RESEARCH CENTRE

Inria Centre
at Université Côte d’Azur

2023
ACTIVITY REPORT

Project-Team

FACTAS

Functional Analysis for ConcepTion and
Assessment of Systems

DOMAIN

Applied Mathematics, Computation and
Simulation

THEME

Optimization and control of dynamic
systems

Contents

Project-Team FACTAS

1 Team members, visitors, external collaborators

2 Overall objectives

1

2

2

3 Research program

3
3
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3.2 Range of inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3.2.1 Elliptic partial differential equations (PDE) . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.2.2 Systems, transfer and scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3.1 Best analytic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Best meromorphic and rational approximation . . . . . . . . . . . . . . . . . . . . . .
9
3.3.3 Behavior of poles of meromorphic approximants . . . . . . . . . . . . . . . . . . . . . 11
3.4 Software tools of the team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4.1 pisa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4.2 DEDALE-HF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4.3 FindSources3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4.4 PRESTO-HF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4.5 RARL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4.6 Sollya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Application domains

15
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Inverse source problems in EEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Inverse magnetization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Structure and stability of active devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.5 Identification of resonating frequencies of compact metallic objects in electromagnetic

inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Social and environmental responsibility

19

6 New results

19
6.1 Inverse problems for Poisson-Laplace and Helmholtz equations . . . . . . . . . . . . . . . . 19
6.1.1 Inverse magnetization issues for planar and volumetric samples . . . . . . . . . . . . 20
6.1.2 Inverse problems in medical imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 Stability assessment of microwave circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3 Hardy-Hodge decomposition with applications to silent sources . . . . . . . . . . . . . . . . 24
6.4 Identification of resonating frequencies of compact metallic objects in electromagnetic

inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.5 Other elliptic PDE problems: from direct to inverse problems . . . . . . . . . . . . . . . . . . 26
6.6 Spectra of Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.7 Tools for numerically guaranteed computations . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Partnerships and cooperations

7.1 International research visitors

28
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.1.1 Visits of international scientists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.2.1 Other european programs/initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.3 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.2 European initiatives

8 Dissemination

29
8.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.1.2 Journal
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.1.3 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.1.4 Contributed talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.1.5 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.2.1 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.2.2 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.2.3 Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.3 Popularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.3.1 Internal or external Inria responsibilities . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.3.2 Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Scientific production

31
9.1 Major publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.2 Publications of the year
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9.3 Cited publications

Project FACTAS

1

Project-Team FACTAS

Creation of the Project-Team: 2019 July 01

Keywords

Computer sciences and digital sciences

A6.1.1. – Continuous Modeling (PDE, ODE)

A6.2.1. – Numerical analysis of PDE and ODE

A6.2.5. – Numerical Linear Algebra

A6.2.6. – Optimization

A6.3.1. – Inverse problems

A6.3.3. – Data processing

A6.3.4. – Model reduction

A6.4.4. – Stability and Stabilization

A6.4.5. – Control of distributed parameter systems

A6.5.4. – Waves

A8.2. – Optimization

A8.3. – Geometry, Topology

A8.4. – Computer Algebra

Other research topics and application domains

B1.2.3. – Computational neurosciences

B2.6.1. – Brain imaging

B3.1. – Sustainable development

B3.3. – Geosciences

B4.5. – Energy consumption

2

Inria Annual Report 2023

1 Team members, visitors, external collaborators

Research Scientists

• Juliette Leblond [Team leader, INRIA, Senior Researcher, HDR]

• Laurent Baratchart [INRIA, Emeritus]

• Sylvain Chevillard [INRIA, Researcher]

• Martine Olivi [INRIA, Researcher, HDR]

• Dmitry Ponomarev [INRIA, ISFP]

Post-Doctoral Fellow

• Cristóbal Villalobos Guillén [INRIA, Post-Doctoral Fellow, until Nov 2023]

PhD Students

• Paul Asensio [INRIA, until Jul 2023]

• Masimba Nemaire [UNIV BORDEAUX, until Jan 2023, (he was paid by INRIA from Feb 2023 until

April 2023)]

• Mubasharah Khalid Omer [UNIV COTE D’AZUR, from Oct 2023]

• Anass Yousfi [UNIV COTE D’AZUR]

Interns and Apprentices

• Mubasharah Khalid Omer [INRIA, Intern, from Apr 2023 until Sep 2023]

Administrative Assistant

• Florence Barbara [INRIA]

Visiting Scientist

• Bernard Hanzon [UCC CORK, from Sep 2023]

External Collaborators

• Vanna Lisa Coli [UNIV COTE D’AZUR]

• Jean-Paul Marmorat [CMA, Mines ParisTech]

• Fabien Seyfert [HighFSolutions, Nice]

2 Overall objectives

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, har-
monic 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.

Project FACTAS

3

The mathematical tools pertain to complex and harmonic analysis, approximation theory, potential
theory, system theory, differential topology, optimization and computer algebra. Targeted applications
mostly concern non-destructive control from field measurements in medical engineering (source recovery
in magneto/electro encephalography), paleomagnetism (determining the magnetization of rock sam-
ples), and since recently, obstacle identification (finding electrical characteristics of an object), for which
an endeavor of the team is to develop algorithms resulting in dedicated software.

3 Research program

3.1 Introduction

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 f can be evaluated
at any point i ω of the imaginary axis from the response to a periodic input at frequency ω. Since f
is holomorphic in the right half-plane, it satisfies there the Cauchy-Riemann equation ¯∂ f = 0, and
recovering f amounts to solve a Dirichlet problem which can be done in principle using, e.g., the Cauchy
formula.

Practice is not nearly as simple, for f is only measured pointwise in the pass-band of the system
which makes the problem ill-posed [79]. Moreover, the transfer function is usually sought in specific
form, displaying the necessary physical parameters for control and design. For instance if f is rational of
degree n, then ¯∂ f = (cid:80)n
1 a j δz j where the z j are its poles and δz j is a Dirac unit mass at z j . Thus, to find
the domain of holomorphy (i.e., to locate the z j ) amounts to solve a (degenerate) free-boundary inverse
problem, this time on the left half-plane. To address such questions, the team has developed a two-step
approach as follows.

Step 1: To determine a complete model, that is, one which is defined at every frequency, in a
sufficiently versatile function class (e.g., Hardy spaces). This ill-posed issue requires regularization,
for instance by adding constraints on the behavior at non-measured frequencies.

Step 2: To compute a reduced order model. This typically consists of rational approximation of the
complete model obtained in step 1, or phase-shift thereof to account for delays. We emphasize that
deriving a complete model in step 1 is crucial to achieve stability of the reduced model in step 2.

Step 1 relates to extremal problems and analytic operator theory, see Section 3.3.1. 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 3.3.2. It also makes contact with the
topology of rational functions, in particular to count critical points and to derive bounds, see Section 3.3.2.
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 3.3.3.

Applying a realization procedure to the result of step 2 yields an identification procedure from
incomplete frequency data which was first demonstrated in [85] 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 [38].

The previous example of harmonic identification quickly suggests a generalization of itself. Indeed,
on identifying C with R2, holomorphic functions become conjugate-gradients of harmonic functions, so
that harmonic identification is, after all, a special case of a classical issue: to recover a harmonic function
on a domain from partial knowledge of the Dirichlet-Neumann data; when the portion of boundary
where data are not available is itself unknown, we meet a free boundary problem. This framework for 2-D
non-destructive control was first advocated in [69] and subsequently received considerable attention. It
makes clear how to state similar problems in higher dimensions and for more general operators than
the Laplacian, provided solutions are essentially determined by the trace of their gradient on part of the
boundary which is the case for elliptic equations 1 [36], [88]. Such questions are particular instances

1There is a subtle difference here between dimension 2 and higher. Indeed, a function holomorphic on a plane domain is defined

4

Inria Annual Report 2023

of the so-called inverse potential problem, where a measure µ has to be recovered from the knowledge
of the gradient of its potential (i.e., the field) on part of a hypersurface (a curve in 2-D) encompassing
the support of µ. For Laplace’s operator, potentials are logarithmic in 2-D and Newtonian in higher
dimensions. For elliptic operators with non constant coefficients, the potential depends on the form of
fundamental solutions and is less manageable because it is no longer of convolution type. Nevertheless it
is a useful concept bringing perspective on how problems could be raised and solved, using tools from
harmonic analysis.

Inverse potential problems are severely indeterminate because infinitely many measures within an
open set of Rn produce the same field outside this set; this phenomenon is called balayage [78]. 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 finite sum of point masses (cid:80)N
1 a j δz j ). The discreteness assumption also prevails in 3-D inverse source
problems, see Section 4.2. Conditions that ensure uniqueness of the solution to the inverse potential
problem are part of the so-called regularizing assumptions which are needed in each case to derive
efficient algorithms.

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 [1], [47, 51]. 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 3.2.1. There, holomorphic functions are replaced by harmonic gradients; applications
are to inverse source problems in neurosciences (in particular in EEG/MEG) and inverse problems in
geosciences.

The approximation-theoretic tools developed by Factas to handle issues mentioned so far are outlined
in Section 3.3. In Section 3.2 to come, we describe in more detail which problems are considered and
which applications are targeted.

We also began to investigate inverse scattering problems of plane waves by obstacles (playing here the
role of a source term), with partners at LEAT. Such problems are again governed by Maxwell’s equations
and, in the time-harmonic regime, these reduce to Helmholtz equations depending on the frequency of
the plane wave. Such issues have applications to detection and identification of metal objects, and this is
part of LEAT research program, but at this early stage our study has remained academic (see Section 6.4).

3.2 Range of inverse problems

3.2.1 Elliptic partial differential equations (PDE)

Participants:

Paul Asensio, Laurent Baratchart,
basharah Khalid Omer,
Dmitry Ponomarev, Cristóbal Villalobos Guillén, Anass Yousfi.

Sylvain Chevillard, Mu-
Juliette Leblond, Masimba Nemaire,

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 E of the
boundary, is equivalent to recover a holomorphic function in the domain from its boundary values on E .
This is the problem raised on the half-plane in step 1 of Section 3.1. It makes good sense in holomorphic
Hardy spaces where functions are entirely determined by their values on boundary subsets of positive
linear measure, which is the framework for Problem (P ) more precisely described in Section 3.3.1.

(P ) Let 1 ≤ p ≤ ∞, K a sub-arc of T , f ∈ Lp (K ), ψ ∈ Lp (T \ K ) and M > 0; find a function
g ∈ H p such that ∥g − ψ∥Lp (T \K ) ≤ M and g − f is of minimal norm in Lp (K ) under this

by its non-tangential limit on a boundary subset of positive linear measure, but there are non-constant harmonic functions in the
3-D ball, C 1 up to the boundary sphere, yet having vanishing gradient on a subset of positive measure of the sphere. Such a “bad”
subset, however, cannot have interior points on the sphere.

Project FACTAS

constraint.

5

Such issues naturally arise in nondestructive testing of 2-D (or 3-D cylindrical) materials from partial
electrical measurements on the boundary. For instance, the ratio between the tangential and the normal
currents (the so-called Robin coefficient) tells one about corrosion of the material. Thus, solving Prob-
lem (P ) where ψ is chosen to be the response of some uncorroded piece with identical shape yields non
destructive testing of a potentially corroded piece of material, part of which is inaccessible to measure-
ments. This was an initial application of holomorphic extremal problems to non-destructive control [62,
64].

Studying Hardy spaces of conjugate Beltrami equations is another interesting topic. For Sobolev-
smooth coefficients of exponent greater than 2, they were investigated in [5], [39]. The case of the critical
exponent 2 is treated in [35], which apparently provides the first example of well-posed 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 L2-class in general, even if the initial problem deals with coefficients of Lr -class for some r > 2.
Such generalizations are now under study within the team, in collaboration with E. Pozzi (Saint Louis
Univ., Missouri, USA) and E. Russ (Univ. J. Fourier, Grenoble), and fairly deep connections between the
regularity of the conformal parametrization of the domain and the range of exponents p for which the
Dirichlet problem is solvable in Lp were brought to light.

Generalized Hardy classes as above are used in [36] where we address the uniqueness issue in the
classical Robin inverse problem on a Lipschitz domain of Ω ⊂ Rn, n ≥ 2, with uniformly bounded Robin
coefficient, L2 Neumann data and conductivity of Sobolev class W 1,r (Ω), r > n. We show that uniqueness
of the Robin coefficient on a subset of the boundary, given Cauchy data on the complementary part, does
hold in dimension n = 2, thanks to a unique continuation result, but needs not hold in higher dimension.
In higher dimension, this raises an open issue on harmonic gradients, namely whether the positivity
of the Robin coefficient is compatible with identical vanishing of the boundary gradient on a subset of
positive measure.

