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 .

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, as well as convex relaxation of hyperbolic approximation problems, see Sections and .

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, Factas 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 inverse source problems in neurosciences (in particular in EEG/MEG) and inverse magnetization problems in geosciences, see Section .

The approximation-theoretic tools developed by Apics and now by Factas 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.

Note that the Inria project-team Apics reached the end of its life cycle by the end of last year. The proposal for our new team Factas is since then being built and examined by the CEP (Comité des Équipes-Projets) of the Research Center, on its way to become a project-team.

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

This was actually carried out in collaboration with CEA (French nuclear agency) and the University of Nice (JAD Lab.), to data from *Tore Supra* in . 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 to the team.

The piece of work we just mentioned requires defining and studying Hardy
spaces of conjugate Beltrami equations, which is an interesting topic. 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. More importantly perhaps,
the exponent 2 is also the key to a corresponding theory on very
general (still rectifiable) domains in the plane, as
coefficients of pseudo-holomorphic functions
obtained by conformal transformation onto a disk are merely of

The 3-D version of step 1 in Section is another
subject investigated by Factas: to recover a harmonic function
(up to an additive constant) in a ball or a half-space from partial knowledge of its
gradient. This prototypical inverse problem
(*i.e.* inverse to the Cauchy problem for the Laplace equation)
often recurs in electromagnetism. At present, Factas 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 and INS (Institut de Neurosciences des Systèmes, http://

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
, .
More precisely, each section cuts out a planar domain, the boundary of which
carries data which can be proved to match an algebraic function. The
singularities of this algebraic function are not located at the 3-D sources,
but are related to them: the section contains a source if and only if some
function of the singularities in that section meets a relative extremum. Using
bisection it is thus possible to determine an extremal place along all sections
parallel to a given plane direction, up to some threshold which has to be
chosen small enough that one does not miss a source. This way, we reduce the
original source problem in 3-D to a sequence of inverse poles and branchpoints
problems in 2-D.
This bottom line generates a steady research activity
within Factas, and again applications are sought to medical imaging and
geosciences, see Sections ,
and .

Conjectures may 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.

Factas 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, Factas 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 deals with this hot topic using
contractive interpolation with constraints on boundary peak points,
within 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.
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 has focused 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 is that
the unstable part of each partial transfer function is rational and can
be computed by analytic projection,
see Section . We now start investigating the
linearized
harmonic transfer-function around a periodic cycle, to check for stability
under non necessarily small inputs. This generalization
generates both doctoral and postdoctoral work by new students in the team.

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 they are 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
a worthy and timely endeavor today. 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. For instance when poles are simple, the McMillan degree is the sum of the ranks of the residues.

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 Factas 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 ,
the original twist by Apics, now Factas, is to consider this issue also 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 given pole cardinality to 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, Factas 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.

Scientific Description

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 coupling matrix synthesis problem (C.M. problem for short) 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-Espana

Functional Description

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 consists in finding all possible electromagnetic coupling values between resonators that yield a realization of given filter characteristics. Solving the latter 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.

Participant: Fabien Seyfert

Contact: Fabien Seyfert

Keywords: Health - Neuroimaging - Visualization - Compilers - Medical - Image - Processing

FindSources3D is a software program dedicated to the resolution of inverse source problems in electroencephalography (EEG). From pointwise measurements of the electrical potential taken by electrodes on the scalp, FindSources3D estimates pointwise dipolar current sources within the brain in a spherical model.

After a first data transmission “cortical mapping” step, it makes use of best rational approximation on 2-D planar cross-sections and of the software RARL2 in order to locate singularities. From those planar singularities, the 3-D sources are estimated in a last step, see .

The present version of FindSources3D (called FindSources3D-bolis) provides a modular, ergonomic, accessible and interactive platform, with a convenient graphical interface and a tool that can be distributed and used, for EEG medical imaging. Modularity is now granted (using the tools dtk, Qt, with compiled Matlab libraries). It offers a detailed and nice visualization of data and tuning parameters, processing steps, and of the computed results (using VTK).

A new version is being developed that will incorporate a first Singular Value Decomposition (SVD) step in order to be able to handle time dependent data and to find the corresponding principal static components.

Participants: Juliette Leblond, Maureen Clerc (team Athena, Inria Sophia), Jean-Paul Marmorat, Théodore Papadopoulo (team Athena).

