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

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

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

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

Factas undergo the departure of its team leader Fabien Seyfert at the end of September. It substantially reduces the manpower of the team, and essentially freezes our research in areas where Fabien was the main driving force. Among our targeted applications, those generating the most intensive technology transfer pertain to the first item and were led by him. He is now running his own consulting company.

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

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

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

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

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

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

Inverse potential problems are severely indeterminate because infinitely many measures within an open set of balayage . In the two steps approach previously described, we implicitly removed this indeterminacy by requiring in step 1 that the measure be supported on the boundary (because we seek a function holomorphic throughout the right half-space), and by requiring in step 2 that the measure be discrete in the left half-plane (in fact: a finite sum of point masses

To recap, the gist of our approach is to approximate boundary data by (boundary traces of) fields arising from potentials of measures with specific support. This differs from standard approaches to inverse problems, where descent algorithms are applied to integration schemes of the direct problem; in such methods, it is the equation which gets approximated (in fact: discretized).

Along these lines, Factas advocates the use of steps 1 and 2 above, along with some singularity analysis, to approach issues of nondestructive control in 2-D and 3-D , , . The team is currently engaged in the generalization to inverse source problems for the Laplace equation in 3-D, to be described further in Section . There, holomorphic functions are replaced by harmonic gradients; applications are to inverse source problems in neurosciences (in particular in EEG/MEG) and inverse problems in geosciences.

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

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 ). Once again, our approach combines steps 1 and 2 above.

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

of the boundary, is equivalent to recover a holomorphic function in the domain from its boundary values on

. This is the problem raised on the half-plane in step 1 of Section

. 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

that we set up in Section

. 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 Problem

where

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 measurements. This was an initial application of holomorphic extremal problems to non-destructive control

,

.

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

This was actually carried out in collaboration with CEA (French nuclear agency) and the Univ. Côte d'Azur (JAD Lab.), to data from Tore Supra in . The procedure is fast because no numerical integration of the underlying PDE is needed, as an explicit basis of solutions to the conjugate Beltrami equation in terms of Bessel functions was found in this case. Generalizing this approach in a more systematic manner to free boundary problems of Bernoulli type, using descent algorithms based on shape-gradient for such approximation-theoretic criteria, is an interesting prospect to the team.

The piece of work we just mentioned requires defining and studying Hardy spaces of conjugate Beltrami equations, which is another interesting topic. For Sobolev-smooth coefficients of exponent greater than 2, they were investigated in , . The case of the critical exponent 2 is treated in , which apparently provides the first example of well-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

The 3-D version of step 1 in Section is another subject investigated by Factas: to recover a harmonic function (up to an additive constant) in a ball or a half-space from partial knowledge of its gradient. This prototypical inverse problem (i.e. inverse to the Cauchy problem for the Laplace equation) often recurs in electromagnetism. At present, Factas is involved with solving instances of this inverse problem arising in two fields, namely medical imaging e.g. for electroencephalography (EEG) or magneto-encephalography (MEG), and paleomagnetism (recovery of rocks magnetization) , , see Section . 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 . 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 Factas) contributed convergence results for certain classes of fields (expressed as Cauchy integrals over extremal contours for the logarithmic potential , , ). Initial schemes to locate cracks or sources via rational approximation on planar domains were obtained this way , , . It is remarkable that finite inverse source problems in 3-D balls, or more general algebraic surfaces, can be approached using these 2-D techniques upon slicing the domain into planar sections , . More precisely, each section cuts out a planar domain, the boundary of which carries data which can be proved to match an algebraic function. The singularities of this algebraic function are not located at the 3-D sources, but are related to them: the section contains a source if and only if some function of the singularities in that section meets a relative extremum. Using bisection it is thus possible to determine an extremal place along all sections parallel to a given plane direction, up to some threshold which has to be chosen small enough that one does not miss a source. This way, we reduce the original source problem in 3-D to a sequence of inverse poles and branchpoints problems in 2-D. This bottom line generates a steady research activity within Factas, and again applications are sought to medical imaging and geosciences, see Sections , and .