The 3-D version of step 1 in Section 3.1 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) [1], [43], see Section 6.1. The question is
considerably more difficult than its 2-D counterpart, due mainly to the lack of multiplicative structure
for harmonic gradients. Still, substantial 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 3.1. 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 the project Apics (predecessor of
Factas2) contributed convergence results for certain classes of fields (expressed as Cauchy integrals over
extremal contours for the logarithmic potential [7], [44, 61]). Initial schemes to locate cracks or sources
via rational approximation on planar domains were obtained this way [47, 51, 62]. 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 [9], [48]. 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

2Actually, Factas started as a team from January 2018, and became a project-team (EPI) from July 2019.

6

Inria Annual Report 2023

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 branch-points 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 4.2, 4.3
and 6.1.

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.

3.2.2 Systems, transfer and scattering

Participants:

Laurent Baratchart, Sylvain Chevillard, Adam Cooman, Martine Olivi,
Fabien Seyfert.

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. 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 2 × 2 unitary-valued matrix function on the
frequency line, say the imaginary axis to fix ideas. In the bandwidth around the resonant frequency, a
modal approximation of the Helmholtz equation in the cavities shows that this matrix is approximately
rational, of Mc-Millan degree twice the number of cavities.

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 3.1 goes
as follows:

1. 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 (P ) in Section 3.2.1, while taking into
account prior knowledge on the decay of the response outside the bandwidth, see [12] for details.

2. 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 [30], [12].

3. 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 [63], 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 3.4.2).

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

Project FACTAS

7

of multi-band response synthesis as a generalization of the classical Zolotarev min-max problem for
rational functions [29, 28]. 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. 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 [84], 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 [32]. Dwelling on this, Factas contributed differential
parametrizations (atlases of charts) of lossless matrix functions [30, 81, 73] which are fundamental to
our rational approximation software RARL2 (see Section 3.4.5). 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. Factas
made significant contributions to this subject [6], in particular within the framework of the (defense
funded) ANR Cocoram.

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 resistors, 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 3.1 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 [10]. We now start investigating the
linearized harmonic transfer-function around a periodic cycle, to check for stability under non necessarily
small inputs.

3.3 Approximation

Participants:

Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Martine Olivi,
Masimba Nemaire, Dmitry Ponomarev, Cristóbal Villalobos Guillén.

3.3.1 Best analytic approximation

In dimension 2, the prototypical problem to be solved in step 1 of Section 3.1 may be described as: given
a domain D ⊂ R2, to recover a holomorphic function from its values on a subset K of the boundary of
D. For the discussion it is convenient to normalize D, which can be done by conformal mapping. So, in
the simply connected case, we fix D to be the unit disk with boundary unit circle T . We denote by H p
the Hardy space of exponent p, which is the closure of polynomials in Lp (T )-norm if 1 ≤ p < ∞ and the
space of bounded holomorphic functions in D if p = ∞. Functions in H p have well-defined boundary
values in Lp (T ), which makes it possible to speak of (traces of) analytic functions on the boundary.

To find an analytic function g in D matching some measured values f approximately on a sub-arc K

of T , we formulate the constrained best approximation problem (P ) as in Section 3.2.1.

There, ψ is a reference behavior capturing a priori assumptions on the behavior of the model off K ,
while M is some admissible deviation thereof. The value of p reflects the type of stability which is sought
and how much one wants to smooth out the data. The choice of Lp classes is suited to handle pointwise
measurements.

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To fix terminology, we refer to (P ) as a bounded extremal problem. As shown in [11], [46, 49, 56], the
solution to this convex infinite-dimensional optimization problem can be obtained when p ̸= 1 upon
iterating with respect to a Lagrange parameter the solution to spectral equations for appropriate Hankel
and Toeplitz operators. These spectral equations involve the solution to the special case K = T of (P ),
which is a standard extremal problem [71]:

(P0) Let 1 ≤ p ≤ ∞ and ϕ ∈ Lp (T ); find a function g ∈ H p such that g − ϕ is of minimal norm
in Lp (T ).

In the case p = 1, partial results are known but computational issues remain open.

Various modifications of (P ) can be tailored to meet specific needs. For instance when dealing with
lossless transfer functions, one may want to express the constraint on T \ K in a pointwise manner:
∞
|g − ψ| ≤ M a.e. on T \ K , see [50]. In this form, the problem comes close to (but still is different from) H
frequency optimization used in control [74, 83]. One can also impose bounds on the real or imaginary
part of g − ψ on T \ K , which is useful when considering Dirichlet-Neumann problems.

In view of our current research on stability of active devices via analyticity of the harmonic transfer
function, on inverse magnetization issues, and on inverse scattering via identification of the frequency
response, bounded extremal problems for analytic functions are receiving renewed interest by the team.
In such issues, a function on an interval of the real line (or an arc of the circle) must be approximated by
the trace of a function holomorphic in the half-plane (or the disk), that meets suitable size constraints.
The analog of Problem (P ) on an annulus, K being now the outer boundary, can be seen as a means
to regularize a classical inverse problem occurring in nondestructive control, namely to recover a har-
monic function on the inner boundary from Dirichlet-Neumann data on the outer boundary (see Sec-
tions 3.2.1, 4.2, 6.1.2). It may serve as a tool to approach Bernoulli type problems, where we are given
data on the outer boundary and we 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 3.2.1 for more general equations than the Laplacian, namely
isotropic conductivity equations of the form div(σ∇u) = 0 where σ is no longer constant (i.e., varies in
the space). Then, the Hardy spaces in Problem (P ) are those of a so-called conjugate Beltrami equation:
¯∂ f = ν∂ f [75], with ν = (1 − σ)/(1 + σ), which are studied for 1 < p < ∞ in [5], [35], [39] and [65]. Expan-
sions of solutions needed to constructively handle such issues in the specific case of linear fractional
conductivities have been expounded in [68].

Though originally considered in dimension 2, Problem (P ) carries over naturally to higher dimensions
where analytic functions get replaced by gradients of harmonic functions. Namely, given some open set
Ω ⊂ Rn and some Rn-valued vector field V on an open subset O of the boundary of Ω, we seek a harmonic
function in Ω whose gradient is close to V on O.

When Ω is a ball or a half-space, a substitute for holomorphic Hardy spaces is provided by the Stein-
Weiss Hardy spaces of harmonic gradients [86]. Conformal maps are no longer available when n > 2, so
that Ω can no longer be normalized. More general geometries than spheres and half-spaces have not
been much studied so far.

On the ball, the analog of Problem (P ) is

(P1) Let 1 ≤ p ≤ ∞ and B ⊂ Rn the unit ball. Fix O an open subset of the unit sphere S ⊂ Rn.
Let further V ∈ Lp (O) and W ∈ Lp (S \ O) be Rn-valued vector fields. Given M > 0, find a
harmonic gradient G ∈ H p (B ) such that ∥G − W ∥Lp (S\O) ≤ M and G − V is of minimal norm
in Lp (O) under this constraint.

When p = 2, Problem (P1) was solved in [1] as well as its analog on a shell, when the tangent component
of V is a gradient (when O is Lipschitz the general case follows easily from this). The solution extends
the work in [46] to the 3-D case, using a generalization of Toeplitz operators. The case of the shell was
motivated by applications to the processing of EEG data. An important ingredient is a refinement of the
Hodge decomposition, that we call the Hardy-Hodge decomposition, allowing us to express a Rn-valued
vector field in Lp (S), 1 < p < ∞, as the sum of a vector field in H p (B ), a vector field in H p (Rn \ B ), and a
tangential divergence free vector field on S; the space of such divergence-free fields is denoted by D(S).
If p = 1 or p = ∞, Lp must be replaced by the real Hardy space or the space of functions with bounded
mean oscillation. More generally this decomposition, which is valid on any sufficiently smooth surface

Project FACTAS

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(see Section 6.1), seems to play a fundamental role in inverse potential problems. In fact, it was first
introduced formally on the plane to describe silent magnetizations supported in R2 (i.e., those generating
no field in the upper half space) [43].

Just like solving problem (P ) appeals to the solution of problem (P0), our ability to solve problem (P1)

will depend on the possibility to tackle the special case where O = S:

(P2) Let 1 ≤ p ≤ ∞ and V ∈ Lp (S) be a Rn-valued vector field. Find a harmonic gradient
G ∈ H p (B ) such that ∥G − V ∥Lp (S) is minimum.

Problem (P2) is simple when p = 2 by virtue of the Hardy-Hodge decomposition together with ortho-
gonality of H 2(B ) and H 2(Rn \ B ), which is the reason why we were able to solve (P1) in this case. Other
values of p cannot be treated as easily and are still under investigation, especially the case p = ∞ which is
of particular interest and presents itself as a 3-D analog to the Nehari problem [82].

Companion to problem (P2) is problem (P3) below.

(P3) Let 1 ≤ p ≤ ∞ and V ∈ Lp (S) be a Rn-valued vector field. Find G ∈ H p (B ) and D ∈ D(S)
such that ∥G + D − V ∥Lp (S) is minimum.

Note that (P2) and (P3) are identical in 2-D, since no non-constant tangential divergence-free vector field
exists on T . It is no longer so in higher dimension, where both (P2) and (P3) arise in connection with
inverse potential problems in divergence form, like source recovery in electro/magneto encephalography
and paleomagnetism, see Sections 3.2.1 and 4.2.

3.3.2 Best meromorphic and rational approximation

The techniques set forth in this section are used to solve step 2 in Section 3.2 and they are instrumental to
approach inverse boundary value problems for the Poisson equation ∆u = µ, where µ is some (unknown)
measure.

Scalar meromorphic and rational approximation We put RN for the set of rational functions with at
most N poles in D. By definition, meromorphic functions in Lp (T ) are (traces of) functions in H p + RN .

A natural generalization of problem (P0) is:

(PN ) Let 1 ≤ p ≤ ∞, N ≥ 0 an integer, and f ∈ Lp (T ); find a function g N ∈ H p + RN such that
g N − f is of minimal norm in Lp (T ).

Only for p = ∞ and f continuous is it known how to solve (PN ) in semi-closed form. The unique
solution is given by AAK theory (named after Adamjan, Arov and Krein), which connects the spectral
decomposition of Hankel operators with best approximation [82].

The case where p = 2 is of special importance for it reduces to rational approximation. Indeed, if we
+ + rN where
from RN in L2(T ). Moreover, rN has no pole outside D, hence it is a stable
. However, in contrast to the case where p = ∞, this best approximant may

− ∈ H 2(C \ D), then g N = f

write the Hardy decomposition f = f
rN is a best approximant to f
−
rational approximant to f
not be unique.

+ ∈ H 2 and f

where f

+ + f

−

−

The Miaou project (predecessor of Apics) already designed a dedicated steepest-descent algorithm
for the case p = 2 whose convergence to a local minimum is guaranteed; the algorithm has evolved over
years and still now, it seems to be the only procedure meeting this property. This gradient algorithm
proceeds recursively with respect to N on a compactification of the parameter space [37]. Although it
has proved to be effective in all applications carried out so far, it is still unknown whether the absolute
minimum can always be obtained by choosing initial conditions corresponding to critical points of lower
degree (as can be done with the RARL2 software, Section 3.4.5).

In order to establish global convergence results, the team 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 [53, 55]. 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) [57] and more generally Cauchy
integrals over hyperbolic geodesic arcs [58]. These are the only results of this kind. Research on this topic

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remained dormant for a while by reasons of opportunity, but revisiting the work [33] in higher dimension
is a worthy and timely endeavor today. Meanwhile, an analog to AAK theory was carried out for 2 ≤ p < ∞
in [56]. Although not as effective computationally, it was recently used to derive lower bounds [3]. When
1 ≤ p < 2, problem (PN ) is still quite open.

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 [60, 59], 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 3.4.3 and 6.1).

In higher dimensions, the analog of Problem (PN ) is best approximation of a vector field by gradients
of discrete potentials generated by N point masses. This basic issue is by no means fully understood,
and it is an exciting field of research. It is connected with certain generalizations of Toeplitz or Hankel
operators, and with constructive approaches to so-called weak factorizations for real Hardy functions [67].
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 D while controlling both
the degree and the extremal points (peak points for the modulus) of the interpolant. Results obtained
by the team in this direction generalize a variant of contractive interpolation with degree constraint as
studied in [72]. We mention that contractive interpolation with nodes approaching the boundary has
been a subsidiary research topic by the team in the past, which plays an interesting role in the spectral
representation of certain non-stationary stochastic processes [42, 45].

Matrix-valued rational approximation 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 F ∈ (H 2)m×l and n an integer; find a
rational matrix of size m × l without poles in the unit disk and of McMillan degree at most n which is
nearest possible to F in (H 2)m×l . Here the L2 norm of a matrix is the square root of the sum of the squares
of the norms of its entries.