Contact: Juliette Leblond

URL: http://

Scientific Description

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).

Functional Description

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

• 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.

Participants: Fabien Seyfert, Jean-Paul Marmorat and Martine Olivi

Contact: Fabien Seyfert

Réalisation interne et Approximation Rationnelle L2

Scientific Description

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 and FindSources3D. 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).

Functional Description

RARL2 is a software for rational approximation. It 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 N Fourier coefficients,

• discretized (uniformly distributed) values on the circle. In this case, a least-square criterion is used instead of the L2 norm.

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.

An appropriate Möbius transformation allows to use the software for continuous-time systems as well.

Participants: Jean-Paul Marmorat and Martine Olivi

Contact: Martine Olivi

Keywords: Numerical algorithm - Supremum norm - Curve plotting - Remez algorithm - Code generator - Proof synthesis

Functional Description

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.

Participants: Sylvain Chevillard, Christoph Lauter, Mioara Joldes and Nicolas Jourdan

Partners: CNRS - ENS Lyon - UCBL Lyon 1

Contact: Sylvain Chevillard

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 .

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 Sections , .

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. The associate team Impinge 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 planar region at small distance from 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

Another timely instance of inverse magnetization problems lies with geomagnetism. Satellites orbiting around the Earth measure the magnetic field at many points, and nowadays it is a challenge to extract global information from those measurements. In collaboration with C. Gerhards (since September with the Geomathematics and Geoinformatics Group, Technische Universität Bergakademie Freiberg, Germany; until then, Univ. of Vienna, Austria), we started to work on the problem of separating the magnetic field due to the magnetization of the globe's crust from the magnetic field due to convection in the liquid metallic core. The techniques involved are variants, in a spherical context, from those developed within the Impinge associate team for paleomagnetism, see Section .

This work is conducted in collaboration with Maureen Clerc and Théo Papadopoulo from the team Athena (Inria Sophia).

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. Our belief
is that such a localization process can add a geometric, valuable piece of
information to the standard temporal analysis of EEG signal records.

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 our partners at INS, hospital la Timone, Marseille.

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 ) in order to get a models for the scattering matrix as an analytic matrix-valued function. This provides us with a scattering matrix known to be close to a rational matrix of order roughly 1/4 of the number of data points.

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

A state space realization of

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 further 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 additionally 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.
The team is also becoming active in this area.

In this connection, 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 . The stability check for
limit cycles is now under investigation.

Keywords: Electrical circuit - Stability

Functional Description: To minimise prototyping costs, the design of analog circuits is performed using computer-aided design tools which simulate the circuit's response as accurately as possible.

Some commonly used simulation tools do not impose stability, which can result in costly errors when the prototype turns out to be unstable. A thorough stability analysis is therefore a very important step in circuit design. This is where pisa is used.

pisa is a Matlab toolbox that allows designers of analog electronic circuits to determine the stability of their circuits in the simulator. It analyses the impedance presented by a circuit to determine the circuit's stability. When an instability is detected, pisa can estimate location of the unstable poles to help designers fix their stability issue.

Release Functional Description: First version

Authors: Adam Cooman, David Martinez Martinez, Fabien Seyfert and Martine Olivi

Contact: Fabien Seyfert

Publications: Model-Free Closed-Loop Stability Analysis: A Linear Functional Approach - On Transfer Functions Realizable with Active Electronic Components

*PUMA-HF: Passive Uniform Matching*

Keyword: Transfer functions

Functional Description: PUMA computes a passive rational 2-port filtering function presenting a reference impedance (i.e. 50 Ohm) at the first port, and the conjugate of the given load impedance within a frequency band at the second port.

Authors: David Martinez Martinez, Adam Cooman, Martine Olivi and Fabien Seyfert

Partners: Xlim - DGA-MI - CNES

Contact: Fabien Seyfert

Publication: Synthesis Method for Matching Filters

This work has been carried out in the framework of the Inria Associate Team Impinge, comprising Cauê Borlina, Eduardo Andrade Lima and Benjamin Weiss from the Earth Sciences department at MIT (Boston, USA) and Douglas Hardin, Edward Saff and Cristobal Villalobos from the Mathematics department at Vanderbilt University (Nashville, USA).