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

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

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

Hardy spaces provide a framework to transform this ill-posed issue into a series of regularized analytic and meromorphic approximation problems. More precisely, the procedure sketched in Section goes as follows:

We actively continue our collaboration with the Chinese Hong Kong University on the topic of frequency depending couplings appearing in the equivalent circuits we compute continuing our work on wide-band design and dispersive coupling, that led to a major publication .

Factas also investigates issues pertaining to design rather than identification. Given the topology of the filter, a basic problem in this connection is to find the optimal response subject to specifications that bear on rejection, transmission and group delay of the scattering parameters. Generalizing the classical approach based on Chebyshev polynomials for single band filters, we recast the problem of multi-band response synthesis as a generalization of the classical Zolotarev min-max problem for rational functions , . Thanks to quasi-convexity, the latter can be solved efficiently using iterative methods relying on linear programming. These were implemented in the software ). Currently, the team is engaged in the synthesis of more complex microwave devices like multiplexers and routers, which connect several filters through wave guides. Schur analysis plays an important role here, because scattering matrices of passive systems are of Schur type (i.e. contractive in the stability region). The theory originates with the work of I. Schur , who devised a recursive test to check for contractivity of a holomorphic function in the disk. The so-called Schur parameters of a function may be viewed as Taylor coefficients for the hyperbolic metric of the disk, and the fact that Schur functions are contractions for that metric lies at the root of Schur's test. Generalizations thereof turn out to be efficient to parametrize solutions to contractive interpolation problems . Dwelling on this, Factas contributed differential parametrizations (atlases of charts) of lossless matrix functions , , which are fundamental to our rational approximation software RARL2 (see Section ). Schur analysis is also instrumental to approach de-embedding issues, and provides one with considerable insight into the so-called matching problem. The latter consists in maximizing the power a multiport can pass to a given load, and for reasons of efficiency it is all-pervasive in microwave and electric network design, e.g. of antennas, multiplexers, wifi cards and more. It can be viewed as a rational approximation problem in the hyperbolic metric, and the team presently deals with this hot topic using contractive interpolation with constraints on boundary peak points, within the framework of the (defense funded) ANR Cocoram, see Sections .

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 can be put to work. The main discovery is that the unstable part of each partial transfer function is rational and can be computed by analytic projection, see Section . We now start investigating the linearized harmonic transfer-function around a periodic cycle, to check for stability under non necessarily small inputs.

To find an analytic function

Here a priori assumptions on the behavior of the model off

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

(

In the case

Various modifications of

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. In the case of least square approximation and constraints on the complementary subset, formulas were recently found to express the solution in important cases, and we currently develop this aspect towards applications of various type, in collaboration with J. Mashreghi (Laval Univ., Canada). We mention related work in collaboration with D. Ponomarev (TU Vienna, Austria), and with E. Pozzi (Saint Louis Univ., Missouri, USA), associated with formulations of inverse magnetization issues.

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

Though originally considered in dimension 2, Problem

When

On the ball, the analog of Problem

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

Just like solving problem

Problem

Companion to problem

Note that

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

We put

A natural generalization of problem

(

Only for

The case where stable rational approximant to not be unique.

The Miaou project (predecessor of Apics) already designed a dedicated steepest-descent algorithm for the case 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 critical points of lower degree (as is done by the RARL2 software, Section ).

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 , . Based on this work, uniqueness or asymptotic uniqueness of the approximant was proved for certain classes of functions like transfer functions of relaxation systems (i.e. Markov functions) and more generally Cauchy integrals over hyperbolic geodesic arcs . These are the only results of this kind. Research on this topic remained dormant for a while by reasons of opportunity, but revisiting the work in higher dimension is a worthy and timely endeavor today. Meanwhile, an analog to AAK theory was carried out for

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

In higher dimensions, the analog of Problem

Besides, certain constrained rational approximation problems, of special interest in identification and design of passive systems, arise when putting additional requirements on the approximant, for instance that it should be smaller than 1 in modulus (i.e. a Schur function). In particular, Schur interpolation lately received renewed attention from the team, in connection with matching problems. There, interpolation data are subject to a well-known compatibility condition (positive definiteness of the so-called Pick matrix), and the main difficulty is to put interpolation points on the boundary of