The scalar approximation algorithm derived in [37] and mentioned in Section 3.3.2 generalizes to the
matrix-valued situation [70]. 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 n. Indeed,
inner matrices play the role of denominators in fractional representations of transfer matrices (using
the so-called Douglas-Shapiro-Shields factorization). The set of inner matrices of given degree is a
smooth manifold that allows one to use differential tools as in the scalar case. In practice, one has to
produce an atlas of charts (local parametrizations) and to handle changes of charts in the course of the
algorithm. Such parametrization can be obtained using interpolation theory and Schur-type algorithms,
the parameters of which are vectors or matrices ([30, 73, 81]). Some of these parametrizations are also
interesting to compute realizations and achieve filter synthesis ([73, 81]). The rational approximation
software “RARL2” developed by the team is described in Section 3.4.5.

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 n or small perturbations thereof (the consistency problem) was solved in [52].
Matrix-valued Markov functions are the only known example beyond this one [34].

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.

Project FACTAS

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3.3.3 Behavior of poles of meromorphic approximants

Participants:

Paul Asensio, Laurent Baratchart.

We refer here to the behavior of poles of best meromorphic approximants, in the Lp -sense on a closed
curve, to functions f defined as Cauchy integrals of complex measures whose support lies inside the
curve. Normalizing the contour to be the unit circle T , we are back to Problem (PN ) in Section 3.3.2;
invariance of the latter under conformal mapping was established in [51]. Research so far has focused on
functions whose singular set inside the contour is polar, meaning that the function can be continued
analytically (possibly in a multiple-valued manner) except over a set of logarithmic capacity zero.

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 3.2.1, 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 4.2). 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 ([7], [51, 44]), 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 [47]. Moreover, the asymptotic density of the poles turns out
to be the Green equilibrium distribution on this cut in D, therefore it charges the singular points if one is
able to approximate in sufficiently high degree (this is where the method could fail, because high-order
approximation requires rather precise data).

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 6.1. The technique is implemented
in the software FindSources3D, see Section 3.4.3.

Another, extremely classical technique to approximate –more accurately: extrapolate– a function
given pointwise values is to compute a rational interpolant of minimal degree to match the values. This
method, know as Padé (or multipoint Padé) approximation has been intensively studied for decades
[31] but fails to produce pointwise convergence, even if the data are analytic. The best it can give in
general is convergence in capacity, at least to functions whose singular set has capacity zero, and this
does not prevent spurious poles of the approximant from wandering about the domain of analyticity of
the approximated function [80]. This phenomenon is standard in numerical practice, and gives rise in
physics and engineering circles to a distinction between “mathematical” and “physical” poles; note that
this distinction ignores the possibility that the function has other singularities than poles (for example
branch-points or essential singularities). A modification of the multipoint Padé technique, where the
degree is kept much smaller than the number of data and only approximate interpolation is performed
in the least-square sense, has become especially popular over the last decade under the name vector
fitting; this is in trend with the soaring development of computational methods in the frequency domain.
Although their behavior looks similar to the one of multipoint Padé approximants from a numerical
viewpoint, there seems to be no convergence result available for such approximate interpolants so far.
Motivated by the behavior of numerical schemes developed at LEAT to recover resonance frequencies
of conductors under electromagnetic inverse scattering (see section 4.5), we started investigating the
behavior of such least-square rational approximants to functions with polar singular set, see section 6.4.

3.4 Software tools of the team

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

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developed by the team or serve as tools to help further research by team members. We present briefly the
most important of them.

3.4.1 pisa

Name: pisa

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 Contributions: First version

URL: https://project.inria.fr/pisa

Publications: hal-01381731, hal-01098616

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

Contact: Fabien Seyfert

3.4.2 DEDALE-HF

Keyword: Microwave filter

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 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 coupling
matrix synthesis problem 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 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.

URL: http://www-sop.inria.fr/apics/Dedale/

Contact: Fabien Seyfert

Participant: Fabien Seyfert

Project FACTAS

3.4.3 FindSources3D

13

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

Scientific Description: Though synthetic data could be static, actual signal recordings are dynamical.
The time dependency is either neglected and the data processed instant by instant, or separated
from the space behavior using a singular value decomposition (SVD). This preliminary step al-
lows to estimate the number of independent activities (uncorrelated sources) and to select the
corresponding quantity of principal static components. After a first data transmission (“cortical
mapping”) step of the static data, using the harmonicity property of the potential in the outermost
layers (solving BEP problems on spherical harmonics bases), FS3D makes use of best rational
approximation on families of 2-D planar cross-sections and of the software RARL2 in order to locate
singularities and to determine the expected quantity of sources. From those planar singularities,
the 3-D sources are finally estimated, together with their moment, in a last clustering step. Through
this process, FS3D is able to recover time correlated sources, which is an important advantage.
When simultaneously available, EEG and MEG data can now be processed together, and this also
improves the recovery performance. In case of dynamical data, a recent additional step is to find the
linear combination of the preliminary selected static components (change of basis) that produces
source estimates which minimize the error w.r.t. data, an original criterion, which allows to improve
the recovery quality.

Functional Description: FindSources3D (FS3D) is a software program written in Matlab dedicated to
the resolution of inverse source problems in brain imaging, electroencephalography (EEG) and
magnetoencephalography (MEG). From data consisting in pointwise measurements of the electrical
potential taken by electrodes on the scalp (EEG), or of a component of the magnetic field taken on
a helmet (MEG), FS3D estimates pointwise dipolar current sources within the brain in a spherical
layered model. Each layer (brain, skull, scalp) is assumed to have a constant conductivity.

URL: http://www-sop.inria.fr/apics/FindSources3D/en/index.html

Contact: Juliette Leblond

Participants: Jean-Paul Marmorat, Juliette Leblond, Maureen Clerc, Nicolas Schnitzler, Théodore Papado-

poulo

3.4.4 PRESTO-HF

Keywords: CAO, Telecommunications, Microwave filter

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 mi-
crowave filters. In order to allow the industrial transfer of our methods, a Matlab-based toolbox has

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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.

URL: https://project.inria.fr/presto-hf/

Contact: Fabien Seyfert

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

3.4.5 RARL2

Name: Réalisation interne et Approximation Rationnelle L2

Keyword: Approximation

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 Find-
Sources3D. 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.

URL: http://www-sop.inria.fr/apics/RARL2/rarl2.html

Contact: Martine Olivi

Participants: Jean-Paul Marmorat, Martine Olivi

Project FACTAS

3.4.6 Sollya

15

Keywords: Computer algebra system (CAS), Supremum norm, Proof synthesis, Code generator, Remez

algorithm, Curve plotting, Numerical algorithm

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 approx-
imation 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.

URL: https://sollya.org/

Contact: Sylvain Chevillard

Participants: Christoph Lauter, Jérôme Benoit, Marc Mezzarobba, Mioara Joldes, Nicolas Jourdan,

Sylvain Chevillard

Partners: CNRS, UPMC, ENS Lyon, LIP6, UCBL Lyon 1, Loria

4 Application domains

4.1 Introduction

Application domains are naturally linked to the problems described in Sections 3.2.1 and 3.2.2, under a
common umbrella of function-theoretic techniques as described in Section 3.3.

4.2 Inverse source problems in EEG

Participants:

Paul Asensio, Laurent Baratchart, Juliette Leblond, Jean-Paul Marm-
orat, Masimba Nemaire.

Solving over-determined Cauchy problems for the Laplace equation on a spherical layer (in 3-D) in
order to extrapolate incomplete data (see Section 3.2.1) is a necessary ingredient of the team’s approach
to inverse source problems, in particular for applications to EEG, see [9]. 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 [7], [48]. 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 3.3.3, 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.

In this connection, a dedicated software FindSources3D (FS3D, see Section 3.4.3) has been developed,
in collaboration with the Inria team Athena (now Cronos) and the CMA - Mines ParisTech. Its Matlab
version now incorporates the treatment of MEG data, the aim being to handle simultaneous EEG–MEG
recordings available from our partners at INS, hospital la Timone, Marseille. Indeed, it is now possible to
use simultaneously EEG and MEG measurement devices, in order to measure both the electrical potential

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Inria Annual Report 2023

and a component of the magnetic field (its normal component on the MEG helmet, that can be assumed
to be spherical). Solving the inverse source problem from joint EEG and MEG data actually improves
accuracy of the source estimation. Note that FS3D takes as inputs actual EEG and MEG measurements,
like time signals, and performs a suitable singular value decomposition in order to separate independent
sources.

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 within rocks, one should consider poles of order five.
Though numerically observed in [9], 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).

Furthermore, other approaches are being studied for EEG and other brain imaging modalities. They
consist in regularizing the inverse source problem by a total variation constraint on the source term (a
measure), added to the quadratic data approximation criterion (see Section 6.1.2) and presently focus on
surface-distributed models. This is similar to the path that is taken for inverse magnetization problems
(see Sections 4.3 and 6.1.1 ).

4.3 Inverse magnetization problems

Participants:

Laurent Baratchart, Sylvain Chevillard, Mubasharah Khalid Omer, Ju-
liette Leblond, Masimba Nemaire, Dmitry Ponomarev, Cristóbal Vil-
lalobos Guillén, Anass Yousfi.

Generally speaking, inverse potential problems, similar to the one appearing in Section 4.2, 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 that reached the end of its term in 2018.

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 aimed at tackling them, experimenting with the SQUID microscope set up in the Paleomag-
netism 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 4.2 and inverse mag-
netization problems described presently amount to recover the (3-D valued) quantity m (primary current
density in case of the brain or magnetization in case of a thin slab of rock) from measurements of the
potential:

V (x) =

(cid:90)

div m(x

) d x

′

′

Ω

|x − x′|

,

(1)

outside the volume Ω of the object. Depending on the geometry of models, the magnetization distribution

Project FACTAS

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m may lie in a volume or spread out on a surface. This results in quite different identifiability properties,
see [43] and Section 6.1.1, but the two situations share a substantial mathematical common core.

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 (Geomathematics and
Geoinformatics Group, Technische Universität Bergakademie Freiberg, Germany), 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 6.1.1.

4.4 Structure and stability of active devices

Participants:

Laurent Baratchart, Sylvain Chevillard, Jean-Paul Marmorat, Mar-
tine Olivi, Fabien Seyfert.

Through contacts with CNES (Toulouse) and UPV (Bilbao), the team got involved in the design of
amplifiers which, unlike filters, are active devices. A prominent issue here is stability. Twenty years ago,
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 [87]. 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). Initial applications by the team have been concerned with the first
type of stability, and emphasis was put on defining and extracting the “unstable part” of the response,
see Section 6.2. Since then, the stability check for limit cycles has made important theoretical advances.
Specifically, the exponential stability of the high frequency limit of a circuit was established last year in
[4], implying that there are at most finitely many unstable poles and no other unstable singularity for
the monodromy operator around the cycle. Furthermore, the links between the monodromy operator
and the (operator-valued) harmonic transfer function of the linearized system along the trajectory were
brought to light in [23]. Numerical algorithms are now under investigation, while important pending
issues involve: (i) whether poles of the harmonic transfer function must be poles of each entry thereof, at
least generically and (ii) describing the stable spectrum of the harmonic transfer function of the linearized
system, in particular understand its continuous and essential part.

4.5 Identification of resonating frequencies of compact metallic objects in electro-

magnetic inverse scattering

Participants:

Paul Asensio, Laurent Baratchart, Juliette Leblond, Martine Olivi,
Dmitry Ponomarev, Fabien Seyfert.

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, magnet-
ically coupled by irises (orthogonal double slits). The electromagnetic wave that traverses the cavities

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Inria Annual Report 2023

Figure 1: Sphere illuminated by an electromagnetic plane wave - measurement of the scattered wave

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.
Study of resonances also led us to another inverse problem. There, the singularity expansion method
featuring the above-mentioned discrete set of wave vectors is instrumental to the object detection. In
this respect, we started an academic collaboration with LEAT (Univ. Côte d’Azur, France, J.-Y. Dauvignac,
N. Fortino, Y. Zaki) on the topic of inverse scattering using frequency dependent measurements. As
opposed to classical electromagnetic imaging where several spatially located sensors are used to identify
the shape of an object by means of scattering data at a single frequency, a discrimination process between
different metallic objects is here being sought for by means of a single, or a reduced number of sensors
that operate on a whole frequency band. For short the spatial multiplicity and complexity of antenna
sensors is here traded against a simpler architecture performing a frequency sweep.

The setting is shown on Figure 1. The total field Et (r, θ, φ) is the sum of the incident field Ei (here a
plane wave) and scattered field Es , that is at every point in space we have Et = Ei + Es . A harmonic time
dependency (e j ωt , where j is the imaginary unit: j 2 = −1 ) is supposed for the incident wave, so that by
linearity of Maxwell equations and after a transient state, the following holds:

Es (ro, θo, φo) = H (s = j ω, θo, φo)Ei (re , θe , φe ).