The overall goal of Impinge was to determine
magnetic properties of rock
samples (*e.g.* meteorites or stalactites), from weak field measurements
close to the sample that
can nowadays be obtained using SQUIDs (superconducting quantum interference
devices).
Depending on the geometry of the rock sample, the magnetization distribution can either be considered to lie in a plane or in a parallelepiped of thickness

We pursued our research efforts towards designing algorithms for net moment recovery. The net moment is the integral of the magnetization over its support, and it is a valuable piece of information to physicists which has the advantage of being determined solely by the field: whereas two different magnetizations can generate the same field, the net moment depends only on the field and not on which magnetization produced it. Hence the goal may be described as to build a numerical magnetometer, capable of analyzing data close to the sample. This is in contrast to classical magnetometers which regard the latter as a single dipole, an approximation which is only valid away from the sample and is not suitable to handle weak fields which get quickly blurred by ambient magnetic sources if one measures the field at a distance from the sample.

The first approach consists in using the fact that the integral of

The second approach attempts to generalize the previous expansions in the case when

A third approach developed during the previous years was to design an alternate procedure to compute a good linear estimator, dwelling on expansions on a family of piecewise affine functions, with a restricted number of pieces. A key point here is that it is possible to derive explicit formulas for the adjoint operator

Concerning full inversion of thin samples, after preliminary experiments on regularization with

Besides, we considered a simplified 2-D setup for magnetizations and magnetic potentials (of which the magnetic field is the gradient). When both the sample and the measurement set are parallel intervals, some best approximation issues related to inverse recovery and relevant BEP problems in Hardy classes of holomorphic functions (see Section ). Note that, in the present case, the criterion no longer acts on the boundary of the holomorphy domain (namely, the upper half-plane), but on a strict subset thereof, while the constraint acts on the support of the approximating function. Both involve functions in the Hilbert Hardy space of the upper half-plane. This is the subject of ongoing work with E. Pozzi (Department of Mathematics and Statistics, St Louis Univ., St Louis, Missouri, USA).

For magnetizations supported in a volume

Other types of inverse magnetization problems can be tackled using such techniques, in particular global Geomagnetic issues which arise in spherical geometry. In collaboration with C. Gerhards from the Technische Universität Bergakademie Freiberg (Germany), we developed a method to separate the crustal component of the Earth's magnetic field from its core component, if an estimate of the field is known on a subregion of the globe . This assumption is not unrealistic: parts of Australia and of northern Europe are considered as fairly well understood from the magnetostatic view point. We are currently working to test the algorithm against real data, in collaboration with Geophysicists.

The team Factas is a partner of the ANR project MagLune on Lunar magnetism, headed by the Geophysics and Planetology Department of Cerege, CNRS, Aix-en-Provence (see Section ). Recent studies let geoscientists think that the Moon used to have a magnetic dynamo for a while. However, the exact process that triggered and fed this dynamo is still not understood, much less why it stopped. The overall goal of the project is to devise models to explain how this dynamo phenomenon was possible on the Moon.

The geophysicists from Cerege went a couple of times to NASA to perform measurements on a few hundreds of 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 samples from NASA facilities. Moreover, the process must be carried out efficiently as a fee is due to NASA by the time when handling these moon samples. Therefore, measurements were performed with some specific magnetometer designed by our colleagues from Cerege. This device measures the components of the magnetic field produced by the sample, at some discrete set of points located on circles belonging to three cylinders (see Figure ). The objective of Factas is to enhance the numerical efficiency of post-processing data obtained with this magnetometer.

Under the hypothesis that the field can be well explained by a single magnetic pointwise dipole, and using ideas similar to those underlying the FindSources3D tool (see Sections and ), we try to recover the position and the moment of the dipole using the available measurements. This is still on-going work which constitutes the main topic of the PhD thesis of K. Mavreas.