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

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

The scalar approximation algorithm derived in and mentioned in Section generalizes to the matrix-valued situation . The first difficulty here is to parametrize inner matrices (i.e. matrix-valued functions analytic in the unit disk and unitary on the unit circle) of given McMillan degree degree

Difficulties relative to multiple local minima of course arise in the matrix-valued case as well, and deriving criteria that guarantee uniqueness is even more difficult than in the scalar case. The case of rational functions of degree

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

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

-sense on a closed curve, to functions

defined as Cauchy integrals of complex measures whose support lies inside the curve. Normalizing the contour to be the unit circle

, we are back to Problem

in Section

; invariance of the latter under conformal mapping was established in

. 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 , the original twist by Apics, now Factas, is to consider this issue also as a means to extract information on singularities of the solution to a Dirichlet-Neumann problem. The general theme is thus: how do the singularities of the approximant reflect those of the approximated function? This approach to inverse problem for the 2-D Laplacian turns out to be attractive when singularities are zero- or one-dimensional (see Section ). It can be used as a computationally cheap initial condition for more precise but much heavier numerical optimizations which often do not even converge unless properly initialized. As regards crack detection or source recovery, this approach boils down to analyzing the behavior of best meromorphic approximants of given pole cardinality to a function with branch points, which is the prototype of a polar singular set. For piecewise analytic cracks, or in the case of sources, we were able to prove (, , ), that the poles of the approximants accumulate, when the degree goes large, to some extremal cut of minimum weighted logarithmic capacity connecting the singular points of the crack, or the sources . Moreover, the asymptotic density of the poles turns out to be the Green equilibrium distribution on this cut in

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

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

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

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

Generally speaking, inverse potential problems, similar to the one appearing in Section

, occur naturally in connection with systems governed by Maxwell's equation in the quasi-static approximation regime. In particular, they arise in magnetic reconstruction issues. A specific application is to geophysics, which led us to form the Inria Associate Team

(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 aims at tackling them, experimenting with the SQUID microscope set up in the Paleomagnetism Laboratory of the department of Earth, Atmospheric and Planetary Sciences at MIT. Typically, pieces of rock are sanded down to a thin slab, and the magnetization has to be recovered from the field measured on a planar region at small distance from the slab.

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

outside the volume

Another timely instance of inverse magnetization problems lies with geomagnetism. Satellites orbiting around the Earth measure the magnetic field at many points, and nowadays it is a challenge to extract global information from those measurements. In collaboration with C. Gerhards (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 .

Solving over-determined Cauchy problems for the Laplace equation on a spherical layer (in 3-D) in order to extrapolate incomplete data (see Section ) is a necessary ingredient of the team's approach to inverse source problems, in particular for applications to EEG, see . Indeed, the latter involves propagating the initial conditions through several layers of different conductivities, from the boundary shell down to the center of the domain where the singularities (i.e. the sources) lie. Once propagated to the innermost sphere, it turns out that traces of the boundary data on 2-D cross sections coincide with analytic functions with branched singularities in the slicing plane , . The singularities are related to the actual location of the sources, namely their moduli reach in turn a maximum when the plane contains one of the sources. Hence we are back to the 2-D framework of Section , and recovering these singularities can be performed via best rational approximation. The goal is to produce a fast and sufficiently accurate initial guess on the number and location of the sources in order to run heavier descent algorithms on the direct problem, which are more precise but computationally costly and often fail to converge if not properly initialized. Our belief is that such a localization process can add a geometric, valuable piece of information to the standard temporal analysis of EEG signal records.

Numerical experiments obtained with our software FindSources3D give very good results on simulated data and we are now engaged in the process of handling real experimental data, simultaneously recorded by EEG and MEG devices, in collaboration with our partners at INS, hospital la Timone, Marseille (see Section ).

Furthermore, another approach is being studied for EEG, that consists in regularizing the inverse source problem by a total variation constraint on the source term (a measure), added to the quadratic data approximation criterion. It is similar to the path that is taken for inverse magnetization problems (see Sections and ), and it presently focuses on surface-distributed models.