The subscripts o and e stand here for «observation point» and «emission point»: the scattered field at
the observation point is therefore related to the emitted planar wave field at the emission point via the
transfer function H (s = j ω, θo, φo). The emission point is here supposed fixed, so the dependency in e is
omitted in H . Under regularity conditions on the scatterer’s boundary the function H can be shown to
admit an analytic continuation into the complex left half plane for the s variable, away from a discrete set
(with a possible accumulation point at infinity) where it admits poles. Thus, H is a meromorphic function
in the variable s. Its poles are called the resonating frequencies of the scattering object. Recovering these
resonating frequencies from frequency scattering measurement, that is measurements of H at particular
s = j ω′

i s is the primary objective of this project.

In order to gain some insight we started a full study of the particular case when the scatterer is a
spherical PEC (Perfectly Electric Conductor). In this case Maxwell equations can be solved «explicitly» by
means of expansions in series of vectorial spherical harmonics. We showed in particular that in this case
H admits following simple structure:

H (ω, θo, φo) = R(s, θo, φo)e

−τ1(θo ,φo )s +C (θo, φo)e

−τ2(θo ,φo )s ,

where R is a meromorphic functions with poles at zeros of the spherical Hankel functions and their
derivatives and C is independent of the frequency. Identification procedures, surprisingly close to the
ones we developed in connection with amplifier stability analysis, are currently being studied to gain
information about the resonating frequencies by means of a rational approximation of the function R

Project FACTAS

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once it has been de-embedded. A preliminary study resulted in a common publication with the LEAT
team [89].

In order to perform the rational approximation of the function H , its behavior outside the range of
measured frequencies, specifically at high frequencies, is being studied for the particular case when the
scatterer is a spherical PEC (Perfectly Electric Conductor). In this case, H can be written as:

H (ω, θ0, φ0) = HO(ω, θ0, φ0) + HC (ω, θ0, φ0)

where HO and HC are respectively the optic and creeping wave parts. Their high-frequency behaviors are
given by:

HO(ω, θ0, φ0) ∼

ω→∞

HC (ω, θ0, φ0) ∼

ω→∞

∞
(cid:88)

n=0
ω 4

An(θ0, φ0)
(i ω)n

ei kω(θ0,φ0),

3 B0ei

π
3 exp

(cid:179)
i B1ω − e

−i

π

6 B2ω1/3(cid:180)

.

where the coefficients B0 ∈ R, B1 ∈ R, B2 ∈ R+
and An ∈ R are identified when the observation point
(ro, θo, φo) and the emission point (re , θe , φe ) coincide. The above asymptotics of HO is called the
Luneberg-Kline expansion; its first terms were analytically computed (solving eikonal and transport
equations).

Numerical simulations showed that even though the creeping wave part is negligible in front of the
optic part at high frequencies, it needs to be taken into account around the band of measured frequencies
for the rational approximation.

Furthermore, the physical interpretation of these two terms leads to consider that the creeping wave
part should carry more information about the scatterer and we want to investigate the conjecture that
the poles of H are those of HC hence that HO is analytic.

We plan in the future to investigate a generalization of this form for other PEC scatterers. See

Section 6.4.

5 Social and environmental responsibility

Participants:

Sylvain Chevillard, Martine Olivi.

• M. Olivi is a member of the CLDD (Commission Locale de Développement Durable) and in charge

of coordination. She is a member of the GDS EcoInfo (CNRS) and of Labos1point5.

• M. Olivi co-animated, with S. Quinton (Inria, SPADES), a workshop SEnS (Science, Environment,

Society) at Toulon university.

• S. Chevillard and M. Olivi organized (together with L. Deneire, S. Icart and G. Urvoy-Keller) a one-
day training program for PhD students (“Formation doctorale”) on the topic “Science, environment
and society”.

6 New results

6.1 Inverse problems for Poisson-Laplace and Helmholtz equations

Participants:

Sylvain Chevillard, Mu-
Paul Asensio, Laurent Baratchart,
basharah Khalid Omer,
Jean-Paul Marmorat,
Masimba Nemaire, Dmitry Ponomarev, Cristóbal Villalobos Guillén,
Anass Yousfi.

Juliette Leblond,

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Inria Annual Report 2023

6.1.1 Inverse magnetization issues for planar and volumetric samples

The goal is to invert magnetizations carried by a rock sample from measurements of the magnetic
field nearby. A typical application is when the sample is shaped into thin slabs, with measurements
taken by a superconducting quantum interference device (SQUID). Figure 2 sketches the corresponding
experimental set up, brought up to our knowledge by collaborators from the Earth and Planetary Sciences
Laboratory at MIT, in the case when the sample is modeled as a parallelepiped.

Figure 2: Schematic view of the SQUID experimental setup. The sample lies on a horizontal plane at
height 0 and its support is included in a parallelepiped (in red). The vertical component B3 of the field
produced by the sample is measured at points of a horizontal region, say a square, at height z (in gray).

When the measurement area is a disk (or even a cylinder, if its thickness is not negligible), the issue of
asymptotic estimates has undergone new developments. As part of A. Yousfi’s thesis work, we derived a
much more straightforward way of establishing the asymptotic estimates of the moments of B3, a key
step to derive asymptotic estimates of the net moment of a sample. This work is still ongoing and a report
on the subject is being written. Meanwhile, a further simplification was achieved in the computations
that we made in 2022 and the previously uploaded preprint [27] was updated accordingly. The work is
now submitted to a journal.

Instability of high-order asymptotic formulas for the net moment and the lack of the measurements
naturally give rise to two aspects: noise filtering and field prolongation. One way to attempt both issues
at once is to use the so-called spectral extrapolation. Such an approach consists in finding a suitable
choice of basis functions which are adapted to problem structure and geometry, but at the same time
are naturally defined outside of the original region. Using an eigenbasis of the suitable integral operator
related to the forward problem, we attempt to construct a field extrapolant which would extend the
available measurements in a way respecting the structure of the problem (such as harmonicity, support
of the source, qualitative behavior at infinity). Restricting the number of terms in the expansion over this
basis also provides a regularization strategy achieving a desired degree of the field denoising.

The work on the field extrapolation initiated at the end of 2022 has been advanced further. In particu-

lar, there were several directions taken to develop a different methodology.
First, after realizing that the previously developed spectral extrapolation procedure based on the forward
operator featuring only one kernel function (the vertical derivative of the Poisson kernel) was somewhat
naive, we have demonstrated that the extrapolation results could be improved by introducing several
local extrapolation steps. Namely, by a procedure reminiscent of the chain method for local analytic
continuation, we gradually increase the extrapolation area up to some intermediate size after which the
previous naive procedure can be applied with a better outcome. The local stepping is computationally
expensive and the details of its particular implementation should be investigated more thoroughly for its
further improvement.
However, we have delved into developing another approach in a somewhat similar fashion. Namely, it
was observed that the spectral extrapolation approach could be made exact if the basis functions are

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chosen as vector-eigenfunctions of a 2 × 2 matrix integral operator. This would invoke a preliminary
solution of a canonical auxiliary problem whose ill-posedness reflects the same phenomenon of the
original problem of the field extrapolation. One would also need to employ this solution to preprocess
the initial measurements. While computationally heavy and difficult to implement numerically, this
approach was verified and showed much better result compared to its more naive counterpart.
A third approach for the field extrapolation is much more in the spirit of previous works of the team.
Here, we use the techniques of constrained approximation in Hilbert spaces. Namely, given a tolerance
level for measurement errors, we search for an extrapolant which agrees in a best possible manner with
the original field data respecting the harmonicity of the field. Currently, we have done it only for the
scalar potential simulated data (a toy problem for the internship work of M. Khalid Omer) rather than the
vertical field component data (that would correspond to the physical SQUID setup at EAPS, MIT). The
extension of this methodology to the latter case must be straightforward.
Finally, we have also started exploring a possibility of obtaining an extrapolant directly from solution
of some integral equation. Even though, we have succeeded obtaining such an equation, no construct-
ive results are yet available here. Any progress in this direction would allow dealing with volumetric
magnetization sources.

Besides, we pursued a joint research effort with E. A. Lima and B. P. Weiss (EAPS dept. MIT, USA),
C. S. Borlina (EAPS dept. Johns Hopkins Univ. USA) and D. Hardin (Vanderbilt Univ. USA), investigating
the question whether is it possible to estimate the moment of an unknown magnetization by the one
of a magnetization producing a field close enough to the observations? In other words, is full-inversion
a demonstrably good (though admittedly costly) method to estimate the moment? We showed in 2022
that the answer is essentially no. More precisely, the field-to-moment map (that exists because silent
magnetizations have zero moment) is discontinuous, even when magnetizations are endowed with a
rather weak topology (like the weak-star topology on distributions of fixed order and given support),
even if models for the magnetization are restricted to a very small class. In 2023, we started studying
the speed of approximation of a given field by the field of a multipole when the degree of the latter goes
large. This speed must be compared with the norm of the operator mapping the field (or the observed
portion thereof) to the moment within the model class of multipoles, in order to decide if full inversion
allows one to efficiently estimate the moment this way. This line of investigation is still in progress, and
an article has been published that describes the method [15].

In collaboration with D. Hardin at Vanderbilt University (Nashville, USA), We also made theoret-
ical progress describing the outcome of regularization techniques that penalize the total variation of
an unknown measure modeling the magnetization, for inverse magnetization problems on compact
samples of zero 3-D Lebesgue measure with connected complement. Namely, we proved that natural
discretizations of the regularized criterion have a unique minimizer in this case, which allows us to
establish the consistency of such schemes. Also when the sample is volumic (like a ball), we showed
that these schemes select a solution on the boundary of the sample (which is the balayage of the “true”
magnetization, whatever the latter is).

In another connection, the development of QDM sensors has heightened the need for 3D-inversion
of field maps. Indeed, while the QDM can be put very close to the surface of the sample which makes for
improved resolution, the thickness of the sample becomes appreciable compared to the distance to the
sensor and can no longer be neglected. In his thesis [20], Masimba Nemaire has taken up the study of
silent magnetization of Lp class on rather general bounded volumes for 1 < p < ∞, see Section 6.3 and
the preprint [26].

6.1.2 Inverse problems in medical imaging

Our software FindSources3D (FS3D, see Sections 3.4.3, 4.2) dedicated to pointwise source estimation in
EEG–MEG is now used by some of our collaborators. Together with M. Darbas (LAGA, Univ. Sorbonne
Paris Nord) and P.-H. Tournier (labo. JLL, Sorbonne Univ), we considered the EEG inverse problem with a
variable conductivity in the intermediate skull layer, in order to model hard / spongy bones, especially
for neonates. Coupled with FS3D, the related transmission step is performed using a mixed variational
regularization and finite elements (FreeFem++) on tetrahedral meshes, and furnishes very promising
results [14].

We studied the uniqueness of a critical point of the quadratic criterion in the electroencephalography

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Inria Annual Report 2023

(EEG) problem for a single dipole situation (PhD of P. Asensio [19], see also [22]). This issue is essential for
the use of descent algorithms. This leads to the study of the following criterion:

J∂Ω : Ω × R3 −→ R+

(cid:90)

−

∂Ω

(cid:182)2

dσ(z),

(x, p) (cid:55)−→

p0 · (x0 − z)
|x0 − z|3

(cid:181) p · (x − z)
|x − z|3
where x and x0 (actual dipole position) are in a smooth open set Ω included in R3 and p, p0 (actual dipole
moment) belong to R3. In the half-space case where Ω = {(x1, x2, x3) ∈ R3, x3 < 0}, we have proven that
there is a unique critical point; i.e., a unique location and moment pair (x, p) where the gradient of J∂Ω
vanishes, which is but (x0, p0) (the absolute minimum). For the spherical case where Ω = B is the ball of
center 0 and radius 1, we only obtained partial results so far, and managed to prove the uniqueness of the
critical point in the specific case where x0 = 0 is located at the center of Ω. Observe that the problem is
more difficult because no translation invariance holds anymore. We also studied the uniqueness of the
critical point of the quadratic criterion in the charge localization problem for a single charge situation.
This leads to the study of the following criterion:

j∂Ω : Ω × R3 −→ R+

x − z
|x − z|3
where x and x0 (actual charge position) are in a smooth open set Ω included in R3 and q, q0 (actual
charge quantity) belong to R. In both cases (the half-space case and the spherical one), we have proven
that there is a unique critical point.

x0 − z
|x0 − z|3

(x, q) (cid:55)−→

dσ(z),

− q0

∂Ω

q

2

(cid:90)

(cid:175)
(cid:175)
(cid:175)
(cid:175)

(cid:175)
(cid:175)
(cid:175)
(cid:175)

We considered a different class of source models, not necessarily dipolar, and related estimation
algorithms. Such models may be supported on the surface of the cortex or in the volume of the encephalon.
We represent sources by vector-valued measures, and in order to favor sparsity in this infinite-dimensional
setting we use a TV (i.e., total variation) regularization term as in Section 6.1.1. The approach follows that
of [8] and is implemented through two different algorithms, whose convergence properties were studied
in the PhD thesis of M. Nemaire [20]. Tests on synthetic data provided good quality results, though they
are quite numerically costly to obtain.