In a given cylinder, using the associated cylindrical system of coordinates, recovering the position of the dipole boils down to determine its height

In practice, due to the many sources of imprecision (the first of all being that the field is not truly generated by a single dipole), the circles do not all truly intersect. This year, we studied three different strategies to estimate the pseudo-intersection point of the circles. In the plane, for a point

finding a point

finding a point

finding a point

The first case turns out to actually be a generalization of two classical concepts: the Fermat point (or Torricelli point) of a triangle, and the Alhazen optical problem. The second case corresponds to a classical notion called the *radical center* of three circles (intersection of the three corresponding radical axes). Finally, the third case does not seem to have a documented solution. We solve it by writing the algebraic system of two equations corresponding to the critical points of the function, after an appropriate change of coordinates in order to reduce the degree. Finally, we get a superset of the solutions by estimating the roots of the resultant of both polynomials. First experiments showed that the third formulation led to the most satisfying estimate of the pseudo-intersection. We also implemented a heuristic numerical procedure (without theoretical formulas for its solution) to estimate the point

Another important part of our work this year has been to extensively test our implementation of the rational approximation procedure which is at the heart of our method (and which is also used for the problem described in Section ). These tests allowed us to detect situations in which the algorithm was falling into an infinite loop or was converging towards a local minimum that was not really the best approximation. It also revealed that all initialization strategies for the iterative optimization algorithm were not equally sensitive to the noise. This led us to redesign our implementation.

Finally, the article that we submitted last year, with a rudimentary approach to recover

In 3-D, functional or clinically 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 ).

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 from dipolar current sources:
for EEG data that correspond to measurements of the electrical potential, one should consider *triple* poles; this will also be the case for MEG – magneto-encephalography – data. However, for (magnetic) field data produced by magnetic dipolar sources, like in Section , one should consider poles of order five. 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 and will be the topic of further study. It is used in order to automatically estimate the “most plausible”
number of sources (numerically: up to 3, at the moment).
Last but not least, the version of the software currently under development
takes as inputs actual EEG measurements, like time signals, and performs a suitable singular value decomposition in order to separate independent sources.

Magnetic data from MEG recently became available along with EEG data, by our medical partners at INS in Marseille; indeed, it is now possible to use simultaneously both measurement devices, in order to measure both the electrical potential and a component of the magnetic field (its normal component on the MEG helmet, that can be assumed to be spherical). This should enhance the accuracy of our source recovery algorithms. We will add the treatment of MEG data as another functionality of the software FS3D.

Concerning dipolar source estimation from EEG, joint work with Marion Darbas (Univ. Picardie Jules Verne, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, LAMFA) is in progress for neonates data and models. Their specificity is that the skull does not have a constant conductivity (at the fontanels location, the bone is spongy). We pursue together a study of the influence of the skull conductivity on the inverse EEG problem, using in particular FS3D, see also .

We also consider non quasi-static models in order to more precisely analyze the time influence on the behavior of the solutions to the inverse source problems in EEG and MEG. This is current work with Iannis Stratis and Atanasios Yannacopoulos (National and Kapodistrian University of Athens, Greece, Department of Mathematics).

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. For example, 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. Antennas are not the only systems where matching is of importance: in multiplexer design, one of the most complicated problems among microwave device synthesis, each filter is plugged at one of its accesses on a load made of the common manifold and all other channel filters. This load is far from being matched, and leads to a complex simultaneous matching problem, of all filters connected via the common manifold junction. 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

Based on our work linking Nevanlinna-Pick interpolation and point-wise matching techniques we tackled the synthesis of a multiplexer on the basis of frequency specifications relative to a triplexer furnished by CNES and deemed to be problematic. Theoretical results guaranty the existence in this case, under a strict contractivity hypothesis on the common junction, of a simultaneous pointwise matching solution of all three channel filters. This result is however not constructive as it relies on Brouwer's fixed point theorem, a purely topological argument. In the context of the PhD of D. Martinez, we developed a continuation algorithm, starting from a completely decoupling junction and ending up with the manifold junction of interest. When the junction decouples all the channels, the overall matching problem results in 3 independent Nevanlinna-Pick interpolation problems as described in : this yield the starting point of our continuation approach. The manifold peaks have also been identified as crucial here: these are resonances taking place in the common junction of the multiplexer, at which transmission of energy becomes impossible between a given channel and the common port. These peaks have been mathematically characterized, and a systematic combinatorial algorithm designed for the synthesis of a “peak free” manifold junction. In particular it was proven, that an extremely compact junction is needed to fit the specifications furnished by CNES: when dispersive effects increase (and they do with the size of the junction) the appearance of manifold peaks has been proven inevitable. Using a continuation approach to compute the channel filters responses and a combinatorial procedure to design a peak-free junction, a complete triplexer has been synthesized that fits the CNES specifications. The synthesis was first considered in terms of circuits and eventually transformed in a real hardware realized in waveguide technology (see Figure. ). The latter is currently being manufactured at XLIM laboratories, where our colleague S. Bila is co-advising D. Martinez' PhD.