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

One of the best training grounds for function-theoretic applications by the team is the identification and design of physical systems whose performance is assessed frequency-wise. This is the case of electromagnetic resonant systems which are of common use in telecommunications.

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

Near the resonance frequency, a good approximation to the Helmholtz equations is given by a second order differential equation. Thus, one obtains an electrical model of the filter as a sequence of electrically-coupled resonant circuits, each circuit being modeled by two resonators, one per mode, the resonance frequency of which represents the frequency of a mode, and whose resistance accounts for electric losses (surface currents) in the cavities.

This way, the filter can be seen as a quadripole, with two ports, when plugged onto a resistor at one end and fed with some potential at the other end. One is now interested in the power which is transmitted and reflected. This leads one to define a scattering matrix

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

In short, the strategy for identification is as follows:

The final approximation is of high quality. This can be interpreted as a confirmation of the linearity assumption on the system: the relative

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

The team further investigates problems relative to the design of optimal responses for microwave devices. The resolution of a quasi-convex Zolotarev problems was proposed, in order to derive guaranteed optimal multi-band filter responses subject to modulus constraints . This generalizes the classical single band design techniques based on Chebyshev polynomials and elliptic functions. The approach relies on the fact that the modulus of the scattering parameter

The filtering function appears to be the ratio of two polynomials

The relative simplicity of the derivation of a filter's response, under modulus constraints, owes much to the possibility of forgetting about Feldtkeller's equation and express all design constraints in terms of the filtering function. This no longer the case when considering the synthesis

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

In this connection, there are two types of stability involved. The first is stability of a fixed point around which the linearized transfer function accounts for small signal amplification. The second is stability of a limit cycle which is reached when the input signal is no longer small and truly nonlinear amplification is attained (e.g. because of saturation). Applications by the team so far have been concerned with the first type of stability, and emphasis is put on defining and extracting the “unstable part” of the response, see Section . The stability check for limit cycles has made important theoretical advances (see , ), and numerical algorithms are now under investigation.

This is a recent activity of the team, linked to image classification in archaeology in the framework of the projects ToMaT and Arch-AI-Story, see Section ; it is pursued in collaboration with L. Blanc-Féraud (project-team Morpheme, I3S-CNRS/Inria Sophia/iBV), D. Binder (CEPAM-CNRS, Nice), in particular.

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

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

In a first step, we devised a method to isolate pores from the 3-D data volumes in binary 3-D images, to which we apply a process named Hough transform (derived from Radon transform). This method, of which the generalization from 2-D to 3-D is quite recent, allows us to evaluate the presence of parallel lines going through the pores. The quantity of such lines and their parallelism furnish good indicators of the “coiling” manufacturing, that they allow to distinguish from the other “spiral patchwork” technique, in particular. These progresses are described in .

Other possibilities of investigation are being analyzed as well, such as machine learning and deep learning techniques.

We continued in 2021 the work engaged in 2020 in coordination with Céline Serrano (in charge of “Sustainable development” at the national level) on setting up methods to evaluate the carbon footprint of research activities at Inria.

First, we discovered and corrected a few minor mistakes in the evaluation that we had made of the 2019 carbon footprint of team Factas. Together with people in charge of monitoring the electricity consumption in the research center, we spotted inconsistencies in the meter readings and improved their reliability. Questions still remain, that cannot be answered yet, due to the very particular case of 2020 (two lock-downs) and of the corresponding data. We hope that 2021 data (coming soon) will be more useful on this respect.

Second, we wrote a complete report, explaining in details both the methodology that we used, and the figures obtained for the carbon footprint of the team Factas in 2019, see . We improved the software tool that we developed for that purpose (see ), and wrote the necessary documentation to make it usable by other teams.

Third, we volunteered to do the carbon footprint of the research center with the . It accounts for the greenhouse gases emissions due to mobility (commute travels and missions), electricity and gas consumption in the buildings. We are still analyzing the results to understand where the efforts should be put to reduce the emissions of the research center in the future. Some preliminary figures are available on (access restricted to the Inria internal network). Overall, more than 80% of the emissions are due to the mobility.

Let us describe new/updated software.