Figure 3: Schematic view of a 3 layered head model. Deep electrodes with sensors along them (red) in Ω0,
current source term µ distributed on S ⊂ Ω0 with normally oriented dipoles.

Progresses were made on the inverse problem of “Stereo” EEG (SEEG), where the potential is measured
by deep electrodes and sensors within the brain as in the scheme of Figure 3. Assuming that the current
source term µ is a R3-valued vector field (or a measure, or a dipolar distribution) supported on a surface
S and normally oriented to S, makes it possible to describe the potential φ(µ) generated by µ as a double
layer potential:

φ(µ)(x) = −

1
4 π

(cid:90)

µ ·

S

x − y
|x − y|3 d y , x ̸∈ S .

The associated forward and inverse problems were solved for both an infinite medium conductor and a
more realistic single model of the brain Ω0 (as first step towards nested domains for the head). Concerning
the latter, it is preliminary assumed that S ⊂ ∂Ω0, whence µ consists in patches on the cortex surface.

Project FACTAS

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The numerical implementation was done by approximating the density µ of the double layer potential
with linear shape functions on triangulations of the involved surface S. For the discretized version, µ is
then approximated on every triangle by a piecewise linear function, and the associated moment is taken
to be located at the center of the triangle, perpendicular to the triangle. Note that explicit expressions of
the double layer potential are also available at points located on S (that supports µ).

The inverse problem for SEEG is ill-posed and a Tychonov regularization is used in order to solve the
problem: find a distribution µ defined on the prescribed surface S such that ∥φ(µ) − φd ∥
+ λ R(µ)
is minimized, where φd is the given data in Ω0, φ(µ) is the “forward model” (measurements of the
double layer potential), R(µ) is an appropriate regularization term that results in µ having some desired
properties (being bounded in L2 norm or T V or L1 on S), and λ a (Lagrange) parameter. Because φd is
only furnished at sensor positions, the L2 term in the above criterion becomes an l 2 term.

L2(Ω0)

We now consider S ⊂ Ω0 within the brain, at the gray / white matters interface and we handle outer
layers Ω1 and Ω2 for the skull and the scalp, as illustrated in Figure 3. Such configurations involve a data
transmission / “cortical mapping” step relating the potential and normal current values at the boundaries
Γi , that the present double layer formulation allows us to solve with the source localization issue. It only
requires to solve the transmission problem for the surface electrical potentials hence this reduces the
complexity of the problem. Numerically, this is a feasible implementation of the problem because the
expressions for double layer potentials do not suffer from numerical singularities, hence this suffices to
give reasonable solutions.

We are now able to handle MEG, EEG, SEEG modalities, simultaneously or not. The simultaneous
handling of the different modalities is made the more straightforward by coupling the source localization
problem and the inverse transmission problem.

The data transmission needs to be considered in the source localization problem so that the prob-
lem remains faithful to the data (electric or magnetic) when realistic head geometries are used in the
computations. This also allows one to exploit properties of the problem that bring more regularity to the
solutions. These considerations warrant that the transmission problem be solved simultaneously with
the source localization problem, but in this connection the vector spaces used to model the sources and
electrical properties need to be handled with care. The mathematical formulation allows for the source
to be a distribution whereas the electric potential is always a function. This leads to the consideration
that the optimal methods to recover the source and electric potential may not be the same. We proposed
an alternating minimization procedure to solve the problem. We managed to show that this procedure
converges to the desired solution and converges to the Tychonov functional linearly.

For more general source terms (vector valued measures, Lp vector-fields), an alternating minimization
procedure between refining the cortical mapping and the source localization is being studied and
implemented. It takes advantage of the fact that both the source and the surface potentials are assumed
to be elements of Banach spaces that have certain smoothness properties which enable the convergence
to a solution as soon as the problem is convex. This alleviates problems that may be encountered when
the regularizing term is non-smooth. This opens up the possibility to consider sources that may exhibit
properties usually associated with distributions rather than functions [13].

6.2 Stability assessment of microwave circuits

Participants:

Laurent Baratchart, Sylvain Chevillard, Fabien Seyfert, Adam Cooman.

This year we continued the study, initiated in [23] in collaboration with J.-B. Pomet from the McTao
team (Inria Sophia Antipolis), of the harmonic transfer function (HTF) of a linear difference delay control
system having finitely many (possibly non-commensurable) delays and periodic coefficients (having the
same period). That is, an input-output system of the form:

y(t ) =

N
(cid:88)

j =1

D j (t ) y(t − τ j ) + u(t ),

t > t0 ,

where τ1 < · · · < τN are positive delays and D1(t ), . . . , D N (t ) real d × d matrices depending periodically on
time t , while u is the Rd -valued input and y(t ) the Rd -valued output (complex coefficients can of course

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Inria Annual Report 2023

be handled in the same way). The HTF, that generalizes the usual transfer function of linear constant
systems, is an analytic function of the Laplace-Fourier variable which is valued in the space of operators
a Hilbert space of periodic functions (the period being that of the system). We were able to construct
a simple nonlinear circuit whose linearization around a periodic trajectory has a spectrum containing
a whole circle; we currently investigate whether the singularities of the Fourier coefficients of the HTF
(that are themselves analytic functions) also contain that circle. Indeed, just like a series of functions may
diverge even though the summands are smooth, it is a priori possible that the HTF has a singularity at
a point whereas its Fourier coefficients do not. Note that these coefficients are all one can estimate by
harmonic balance techniques, and therefore the above question is of great practical relevance. We also
investigate the relation between our stability criterion (that the HTF should be bounded and analytic on
a “vertical” half plane containing the origin), and the weaker requirement that the HTF exists pointwise
in a half-plane. Connections with the pseudo-periodic Schrödinger oscillator are a motivation for such a
study, see [24].

6.3 Hardy-Hodge decomposition with applications to silent sources

Participants:

Laurent Baratchart, Juliette Leblond, Masimba Nemaire, Cristóbal Vil-
lalobos Guillén.

In a joint work with T. Qian and P. Dang from the university of Macao [54], we proved that on a
compact hypersurface Σ embedded in Rn, a Rn-valued vector field of Lp class decomposes as the sum
of a harmonic gradient from inside Σ, a harmonic gradient from outside Σ, and a tangent divergence-
free field, provided that 2 − ε < p < 2 + ε′
, where ε and ε′
depend on the Lipschitz constant of the
surface. We also showed that the decomposition is valid for 1 < p < ∞ when Σ is V MO-smooth (i.e., Σ is
locally the graph of Lipschitz function with derivatives in V MO). By projection onto the tangent space,
this gives us a Helmholtz-Hodge decomposition for vector fields on a compact Lipschitz hypersurface,
which is apparently new since existing results deal with smooth surfaces. In fact, the Helmholtz-Hodge
decomposition holds on Lipschitz surfaces (not just hypersurfaces), In [40], we had also showed that
L2-magnetizations (i.e. Rn-valued vector fields) on a compact connected Lipschitz surface embedded
in Rn that are silent outside (i.e., that produce no magnetic field in the unbounded component of the
complement of the surface) form the orthogonal space to harmonic gradients inside the surface. This
result teams up with the Hardy-Hodge decomposition to produce a description of silent magnetizations
distributions of Lp -class on a surface as solutions to a certain spectral equation for the double layer
potential (for restricted range of p when the surface is only Lipschitz). This last piece of research was a
collaboration with C. Gerhards and A. Kegeles from TU Freiberg.

Incented by these works, we developed a version of the Hardy-Hodge decomposition for the Helmholtz
equation, in collaboration with H. Haddar from the team IDEFIX (Inria Saclay & ENSTA-Paris Tech), [25].
Our motivation to study such issues stems from a linearized model for an acoustic scattering problem
with anisotropic layer, in the limit as the thickness of the layer goes to 0. From the mathematical point of
view, the problem generalizes the one of recovering magnetizations mentioned above. Here, we work
with sources M belonging to [L2(∂Ω)]3, where Ω is a bounded Lipschitz domain in R3. The generalization
comes from taking the Helmholtz equation instead of the Poisson equation, that is:

∆w + k2w = ∇ · (M σ),

(2)

where k > 0 and σ is the restriction of 2-dimensional Hausdorff measure to ∂Ω. Since the inverse problem
of recovering M is non uniquely solvable with kernel of infinite dimension, decomposing the space of
sources in a manner that singles out silent ones on either side of ∂Ω is useful, for it allows one to make
ansatz on the form of the solutions and to discuss constraints that may restore uniqueness. More precisely,
take ν to be the exterior unit normal to Ω and Mν the normal component of M ∈ L2(∂Ω, R3), with MT its
tangential component. Define three spaces as follows:

M0 = {M ∈ [L2(∂Ω)]3 : (Mν, ∇T · MT ) = (0, 0)},

M+ = {0} ∪ {M ∈ M ⊥
0

∩ [L2(∂Ω)]3 : w|R3\Ω = 0, w|Ω ̸= 0},

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M− = {0} ∪ {M ∈ M ⊥
0

∩ [L2(∂Ω)]3 : w|Ω = 0, w|R3\Ω ̸= 0}.

Like in the Poisson case, it can be shown that M ∈ M0 if and only if w = 0. However, unlike the Poisson
case, there is a nontrivial (but still finite-dimensional) space MH whose nonzero elements generate
a field that cannot be decomposed as the sum of fields each of which vanishes on one component of
Rn \ Ω. The dimension of this space depends on the wave number k in (2) and is at most equal to the
number of connected components of ∂Ω (it is in fact, the dimension of the vector space generated by
piecewise constant and continuous normal functions on ∂Ω which are non-silent). Then, it holds that
M0 ⊕ M+ ⊕ M− ⊕ MH = [L2(∂Ω)]3. Using this decomposition it is clear that the problem of recovering
sources when measurements are merely taken in the unbounded component of the complement of the
sample (this is the only accessible component) is solvable only up to an element of M0 ⊕ M+, and that
any approach to the inverse problem should take this into consideration.

We also pursued our work on a direct characterization of silent [Lp (Ω)]n, (n ≥ 3) vector-fields; that
is, vector fields representing magnetizations whose field is zero almost everywhere off Ω. Note that the
present definition of a silent magnetization is different, when the boundary of Ω is not connected, from
the one defined in [41]. Specifically, we showed that a vector field carried by a bounded open set Ω is silent
outside Ω if and only if, in its Helmholtz decomposition as the sum of a Lp gradient and a divergence-free
Lp field on Rn, both summands vanish a.e. outside of Ω. It should be noticed that this holds for 1 < p < ∞,
even though the Helmholtz decomposition in Ω may not hold for every Lp -field (in general, even for
Lipschitz open sets, it only holds for p in a neighborhood of [2/3, 3]). Moreover, we showed that under
mild capacitary conditions on Ω (its boundary should be of zero Lebesgue measure, if the closure of a
connected component of Rn\Ω meets the closure of the union of other components then the intersection
has positive Bessel capacity, moreover each component of Rn\Ω is non thin at quasi every boundary
point), a Lp gradient which is zero a.e. outside Ω lies in the closure of smooth compactly supported
gradients in Ω. We also showed that if Ω satisfies the segment condition, or else if it is quasi convex and
Gromov hyperbolic, a Lp divergence-free field which is zero a.e. outside Ω lies in the closure of smooth
compactly supported divergence-free fields in Ω. When Ω satisfies both properties we say that it is regular
for the zero extension, and in this case one can describe the norm-minimizing equivalent vector-field
to a given vector-field M ∈ [Lp (Ω)]n. Here, two vector-fields are equivalent if their magnetic potentials
outside Ω differ locally by a constant (possibly different constants in different connected components
of the complement), or equivalently if the vector-fields differ by a silent source. We showed that the
norm-minimizing equivalent source can be written as 〈M , ∇φ〉|∇φ|q−2∇φ with φ a harmonic function
with Lp -gradient in Ω. This yields in passing a nonlinear Helmholtz-like decomposition, valid for all
1 < p < ∞, containing an extra p-divergence-free term.