The internal representation of our problem, relies on the use of a Pick matrix

This is an important result, as it justifies the use of Lagrangian relaxation for our problem that appears to be a non-linear, convex, semi-definite optimization program (NLSDP), the hardest class among current convex optimization. We plan to build on this result, in order to obtain a critical point equation to our problem, in terms of extremal points of the involved polynomials.

The software implementation developed within the PhD of D. Martinez, combining logarithmic barrier functions and Lagrangian relaxation techniques has been made available for practitioners as Matlab library called Puma . Results obtained thank to this tool, have been presented in , and on a dedicated workshop at the conference IMS2018. Design of use cases, for antenna matching problems relevant in 5G and IOT applications are currently being analyzed with LEAT, within the context of G. Bose's PhD.

The goal is here to help design amplifiers, in particular to detect instability at an early stage of the design. This activity has gained momentum with the doctoral work of (S. Fueyo), co-advised with J.-B. Pomet (from the McTao Inria project-team) and the postdoctoral stay of (A. Cooman) that eventually resulted in substantial software developments. Application of our work to oscillator design methodologies started recently in collaboration with Smain Amari from the Royal Military College of Canada (Kingston, Canada).

As opposed to Filters and Antennas, Amplifiers and Oscillators are active components that intrinsically entail a non-linear functioning. The latter is due to the use of transistors governed by electric laws exhibiting saturation effects, and therefore inducing input/output characteristics that are no longer proportional to the magnitude of the input signal. Hence, they typically produce non-linear distortions. A central question arising in the design of amplifiers is to assess stability. The latter may be understood around a functioning point when no input but noise is considered, or else around a periodic trajectory when an input signal at a specified frequency is applied. For oscillators, a precise estimation of their oscillating frequency is crucial during the design process. As regards devices devised to operate at relative low frequencies, time domain simulations, based on the integration of the underlying non-linear dynamical system, answers these questions satisfactorily. For complex microwave amplifiers and oscillators, the situation is however drastically different: the time step necessary to integrate the transmission line's dynamical equations (which behave like simple electrical wire at low frequency) becomes so small that simulations are intractable in reasonable time. In addition to this problem, most linear components of these circuits are known through their frequency response, and require therefore a preliminary, numerically unstable step to obtain their impulse response, prior to any time domain simulation.

For all these reasons it is widely preferred to perform the analysis of such systems in the frequency domain. In the case of stability issues around a functioning point, where only small input signals are considered, the stability of the linearized system obtained by a first order approximation of each non-linear dynamic is considered. This is done by means of the analysis of transfer impedance functions computed at some ports of the circuit. We have shown, that under some realistic hypothesis on the building blocks of the circuit, these transfer functions are meromorphic functions of the frequency variable

Dwelling on the unstable/stable decomposition in Hardy Spaces, we developed a procedure to assess the stability or instability of the transfer functions at hand, from their evaluation on a finite frequency grid , that we further improved in to address the design of oscillators, in collaboration with Smain Amari. The evaluation of the admittance function of interest is furnished, on a finite frequency band, by a circuit simulator. Progress were made on the interpolation procedure and the determination of a filtering function that are used to obtain a functional representation of high order of the unstable part of the admittance function to be analyzed. The latter was tested on a time-delayed Chua oscillator circuit: the analytical model of this circuit is known in closed form, and using continuation techniques on the involved delay components it is possible to compute the exact unstable poles of this circuit: two being exactly at the oscillators frequency, while two others spurious poles at DC frequency are present and are usually hard to detect with classical methods. Our approximation based procedure, which was fed with incomplete frequency data estimation of the admittance, was able to recover all poles within a relative error of less than

When stability is studied around a periodic trajectory, which is
determined in practice by Harmonic Balance algorithms, linearization yields a linear time varying dynamical system with periodic coefficients
and a periodic trajectory thereof. While in finite dimension the stability of such systems is well understood
via the Floquet theory, this is no longer the case in the infinite dimensional setting when delays are considered. Dwelling on the theory of retarded systems,
S. Fueyo's PhD work showed in previous years that, for certain simple circuits with
properly positioned resistors,
the monodromy operator is a compact perturbation of a stable operator,
and that only finitely many unstable point of its spectrum can occur.
This year, we proved a similar result for general circuits, provided that they are
passive at very high frequency. For this, we use Lyapunov functions for the
transmission lines to establish exponential

(This is joint work with T. Qian and P. Dang from the university of Macao.)
It was proven in previous year that on a smooth compact hypersurface *i.e.*

This is a very recent activity of the team, linked to images classification in archaeology in the framework of the project ToMaT, “Multiscale Tomography: imaging and modeling ancient materials, technical traditions and transfers” (see Section ), and to the post-doctoral stay of V. L. Coli ; it is pursued in collaboration with L. Blanc-Féraud (project-team Morpheme, I3S-CNRS/Inria Sophia/iBV), D. Binder (CEPAM-CNRS, Nice), in particular.

The pottery style is classically used as the main cultural marker within Neolithic studies. Archaeological analyses focus on pottery technology, and particularly on the first stages of pottery manufacturing processes. These stages are the most demonstrative for identifying the technical traditions, as they are considered as crucial in apprenticeship processes.
Until now, the identification of pottery manufacturing methods was based on macro-traces analysis, i.e. surface topography, breaks and discontinuities indicating the type of elements (coils, slabs...) and the way they were put together for building the pots.
Overcoming the limitations inherent to the macroscopic pottery examination requires a complete access to the internal structure of the pots. Micro-computed tomography (

The main challenge of our current analyses aims to overcome the lack of existing protocols to apply in order to quantify observations. In order to characterize the manufacturing sequences, the mapping of the paste variability (distribution and composition of temper) and the discontinuities linked to different classes of pores, fabrics and/or organic inclusions appears promising. The totality of the acquired data composes a set of 2-D and 3-D surface and volume data at different resolutions and with specific physical characteristics related to each acquisition modality (multimodal and multi-scale data). Specific shape recognition methods need to be developed by application of robust imaging techniques and 3-D-shapes recognition algorithms.

In a first step, we devised a method to isolate pores from the 3-D data volumes; we are currently focusing our investigation on 2-D slices displaying pores locations and we considering several data processing treatments, such as multiresolution processing and Hough transform (derived from Radon transform), in order to evaluate their outcome when applied to these very particular images. Different possibilities of investigation will be analyzed as well, such as “a contrario” analysis and deep learning techniques.

We proved this year that if a function is holomorphic outside a disk of radius

The overall and long-term goal is to enhance the quality of numerical computations. The software tool Sollya (see Section ), developed together with C. Lauter (Université Pierre et Marie Curie) intends to provide an interactive environment for performing numerically rigorous computations. It comes as a standalone tool and also as a C library that allows one to benefit from all the features of the tool in C programs.

In September 2018, we released version 7.0. Among other things, this release fixes some bugs and improves the way base functions are internally handled. Another important novelty of 2018 is that Sollya now comes with the companion `pythonsollya` `pythonsollya`.

This contract (reference Inria: 11282) accompanies the PhD of David Martinez and focuses on the development of efficient techniques for the design of matching network tailored for frequency varying loads. Applications of the latter to the design output multiplexers occurring in space applications will be considered.

This is a scientific consulting activity for the start-up company SKAVENJI. The latter develops an innovative and communicative device to facilitate the production and home consumption of small amounts of energy, produced by one or more local sources of renewable energy. Our contribution consisted in designing a simple controller improving the energy efficiency of the energy production while minimizing the number of charge and discharge cycles of the associated battery. The retained control strategy was based on consumption and production profiles.

The team participates in the project WIMAG (Wave IMAGing) funded by the Idex

The team co-advises a PhD (G. Bose) with the CMA team of LEAT (http://

The team participates in the transverse action C4PO funded by the Idex

The team also participates in the project ToMaT, “Multiscale Tomography: imaging and modeling ancient materials, technical traditions and transfers”, funded by the Idex

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, then Factas 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. Factas 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.