∗

∗

1,p
R∂Ω

with f

When Ω is Lipschitz, it is a fortiori regular for the zero extension and all regularity issues are easily
unwound, entailing that silent vector-fields M ∈ [Lp (Ω)]n (i.e., vector fields representing magnetizations
(Ω) (the
that produce the zero magnetic field outside Ω) are of the form M = ∇ψ + D with ψ ∈ W
familiar space of Sobolev functions with locally constant trace, with possibly different constant on
different connected components of the boundary) and D divergence-free with zero normal component
on the boundary ∂Ω [26]. Further, one may use in this case that the harmonic function φ appearing
in the expression of the minimal norm equivalent magnetization M can be expressed as a single layer
potential, say S f
maximizing the duality pairing 〈PM (y), f (y)〉 on ∂Ω (here PM is the Newton
potential of the distributional divergence of M ) under appropriate constraints. Thus, a Galerkin-type
method can be used to approach the problem numerically, using a frame of the space of distributions
−1
q ,q (∂Ω) that can be constructed as in [77]. Alternatively, in the case where n = 3 and Ω is a C 1,1 simply
W
connected domain, a minimization procedure that alternates between approximating M ∈ [W 1,p (Ω)]3
by a silent gradient and a silent divergence-free vector-field can be used. This alternating minimization
procedure makes connection with a non-linear stationary Maxwell system and a q-Laplace boundary
value problem, both of which can be solved via Galerkin methods with the use of finite elements methods.
Of course, with sparsity properties in mind, it would be very interesting to derive counterparts of these
results in the case p = 1, or even when n = 3 and M is a R3-valued measure. The previous results show
that the characterization as the sum of a gradient and a divergence free distribution supported on Ω s still
valid, however these summands are generally no measures but distributions of higher order.

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Inria Annual Report 2023

6.4 Identification of resonating frequencies of compact metallic objects in electro-

magnetic inverse scattering

Participants:

Paul Asensio, Laurent Baratchart, Juliette Leblond, Martine Olivi, Fa-
bien Seyfert, Dmitry Ponomarev.

Some fundamental aspects of the temporal behavior of the solution of the time-domain wave equation
(in spatial dimensions 1, 2 and 3) have been studied in a recent work [21] (previously a preprint, now
completed and submitted for journal publication) in connection with designing an efficient time-domain
solver for the Helmholtz equation in high-frequency regime and with smooth coefficients. In physical
3D settings, the rate of the stabilization of the solution is linked to the location and the strength of the
resonances (poles of meromorphic continuation of the resolvent operator). Since the information on
the resonances is useful for the inverse problems (object identification), some of the already developed
numerical methods could be employed for their efficient computation. Those methods would often rely
on the complex-scaling (PML) approach to the open-domain problems [76] and have yet to be compared
with the present rational-approximation viewpoint adopted by the team.
Another connection between time-domain methods for elliptic PDE problems and inverse problems is
the so-called controllability approach which has recently received a renewed interest [66]. In such an
approach, one seeks to accelerate temporal convergence to the stationary solution of the time-domain
wave equation with a periodic source term by minimizing the difference of the initial solution and
the solution after one time-period. Once properly set up, this would yield the solution to the original
Helmholtz equation without large computational costs of simulating the time-domain problem for long
times. Potential extension of such methods could be a possibility for future joint work with Atlantis team.
Related to our collaboration with LEAT, the rational approximation of the transfer function H defined
in Section 4.5 is performed with a least-squares version of multipoint Padé approximant: Rm,n[H ] =
pm,n/qm,n given by:

(pm,n, qm,n) = argminp∈P n , q∈P 1

m

(cid:175)
(cid:175)p(zi ) − H (zi )q(zi )

(cid:175)
(cid:175)

2 ,

N
(cid:88)

i =1

(3)

where n, m and N are integers such that n +m +1 ≤ N and (zi )i =1...N is a collection of points at which H is
analytic. Here, P n is the set of polynomials of degree less than n and P 1
m is the set of monic polynomials
of degree m. We seek an extension of the Nuttall-Pommerenke theorem for this approximation scheme;
i.e., that the capacity of the set where the pointwise error is bigger than some ε > 0 goes to zero as N ,
n and m go to infinity, while meeting some constraints to be made precise (expressing typically that
N remains much larger than n + m). Such a theorem would be very interesting, as it would imply that
certain poles of H can indeed be retrieved using this method while explaining the chaotic behavior of the
so-called spurious poles that wander about the domain of analyticity.

This year we were able to obtain an analog of the Nuttal-Pommerenke theorem on the converge in
capacity, not for problem (3) as we seek in the framework of our collaboration with LEAT, but for a least
square analog of Padé approximation that goes as follows:

(pm,n, qm,n) = argminp∈P n , q∈P 1

m

(cid:175)
(cid:175)[p − H q]i

(cid:175)
(cid:175)

2 ,

N
(cid:88)

i =1

where [·]i indicates the i -th coefficient in the Taylor expansion at the origin (or any other point in the
domain of analyticity of f ), see [19]. An article is being written to explain our result. This problem is
also a classical version of the vector fitting method, and it is the first time a convergence result is proved
regarding the latter. We hope that our method of proof will pave the way to the multipoint case.

6.5 Other elliptic PDE problems: from direct to inverse problems

Participants:

Juliette Leblond, Dmitry Ponomarev.

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Some direct problems have been considered from an analytical viewpoint that could shed light on

their inverse counterparts.
The work [18] (previously a preprint, now a published work) deals with the extension of a classical model
of contact problem with sliding punch in presence of material wear. The semi-explicit form of the
obtained results and the generality of the model call for looking into relevant inverse problems. These
could include, for example, determining an optimal initial shape of the punch profile that would minimize
the worn volume.

In the joint work [17] with E. Pozzi (Saint Louis Univ., Missouri, USA), we present an operator theoret-
ical approach of some inverse problems. It gives an overview of the solution to linear forward and inverse
problems for a class of elliptic PDE in two-dimensional domains, in the framework of Banach spaces
and operators. We focused on the conductivity equation, with smooth enough conductivity (see Section
3.2.1), and with Dirichlet or Neumann or mixed boundary conditions. For domains with Dini-smooth
boundary, the approach is based on the properties of generalized Hardy spaces.

6.6 Spectra of Toeplitz operators

Participants:

Juliette Leblond, Masimba Nemaire.

When solving fixed point problems using iterative methods, the rate of convergence is determined
by the spectral radius of the operator and how quickly the iterative methods settle to the convergence
rates is determined by the growth of the resolvent of the operator. It is therefore of interest to study the
spectra of operators, here we studied the spectra of Toeplitz operators, together with S. Kupin (IMB, Univ.
Bordeaux) and L. Golinskii (B. Verkin Institute for Low Temperature Physics and Engineering, Kharkiv,
Ukraine).

Let D be the unit disk in the complex plane, C and let T = ∂D. We showed that the essential spectrum
of a Toeplitz operator with symbol φ ∈ W 1,2(T) is equal to the essential spectrum of a Toeplitz-Bergman
operator with symbol that is the harmonic extension of φ to D, [16].

6.7 Tools for numerically guaranteed computations

Participants:

Sylvain Chevillard.

The overall and long-term goal is to enhance the quality of numerical computations.
This year, we worked on improving the implementation of the sin function in the glibc for the ARM
architecture. Indeed, a vectorized implementation has been proposed by J. Ramsay during the summer
on the mailing-list of the glibc, based on two steps: first, a range reduction to reduce the problem to the
evaluation of sin on the interval [0, π/2], and second the evaluation of a polynomial approximation on
the reduced range.

We did a full error analysis of the proposed implementation, including an exhaustive search on some
selected sub-intervals to find the actual worst cases for the range reduction, the evaluation error and the
combined error. We also proposed new values for the coefficients of the polynomial, in order to make it
practically optimal among all polynomials in term of the maximal total error (i.e., the error combining
the range reduction error, the approximation error and the evaluation error). This work was done in
collaboration with G. Melquiond (Toccata, Inria) and P. Zimmermann (Caramba, Inria). S. Chevillard
presented it at the “Rencontres nationales du groupe de travail Arith du GDR IM” (RAIM 2023) and a
report is currently being written on the subject.

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Inria Annual Report 2023

7 Partnerships and cooperations

7.1 International research visitors

7.1.1 Visits of international scientists

Other international visits to the team

Maxim Yattselev

Status: Associate Professor

Institution of origin: Indiana University-Purdue University Indianapolis

Country: Indiana, USA

Dates: June 2023

Context of the visit: This is a long term research effort to describe the behavior of poles and zeros of
asymptotically optimal meromorphic approximant to functions with singular set of capacity zero.

Mobility program/type of mobility: Research stay

Bernhard Hanzon

Status: Professor

Institution of origin: University College Cork

Country: Ireland

Dates: September to December 2023

Context of the visit: Writing a book on former joint research with members of the team

Mobility program/type of mobility: Research stay, lecture

7.2 European initiatives

7.2.1 Other european programs/initiatives

Participants:

Paul Asensio, Laurent Baratchart, Masimba Nemaire, Martine Olivi,
Cristóbal Villalobos Guillén.

Factas 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.

7.3 National initiatives

Participants:

Paul Asensio, Laurent Baratchart, Sylvain Chevillard, Juliette Leblond,
Masimba Nemaire, Martine Olivi, Dmitry Ponomarev, Cristóbal Vil-
lalobos Guillén, Anass Yousfi.

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ANR REPKA ANR-18-CE40-0035, “REProducing Kernels in Analysis and beyond” (2019–2023).

Led by Aix-Marseille Univ. (IMM), involving Factas team, together with Bordeaux (IMB), Paris-Est,

Toulouse Universities.

The project consists of several interrelated tasks dealing with topical problems in modern complex
analysis, operator theory and their important applications to other fields of mathematics including
approximation theory, probability, and control theory. The project is centered around the notion of the
so-called reproducing kernel of a Hilbert space of holomorphic functions. Reproducing kernels are very
powerful objects playing an important role in numerous domains such as determinantal point processes,
signal theory, Sturm-Liouville and Schrödinger equations.

This project supported the PhD of M. Nemaire within Factas, co-advised by IMB partners.

GDR AFHP GDR “Analyse Fonctionnelle, Harmonique et Probabilités”.

Led by Gustave Eiffel Univ. (LAMA), involving Factas team, together with several universities.
The GDR is concerned with five main axes: linear dynamics, Banach spaces and their operators,
holomorphic dynamics, harmonic analysis, analysis and probability, and with the interactions between
them.

8 Dissemination

Participants:

Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Martine Olivi,
Dmitry Ponomarev.

8.1 Promoting scientific activities

8.1.1 Scientific events: organisation

General chair, scientific chair

• J. Leblond was a member of the scientific committee of the GDR Analyse Fonctionnelle, Harmo-
nique et Probabilités (AFHP) days, October 2023, Porquerolles. She was also within the scientific
committee of the 1st international conference on artificIAl Intelligence and applied MAthematics
for History and Archaeology (IAMAHA), November 2023, Nice.

Member of the organizing committees

• F. Barbara, S. Chevillard, and J. Leblond were members of the organizing committee of the GDR

AFHP days, October 2023, Porquerolles.

Member of the conference program committees L. Baratchart is a member of the program committee
of the conference Inverse problems: modeling and simulation (IPMS). “

8.1.2 Journal

Member of the editorial boards L. Baratchart is a member of the editorial board of the journals Compu-
tational Methods and Function Theory (CMFT) and Complex Analysis and Operator Theory (CAOT).

Reviewer - reviewing activities

• S. Chevillard was a reviewer for Reliable Computing.

• J. Leblond was a reviewer for Arima.

• M. Olivi was a reviewer for Enseigner les enjeux socio-écologiques dans le supérieur.

• D. Ponomarev was a reviewer for Complex Analysis and Operator Theory (CAOT) and Mathematical

Reviews.

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Inria Annual Report 2023

8.1.3 Invited talks

• L. Baratchart was an invited speaker at the workshop “Orthogonal polynomials and special func-
tions” of the Foundations of Computational Mathematics (FoCM) conference, June 19-21, 2023,
Paris.

• L. Baratchart and D. Ponomarev were invited speakers at the GDR AFHP days, October 24-27, 2023,

Porquerolles.

• D. Ponomarev was an invited speaker at Journée Analyse Harmonique et EDP, 27 March 2023,
Bordeaux, and at the minisymposium “Inverse Problems in Imaging and Tomographic Issues,
Theory and Practice” of SIAM GS23 conference, June 19-22, Bergen, Norway.

8.1.4 Contributed talks

• S. Chevillard gave a presentation at the “Rencontres nationales du groupe de travail Arith du

GDR IM” (RAIM 2023), November 2023, Nancy.

• M. Nemaire gave a communication at the workshop Mathematics of electrical imaging: modeling,

theory and implementation, June 2023, Toulouse.