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 and ended October 2018. We were associated with three other teams from XLIM (Limoges University), geared respectively towards filters, antennas and amplifiers
design. The core idea of the project was to
realize dual band
reception an emission chains by co-conceiving the antenna, the filters, and the amplifier. A complete chain has been synthesized using matching filters placed after each of the accesses of a network of 4 bi-polarized antennas, resulting in a substantial gain in energy effectiveness of

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):

Massachusetts Institute of Technology (United States) - Department of Earth, Atmospheric and Planetary Sciences - Benjamin P. Weiss

Start year: 2016

See also: http://

This proposal is concerned with the inverse problem of recovering a magnetization distribution from measurements of the magnetic field in a portion of space nearby. The application domain is to Earth and planetary sciences. Indeed, the remanent magnetization of rocks provides valuable information on their history. The proposal aims at renewing the existing “Équipe Associée” Impinge ending 2015, between Apics (now Factas) team at Inria and the Department of Earth, Atmospheric and Planetary Sciences at MIT (Cambridge, MA, USA), with the Department of Mathematics at Vanderbilt University (Nashville, TN, USA) as a secondary partner. Several research paths were broken towards magnetization recovery and promising numerical experiments have been conducted. This initial effort must be continued to achieve a reasonably complete methodology for reconstructing magnetizations and checking for hypotheses by geophysicists (e.g., unidirectionality of magnetization distributions).

**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).

Axel Ringh (KTH Royal Institute of Technology, Stockholm, Sweden), June 11 -July 10.

Elodie Pozzi (St Louis Univ., St. Louis, Missouri, USA), Brett Wick (Washington University, St. Louis, Missouri, USA), July 25.

Cristobal Villalobos, Douglas Hardin, Edward Saff (Vanderbilt University, Nashville, Tennessee, USA), December, 6-19, 13-19, and 13-20 respectively.

M. Olivi proposed with B. Hanzon and R. Peeters a Research in Pair event at CIRM « A state-space approach to parametrization of lossless and stable systems ». which was accepted and took place on August 27-31 https://

Collaboration under contract with Thales Alenia Space (Toulouse, Cannes, and Paris), CNES (Toulouse), XLIM (Limoges), LEAT (Sophia Antipolis), University of Bilbao (Universidad del País Vasco / Euskal Herriko Unibertsitatea, Spain), 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, ENSTA), IMT Atlantique (Institut Mines-Télécom., Brest), 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, St Louis University and Washington University (St Louis, Missouri, USA), Cinvestav (Queretaro, Mexico), 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 and Universität Wien (Austria), TFH-Berlin and Technische Universität Bergakademie Freiberg (Germany), ENIT (Tunis), KTH (Stockholm), University of Cyprus (Nicosia, Cyprus), University of Macau (Macau, China), SIAE Microelettronica (Milano), Université Picardie Jules Verne (France), National and Kapodistrian University of Athens (Greece).

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).

V. L. Coli presented oral communications at the
“Journées Idex UCA

J. Leblond and K. Mavreas presented oral communications at the 5th International Conference “Modern Mathematical Methods in Science and Technology 2018 (M3ST '18)”, September 2-4, Kalamata, Greece.

K. Mavreas gave a talk at the PhD Seminar at Inria Sophia, France, November 13, https://

G. Bose, S. Fueyo and D. Martinez Martinez presented posters at the Workshop ERNSI 2018, http://

G. Bose gave a talk at the 23rd International Symposium on Mathematical Theory of Networks and Systems http://

D. Martinez Martinez gave a talk at the 7th International Workshop on Microwave Filters https://

A. Cooman gave a talk at IMS 2018 https://

F. Seyfert gave a talk at IMS 2018 https://

L. Baratchart, S. Chevillard, and J. Leblond organized (scientifically and practically) the Spring School/Workshop “Inverse problems and approximation techniques in planetary sciences”

L. Baratchart and J. Leblond organized a mini-symposium

Fabien Seyfert organized a workshop at IMS 2018, on “Advanced Synthesis Techniques for reduced size filtering techniques”, https://

L. Baratchart sits on the Program Committee of
*Applied Inverse Problems 2019*, to be held in Grenoble, France.

L. Baratchart sits on the editorial boards of “Constructive Methods in Function Theory” and “Complex Analysis and Operator Theory”.

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

J. Leblond was a reviewer for the journals *Numerical Algorithms, DMV Jahresberichte*.

M. Olivi was a reviewer for the journal *Automatica* and for the *18th IFAC Symposium on System Identification* (SYSID 2018).

F. Seyfert is a reviewer of *IEEE Transactions on Microwave Theory and Techniques*.

L. Baratchart was a reviewer for the seed fund program at MIT.

L. Baratchart was an invited speaker at the SIGMA day organized by SMAI at Université Pierre et Marie Curie, November 30.

L. Baratchart and J. Leblond were invited to give talks at the IPMS 2018 conference, Malta, May (see Section ). They were invited speakers at the Conference “Advances in Operator Theory with Applications to Mathematical Physics”,
http://

L. Baratchart and K. Mavreas gave talks at the workshop “Inverse problems and approximation techniques in planetary sciences”, Sophia Antipolis, France, May.

L. Baratchart is sitting on committee 40 of the ANR.

J. Leblond was a reviewer for the Strategic Research Programmes of the Research Council of the Vrije Universiteit Brussel (VUB, Brussel, Belgium).

F. Seyfert is a member of the IEEE MTT-8 committee for filters and passive components.

L. Baratchart sits on the committee “Mathématiques et Informatique” of the French Agency for research (ANR).

J. Leblond is an elected member of the “Conseil Scientifique” and of the “Commission Administrative Paritaire” of Inria.

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

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

PhD in progress: D. Martinez Martinez, *Méthodologie et Outils de Synthèse pour des Fonctions de Filtrage Chargées par des Impédances complexes*, since October 2015, advisors: L. Baratchart, F. Seyfert, M. Olivi.

PhD in progress: G. Bose, *Filter Design to Match Antennas*, since
December 2016, advisors: F. Ferrero, F. Seyfert and M. Olivi.

PhD in progress: S. Fueyo, *Cycles limites et stabilité dans les circuits*, since October 2016, advisors: L. Baratchart and J.-B. Pomet (Inria Sophia, McTao).

Post-doc. in progress: V. L. Coli, *Multiscale Tomography: imaging and modeling ancient materials*, since March 2018, advisors: J. Leblond, L. Blanc-Féraud (project-team Morpheme, I3S-CNRS/Inria Sophia/iBV), D. Binder (CEPAM-CNRS, Nice).

Post-doc.: A. Cooman, from January 2017 until June 2018, advisors: F. Seyfert and M. Olivi.

J. Leblond was a member of the “Comités de Suivi Doctoraux” (preliminary evaluation committees, after one year of PhD) for the ongoing PhDs of Kostiantyn Maksymenko (project-team Athena, doctoral school STIC, UCA), and for Arne Bechensteen (project-team Morpheme, I3S-CNRS), May. She was a member of a “Comité de Sélection” for professors at the University Paris Descartes (MAP5).

M. Olivi was a member of the “jury d'admission du concours CR” of the Inria Research Center.

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

submit applications to get founding for 2019: “cordées de la réussite” (accepted) et “APOCS région”.

co-organize 10 robotic sessions for 2 classes of middle school students (device
“MEDITES” http://

co-organize the “stage MathC2+” https://

co-organize Inria participation to the event “Le Village des Sciences et de l'Innovation” in Antibes (October 20-21, 10000 people), and the event “Mouans-Sartoux fête les sciences du quotidien” http://

co-organize about 10 “cafés scientifiques” (c@fé-in's and cafés Techno, 30 to 80
participants each) https://

M. Olivi animated a half-day workshop session “activités débranchées” at “l'ESPE de Nice” for primary school students (April 10).

National events: M. Olivi participated to the event “Le Village des Sciences et de l'Innovation” in Antibes (October 20-21, 8000 people) and to the event “Mouans-Sartoux fête les sciences du quotidien” (October 13, 1000 people).

In educational institutions: M. Olivi participated to the event “bilan Medium's” at the “collège Émile Roux, Le Cannet”.

Welcoming of schoolchildren or the general public in an Inria center: M. Olivi animated a workshop session during the “stage MathC2+”, a four-day internship for high-school students organized by the Committee MASTIC and its partners (June 19-22).

M. Olivi presented the scientific objects produced during the year at the CaféIn of the Research Center, https://

J. Leblond together with members of the SED gave a communication at the CaféIn of the Research Center, http://

M. Olivi participated in the creation of the web pages https://