• D. Ponomarev gave presentations (poster and oral, respectively) at the conference IEEE AIM 2023
Advances in Magnetics (January 15-18, 2023, Moena, Italy) and at the workshop Complex Analysis:
Techniques, Applications and Computations - Perspectives 2023 (July 24-28, 2023, Cambridge, UK).

• C. Villalobos Guillén gave presentations at the Journées du GDR AFHP, October 2023, Porquer-
olles, and at the Winter-School Sound and Fury of Modeling, November 2023, Arpino, Italy. She
also presented a poster at the Conference: 30 ans de mathématiques pour l’imagerie optique,
September 2023, Marseille.

8.1.5 Research administration

• J. Leblond is a member of the “Conseil Scientifique” and of the “Commission Administrative

Paritaire” of Inria.

• M. Olivi is a member of the CLDD (Commission Locale de Développement Durable) and in charge

of coordination.

• During the period March-November 2023, the team members were actively involved in an ex-
perimental project led by the Direction of the Research Center for the design of new Inria office
spaces. This included participation in a series of workshops “Ateliers de co-conception de l’Espace
UX” (S. Chevillard, D. Ponomarev), complemented by discussions within the “Comité de pilotage”
(D. Ponomarev) and general meetings with the other concerned teams.

8.2 Teaching - Supervision - Juries

8.2.1 Teaching

• S. Chevillard gives “Colles” (oral examination preparing undergraduate students for the competitive
examination to enter French Engineering Schools) at Centre International de Valbonne (CIV) (2
hours per week). In 2023, he also gave a course on Carbon Footprints at Polytech Sophia Antipolis,
as a participant of the course Environmental Issues addressed to all students of Polytech, whatever
their pathway (level L3; the course on Carbon Footprints was 2 hours long and represented a
fifth of the whole course; it was repeated 4 times, due to the year group being divided into four
sub-groups).

• S. Chevillard and M. Olivi gave a one-day Doctoral Workshop on Science, Environment and Society
(in collaboration with G. Urvoy-Keller, S. Icart and L. Deneire, I3S). The first half-day was inspired
by the workshop SEnS, created by S. Quinton (Inria, SPADES) and E. Tannier (Inria, BEAGLE).

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The second half-day was more specifically dedicated to the environmental crisis and the study of
prospective scenarios.

8.2.2 Supervision

• PhD: P. Asensio, Inverse source estimation problems in EEG and MEG, defended September 2023,

advisers: L. Baratchart, J. Leblond.

• PhD: M. Nemaire, Inverse potential problems with application to quasi-static electromagnetics,

defended March 2023, advisers: L. Baratchart, J. Leblond, S. Kupin (IMB, Univ. Bordeaux).

• PhD in progress: A. Yousfi, Methods to estimate the net magnetic moment of rocks, since Octo-

ber 2022, advisers: S. Chevillard, J. Leblond.

• PhD in progress: M. Khalid Omer, Field preprocessing and treatment of complex samples in the

paleomagnetic context, since October 2023, advisers: J. Leblond, D. Ponomarev.

• Post-doc.: C. Villalobos Guillén, Inverse source problems for magnetization, primary current and

inverse scattering, adviser: L. Baratchart.

• Internship: M. Khalid Omer, Field preprocessing in inverse magnetisation problems, April-August 2023,

advisers: J. Leblond, D. Ponomarev.

8.2.3 Juries

• J. Leblond was a member of the examining committees for the defense of the PhD theses of B. Laville

(Univ. Côte d’Azur, ED STIC, September) and of N. Nasr (Univ. Bordeaux, ED MI, December).

• M. Olivi was a member of the “Comité de sélection MCF61” of Centrale Lyon EEA/Ampère.

8.3 Popularization

8.3.1 Internal or external Inria responsibilities

J. Leblond and M. Olivi are members of Terra Numerica.

8.3.2 Interventions

• M. Olivi participated in a meeting with scholars at Collège la Chênaie, in the context of “la cordée
de la réussite” : Un parcours scientifique sans préjugés : brisons le plafond de verre !. She gave four
conferences “CHICHE” at Lycée Simone Veil in Valbonne. She gave a the c@fé-in of the research
center, Martine à la COP : au delà des slogans, quelles solutions ?, and a general audience conference
at Terra Numerica on the same subject.

• D. Ponomarev gave a presentation at C@fé ADSTIC, 30 November 2023, Sophia Antipolis.

9 Scientific production

9.1 Major publications

[1] B. Atfeh, L. Baratchart, J. Leblond and J. R. Partington. ‘Bounded extremal and Cauchy-Laplace
problems on the sphere and shell’. In: J. Fourier Anal. Appl. 16.2 (2010). Published online Nov. 2009,
pp. 177–203. URL: http://dx.doi.org/10.1007/s00041-009-9110-0.

[2] L. Baratchart, S. Chevillard, D. P. Hardin, J. Leblond, E. A. Lima and J.-P. Marmorat. ‘Magnetic
moment estimation and bounded extremal problems’. In: Inverse Problems and Imaging 13.1 (Feb.
2019), p. 29. DOI: 10.3934/ipi.2019003. URL: https://hal.inria.fr/hal-01623991.
[3] L. Baratchart, S. Chevillard and T. Qian. ‘Minimax principle and lower bounds in H 2-rational

approximation’. In: Journal of Approximation Theory 206 (2015), pp. 17–47.

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Inria Annual Report 2023

[4] L. Baratchart, S. Fueyo, G. Lebeau and J.-B. Pomet. ‘Sufficient Stability Conditions for Time-varying
Networks of Telegrapher’s Equations or Difference Delay Equations’. In: SIAM Journal on Mathem-
atical Analysis 53 (2021), pp. 1831–1856. DOI: 10.1137/19M1301795. URL: https://hal.inria
.fr/hal-02385548.

[5] L. Baratchart, J. Leblond, S. Rigat and E. Russ. ‘Hardy spaces of the conjugate Beltrami equation’.
In: Journal of Functional Analysis 259.2 (2010), pp. 384–427. URL: http://dx.doi.org/10.1016
/j.jfa.2010.04.004.

[6] L. Baratchart, M. Olivi and F. Seyfert. ‘Boundary Nevanlinna-Pick interpolation with prescribed
peak points. Application to impedance matching’. In: SIAM Journal on Mathematical Analysis
(2017). DOI: 10.1137/16M1085577. URL: https://hal.inria.fr/hal-01377782.

[7] L. Baratchart, H. Stahl and M. Yattselev. ‘Weighted Extremal Domains and Best Rational Approxim-
ation’. In: Advances in Mathematics 229 (2012), pp. 357–407. URL: http://hal.inria.fr/hal-0
0665834.

[8] L. Baratchart, C. Villalobos Guillén, D. Hardin, M. Northington and E. Saff. ‘Inverse Potential
Problems for Divergence of Measures with Total Variation Regularization’. In: Foundations of
Computational Mathematics 20 (2020), pp. 1273–1307. DOI: 10.1007/s10208-019-09443-x. URL:
https://hal.inria.fr/hal-02424672.

[9] M. Clerc, J. Leblond, J.-P. Marmorat and T. Papadopoulo. ‘Source localization using rational approx-
imation on plane sections’. In: Inverse Problems 28.5 (May 2012), Article number 055018 –24 pages.
DOI: 10.1088/0266-5611/28/5/055018. URL: https://hal-mines-paristech.archives-o
uvertes.fr/hal-00699541.

[10] A. Cooman, F. Seyfert, M. Olivi, S. Chevillard and L. Baratchart. ‘Model-Free Closed-Loop Stability
Analysis: A Linear Functional Approach’. In: IEEE Transactions on Microwave Theory and Techniques
(2017). https://arxiv.org/abs/1610.03235. URL: https://hal.inria.fr/hal-01381731.

[11]

J. Leblond and D. Ponomarev. ‘On some extremal problems for analytic functions with constraints
on real or imaginary parts’. In: Advances in Complex Analysis and Operator Theory: Festschrift in
Honor of Daniel Alpay’s 60th Birthday. Birkhauser, 2017, pp. 219–236. DOI: 10.1007/978-3-319-
62362-7_8. URL: https://hal.archives-ouvertes.fr/hal-01623281.

[12] M. Olivi, F. Seyfert and J.-P. Marmorat. ‘Identification of microwave filters by analytic and rational
H2 approximation’. In: Automatica 49.2 (15th Jan. 2013), pp. 317–325. DOI: 10.1016/j.automati
ca.2012.10.005. URL: https://hal.inria.fr/hal-00753824.

9.2 Publications of the year

International journals

[13] P. Asensio, J.-M. Badier, J. Leblond, J.-P. Marmorat and M. Nemaire. ‘A layer potential approach to
inverse problems in brain imaging’. In: Journal of Inverse and Ill-posed Problems (Feb. 2023). DOI:
10.1515/jiip-2023-0041. URL: https://inria.hal.science/hal-03977724.

[14] M. Darbas, J. Leblond, J.-P. Marmorat and P.-H. Tournier. ‘Numerical resolution of the inverse source
problem for EEG using the quasi-reversibility method’. In: Inverse Problems 39.11 (2023), p. 115003.
DOI: 10.1088/1361-6420/acf9c6. URL: https://inria.hal.science/hal-03880526.

[15] E. Lima, B. Weiss, C. Borlina, L. Baratchart and D. Hardin. ‘Estimating the Net Magnetic Moment
of Geological Samples From Planar Field Maps Using Multipoles’. In: Geochemistry, Geophysics,
Geosystems 24.7 (26th July 2023), p. 38. DOI: 10.1029/2022GC010724. URL: https://inria.hal
.science/hal-04379703.

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Scientific book chapters

[16] L. Golinskii, S. Kupin, M. Nemaire and J. Leblond. ‘On discrete spectra of Bergman-Toeplitz operat-
ors with harmonic symbols’. In: From Complex Analysis to Operator Theory: A PanoramaIn Memory
of Sergey Naboko. Vol. 291. Operator Theory: Advances and Applications. Springer International
Publishing, 2023. DOI: 10.1007/978-3-031-31139-0. URL: https://inria.hal.science/ha
l-03650364.

[17]

J. Leblond and E. Pozzi. ‘An operator theoretical approach of some inverse problems’. In: Function
Spaces, Theory and Applications. Fields Institute Communications ; vol 87. Springer, 2023. DOI:
10.1007/978-3-031-39270-2_3. URL: https://inria.hal.science/hal-04355468.

[18] D. Ponomarev. ‘Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load’. In:
Integral Methods in Science and Engineering: Analytic and Computational Procedures. Birkhäuser,
1st Nov. 2023. DOI: 10.1007/978-3-031-34099-4_21. URL: https://hal.science/hal-0381
3510.

Doctoral dissertations and habilitation theses

[19] P. Asensio. ‘Inverse problems of source localization with applications to EEG and MEG’. Université

cote d’azur, 21st Sept. 2023. URL: https://hal.science/tel-04366056.

[20] M. Nemaire. ‘Inverse potential problems, with applications to quasi-static electromagnetics’. Uni-
versité de Bordeaux, 28th Mar. 2023. URL: https://theses.hal.science/tel-04083678.

Reports & preprints

[21] A. Arnold, S. Geevers, I. Perugia and D. Ponomarev. On the limiting amplitude principle for the wave
equation with variable coefficients. 29th Dec. 2023. URL: https://hal.science/hal-03900877.

[22] P. Asensio and J. Leblond. Critical points for least-squares estimation of dipolar sources in inverse
problems for Poisson equation. 3rd Jan. 2023. URL: https://hal.science/hal-03901123.

[23] L. Baratchart, S. Fueyo and J.-B. Pomet. Exponential stability of linear periodic difference-delay

equations. 2023. URL: https://inria.hal.science/hal-03500720.

[24] L. Baratchart, S. Fueyo and J.-B. Pomet. Integral representation formula for linear non-autonomous

difference-delay equations. 2023. URL: https://hal.science/hal-04204500.

[25] L. Baratchart, H. Haddar and C. Villalobos Guillén. Silent sources on a surface for the Helmholtz
equation and decomposition of L² vector fields. 30th Dec. 2023. URL: https://hal.science/hal-
04367726.

[26] L. Baratchart, J. Leblond and M. Nemaire. Silent sources in Lp and Helmholtz-type decompositions.

2023. URL: https://inria.hal.science/hal-03915548.

[27] D. Ponomarev. Magnetisation Moment of a Bounded 3D Sample: Asymptotic Recovery from Planar
Measurements on a Large Disk Using Fourier Analysis. 28th Sept. 2023. URL: https://hal.scienc
e/hal-03813559.

9.3 Cited publications

[28] V. Lunot, F. Seyfert, S. Bila and A. Nasser. ‘Certified Computation of Optimal Multiband Filtering
Functions’. In: IEEE Transactions on Microwave Theory and Techniques 56.1 (2008), pp. 105–112.
URL: https://dx.doi.org/10.1109/TMTT.2007.912234.

[29] N. I. Achieser. Elements of the Theory of Elliptic Functions. AMS, 1990.

[30] D. Alpay, L. Baratchart and A. Gombani. ‘On the Differential Structure of Matrix-Valued Rational

Inner Functions’. In: Operator Theory : Advances and Applications 73 (1994), pp. 30–66.

[31] G. A. Baker and P. Graves-Morris. Padé approximants. 2d. Cambridge University Press, 2010.

[32]

J. A. Ball, I. Gohberg and L. Rodman. Interpolation of rational matrix functions. Birkhäuser, 1990.

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[33] L. Baratchart. ‘A remark on uniqueness of best rational approximants of degree 1 in L2 of the circle’.

In: Elec. Trans.on Numerical Anal. 25 (2006), pp. 54–66.

[34] L. Baratchart. ‘On the H 2 Rational Approximation of Markov Matrix-Valued Functions’. In: Proc.
17th Symposium on Mathematical Theory of Networks and Systems (MTNS). Kyoto, Japon, 2006,
pp. 180–182.

[35] L. Baratchart, A. Borichev and S. Chaabi. ‘Pseudo-holomorphic functions at the critical exponent’.
In: Journal of the European Mathematical Society 18.9 (2016). DOI: 10.4171/JEMS. URL: https:
//hal.inria.fr/hal-00824224.

[36] L. Baratchart, L. Bourgeois and J. Leblond. ‘Uniqueness results for inverse Robin problems with
bounded coefficient’. In: Journal of Functional Analysis (2016). DOI: 10.1016/j.jfa.2016.01.01
1. URL: https://hal.inria.fr/hal-01084428.

[37] L. Baratchart, M. Cardelli and M. Olivi. ‘Identification and rational L2 approximation: a gradient

algorithm’. In: Automatica 27 (1991), pp. 413–418.

[38] L. Baratchart, M. Chyba and J.-B. Pomet. ‘A Grobman-Hartman theorem for control systems’. In: J.

Dyn. Differential Eqs. 19 (2007), pp. 75–107.

[39] L. Baratchart, Y. Fischer and J. Leblond. ‘Dirichlet/Neumann problems and Hardy classes for the
planar conductivity equation’. In: Complex Variables and Elliptic Equations (2014), p. 41. DOI: 10.1
080/17476933.2012.755755. URL: https://hal.archives-ouvertes.fr/hal-00909577.

[40] L. Baratchart, C. Gerhards and A. Kegeles. ‘Decomposition of L2-vector fields on Lipschitz surfaces:
characterization via null-spaces of the scalar potential’. In: SIAM Journal on Mathematical Analysis
53.4 (2021), pp. 4096–4117. DOI: 10.1137/20M1387754. URL: https://hal.inria.fr/hal-030
86446.

[41] L. Baratchart, C. Gerhards, A. Kegeles and P. Menzel. ‘Unique reconstruction of simple magnet-
izations from their magnetic potential’. In: Inverse Problems 37.10 (Sept. 2021), p. 105006. DOI:
10.1088/1361-6420/ac1e82. URL: https://hal.inria.fr/hal-03434504.

[42] L. Baratchart, L. Golinskii and S. Kupin. ‘Orthogonal rational functions and nonstationary stochastic
processes: a Szeg˝o theory’. In: Proc. 19th Symposium on Mathematical Theory of Networks and
Systems. Budapest, 2010.

[43] L. Baratchart, D. P. Hardin, E. A. Lima, E. B. Saff and B. Weiss. ‘Characterizing kernels of operators
related to thin-plate magnetizations via generalizations of Hodge decompositions’. In: Inverse
Problems 29.1 (2013). DOI: 10.1088/0266-5611/29/1/015004. URL: https://hal.inria.fr/h
al-00919261.

[44] L. Baratchart, R. Kuestner and V. Totik. ‘Zero distributions via orthogonality’. In: Annales de l’Institut

Fourier 55.5 (2005), pp. 1455–1499.

[45] L. Baratchart, S. Kupin, V. Lunot and M. Olivi. ‘Multipoint Schur algorithm and orthogonal rational
functions: convergence properties, I’. In: Journal d’Analyse 112 (2011), pp. 207–255. URL: https:
//arxiv.org/abs/0812.2050v3.

[46] L. Baratchart and J. Leblond. ‘Hardy approximation to Lp functions on subsets of the circle with

1 ≤ p < ∞’. In: Constructive Approximation 14 (1998), pp. 41–56.

[47] L. Baratchart, J. Leblond, F. Mandréa and E. B. Saff. ‘How can meromorphic approximation help
to solve some 2D inverse problems for the Laplacian?’ In: Inverse Problems 15.1 (1999), pp. 79–90.
URL: https://dx.doi.org/10.1088/0266-5611/15/1/012.

[48] L. Baratchart, J. Leblond and J.-P. Marmorat. ‘Sources identification in 3D balls using meromorphic
approximation in 2D disks’. In: Electronic Transactions on Numerical Analysis (ETNA) 25 (2006),
pp. 41–53.

[49] L. Baratchart, J. Leblond and J. R. Partington. ‘Hardy approximation to L
the circle’. In: Constructive Approximation 12 (1996), pp. 423–435.

∞

functions on subsets of

[50] L. Baratchart, J. Leblond and F. Seyfert. ‘Constrained extremal problems in H2 and Carleman’s
formulas’. In: Matematicheskii Sbornik 209.7 (2018), p. 36. URL: https://hal.archives-ouvert
es.fr/hal-01625840.

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[51] L. Baratchart, F. Mandréa, E. B. Saff and F. Wielonsky. ‘2D inverse problems for the Laplacian: a
meromorphic approximation approach’. In: Journal de Math. Pures et Appliquées 86 (2008), pp. 1–
41.

[52] L. Baratchart and M. Olivi. ‘Critical points and error rank in best H 2 matrix rational approximation

of fixed McMillan degree’. In: Constructive Approximation 14 (1998), pp. 273–300.

[53] L. Baratchart and M. Olivi. ‘Index of critical points in l 2-approximation’. In: System and Control

Letters 10 (1988), pp. 167–174.

[54] L. Baratchart, D. Pei and T. Qian. ‘Hardy-Hodge decomposition of vector fields on compact Lipschitz

hypersurfaces’. in preparation.

[55] L. Baratchart, E. B. Saff and F. Wielonsky. ‘A criterion for uniqueness of a critical point in H 2 rational

approximation’. In: Journal d’Analyse 70 (1996), pp. 225–266.

[56] L. Baratchart and F. Seyfert. ‘An Lp analog to AAK theory for p ≥ 2’. In: Journal of Functional Analysis

191.1 (2002), pp. 52–122.

[57] L. Baratchart, H. Stahl and F. Wielonsky. ‘Asymptotic uniqueness of best rational approximants of
given degree to Markov functions in L2 of the circle’. In: Constr. Approx. 17.1 (2001), pp. 103–138.

[58] L. Baratchart and M. Yattselev. ‘Asymptotic uniqueness of best rational approximants to complex
Cauchy transforms in L2 of the circle’. In: Recent trends in orthogonal polynomials and approxima-
tion theory. Vol. 507. Contemp. Math. Providence, RI: Amer. Math. Soc., 2010, pp. 87–111.

[59] L. Baratchart and M. Yattselev. ‘Convergent interpolation to Cauchy integrals over analytic arcs’. In:

Found. Comp. Math. 9.6 (2009), pp. 675–715.

[60] L. Baratchart and M. Yattselev. ‘Convergent Interpolation to Cauchy Integrals over Analytic Arcs
with Jacobi-Type Weights’. In: International Mathematics Research Notices 2010.22 (2010), pp. 4211–
4275. URL: https://hal.inria.fr/hal-00508314.

[61] L. Baratchart and M. Yattselev. ‘Meromorphic approximants for complex Cauchy transforms with

polar singularities’. In: Mat. Sbornik 200.9 (2009), pp. 3–40.

[62] A. Ben Abda, F. Ben Hassen, J. Leblond and M. Mahjoub. ‘Sources recovery from boundary data: a
model related to electroencephalography’. In: Mathematical and Computer Modelling 49.11–12
(2009), pp. 2213–2223. URL: https://dx.doi.org/10.1016/j.mcm.2008.07.016.

[63] R. Cameron, J.-C. Faugère, F. Rouillier and F. Seyfert. ‘Exhaustive approach to the coupling matrix
synthesis problem and application to the design of high degree asymmetric filters’. In: International
Journal of RF and Microwave Computer-Aided Engineering 17.1 (2007), pp. 4–12. URL: https://ha
l.inria.fr/hal-00663777.

[64]

S. Chaabane, I. Fellah, M. Jaoua and J. Leblond. ‘Logarithmic stability estimates for a Robin coeffi-
cient in 2D Laplace inverse problems’. In: Inverse Problems 20.1 (2004), pp. 49–57. URL: https://d
x.doi.org/10.1088/0266-5611/20/1/003.

[65] S. Chaabi. ‘Analyse complexe et problèmes de Dirichlet dans le plan : équation de Weinstein et
autres conductivités non bornées’. PhD thesis. Mathématiques et Informatique de Marseille, 2013.

[66] T. Chaumont-Frelet, M. Grote, S. Lanteri and J. H. Tang. ‘A controllability method for Maxwell’s

equations’. In: arXiv preprint arXiv:2106.02858 (2021).

[67] R. Coifman, R. Rochberg and G. Weiss. ‘Factorization theorems for Hardy spaces in several variables’.

In: Ann. Math. 103 (1976), pp. 611–635.

[68] Y. Fischer. ‘Approximation des des classes de fonctions analytiques généralisées et résolution
de problèmes inverses pour les tokamaks’. PhD thesis. Univ. Nice Sophia Antipolis, 2011. URL:
https://tel.archives-ouvertes.fr/tel-00643239/.

[69] A. Friedman and M. Vogelius. ‘Determining cracks by boundary measurements’. In: Indiana Univ.

Math. J. 38.3 (1989), pp. 527–556.

[70] P. Fulcheri and M. Olivi. ‘Matrix rational H 2-approximation: a gradient algorithm based on Schur

analysis’. In: SIAM J. on Control & Optim. 36 (1998), pp. 2103–2127.

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[81]

J.-P. Marmorat and M. Olivi. ‘Nudelman Interpolation, Parametrization of Lossless Functions and
balanced realizations’. In: Automatica 43 (2007), pp. 1329–1338. URL: https://dx.doi.org/10.1
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[85] F. Seyfert, L. Baratchart, J.-P. Marmorat, S. Bila and J. Sombrin. ‘Extraction of coupling parameters
for microwave filters: determinati on of a stable rational model from scattering data’. In: 2003 IEEE
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[89] Y. Zaky, N. Fortino, J.-Y. Dauvignac, F. Seyfert, M. Olivi and L. Baratchart. ‘Comparison of SEM
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<scratchpad>
- Main topic: Inria Centre at Université Côte d’Azur, focusing on inverse problems in mathematics and applications.
- Key points:
  - Development of constructive, function-theoretic approaches to inverse problems.
  - Research on inverse source problems in EEG, magnetization, and electromagnetic inverse scattering.
  - Application of Hardy spaces and harmonic analysis to solve inverse problems.
  - Development of software tools for rational approximation and identification of resonating frequencies.
  - Collaboration with international partners on research projects and applications.
- Structure: The document is divided into sections on research center, activity report, project-team, domain, theme, contents, project, application domains, social and environmental responsibility, new results, partnerships and cooperations, dissemination, scientific production, and more.
- Potential artifacts to ignore: Links, HTML tags, and web-related elements that do not provide relevant information about the content.
</scratchpad>

<final_summary>
The Inria Centre at Université Côte d’Azur focuses on inverse problems in mathematics and applications. The team develops constructive, function-theoretic approaches to inverse problems, including research on inverse source problems in EEG, magnetization, and electromagnetic inverse scattering. Hardy spaces and harmonic analysis are used to solve these inverse problems. The team also develops software tools for rational approximation and identification of resonating frequencies, and collaborates with international partners on research projects and applications. These efforts aim to advance the field of inverse problems and their applications in various domains, including medicine, geophysics, and telecommunications.
The Inria Centre at Université Côte d’Azur focuses on inverse problems in mathematics and applications. The team develops constructive, function-theoretic approaches to inverse problems, including research on inverse source problems in EEG, magnetization, and electromagnetic inverse scattering. Hardy spaces and harmonic analysis are used to solve these inverse problems. The team also develops software tools for rational approximation and identification of resonating frequencies, and collaborates with international partners on research projects and applications. These efforts aim to advance the field of inverse problems and their applications in various domains, including medicine, geophysics, and telecommunications.