The goal is to invert magnetizations carried by a planar set in Euclidean space from measurements of the magnetic field nearby. A typical application is to paleomagnetism, to determine magnetic properties of rock samples, shaped into thin slabs, with measurements taken by a superconducting quantum interference device (SQUID). Figure sketches the corresponding experimental set up, brought up to our knowledge by collaborators from the Earth and Planetary Sciences Laboratory at MIT.

We pursued our research program on the recovery of
magnetizations modeled by signed measures on thin samples, which is an instance of Poisson inverse problem with right hand side in divergence form
(the divergence of a

We focused this year on consistent discretization schemes, meaning
discrete versions of the continuous regularized problem whose solutions converge to the solution of this continuous regularized problem.
We were able to show that, for a fairly large family of discretizations based on weak-star dense collections of mutually singular measures, the solutions to the
regularized discretized problem are asymptotically supported on a system of
analytic arcs that contains the solution to the continuous problem,
regularized with the same value of the regularizing parameter. We are currently analyzing the significance of this system of arcs, in connection with the maximizing function of a
certain extremal problem for the measure generating the data when it exists; i.e., when we assume that the data are exact. This extremal problem, that recurs in inverse magnetization problems for planar samples, consists in best approximation by a divergence-free measure; such measures form
the kernel of the forward operator.
An implementation of a variant of the FISTA algorithm
is used to solve the discrete regularized problem when the discrete model
consists of dipoles or else of uniformly magnetized squares,
in order to substantiate our results in this case.
This is collaborative research with D. Hardin from Vanderbilt university
in the course of the postdoctoral program of C. Villalobos Guillén.

In 3-D, functional or clinically active regions in the cortex are often modeled by pointwise sources that have to be localized from measurements, taken by electrodes on the scalp, of an electrical potential satisfying a Laplace equation (EEG, electroencephalography). In the works , on the behavior of poles in best rational approximants of fixed degree to functions with branch points, it was shown how to proceed via best rational approximation on a sequence of 2-D disks cut along the inner sphere, for the case where there are finitely many sources (see Section ).

In this connection, a dedicated software FindSources3D (FS3D, see Section ) is being developed, in collaboration with the Inria team Athena 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 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.

From synthetic data simulated with , that consist in two asynchronous source patches (in the visual cortex), FS3D furnishes the results shown in Figure where they are mapped in a realistic head.

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 , 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).

FS3D 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 recently began to handle 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.

Another aspect is the influence of the time dependency within the PDE on the behavior of the solution, which we study with I. Stratis and A. Yannacopoulos from National & Kapodistrian Univ., Athens. It seems that the time derivative of the electric field is not so small within the brain, compared to that of the magnetic field in particular, whence the quasi-static assumption may not be valid.

We also studied the uniqueness of the critical point of the quadratic criterion in the electroencephalography problem for a single dipole situation (PhD of P. Asensio). This issue is essential for the use of descent algorithms. This leads to the study of the following criterion:

where i.e., a unique point

We started considering a different class of 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 . The approach follows that of and is implemented through two different algorithms, whose convergence properties are currently being studied. Tests on synthetic data from a few dipolar sources provide results of different qualities that need to be better understood. In particular, a weight is being added in the TV term in order to better identify deep sources. This is the topic of the PhD researches of P. Asensio and M. Nemaire. Ultimately, the results will be compared to those of FS3D and other available software tools.

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 . Assuming that the
current source term

The associated forward and inverse problems were solved for both an infinite medium conductor and a more realistic single model of the brain

The numerical implementation was done by approximating the density

The inverse problem for SEEG is ill-posed and a Tychonov regularization is used in order to solve the problem: find a distribution

We now consider

We are now able to handle MEG, EEG, SEEG modalities, simultaneously or not.

For more general source terms (vector valued measures,

On the topic of uniform matching, Gibin Bose defended his thesis «Approximation th 2021. Triggered by remarks of one of the reviewers the PUMA code was extended to treat the presence of one transmission zero at infinity in the antenna's response and compare results obtained by our code with the classical Fano bound.

The Fano bound is very popular in the antenna community because of its simplicity. Under the realistic hypothesis that the load is totally reflective at high frequency, it states that: if

holds. Here

where

This work was pursued in collaboration with Ke-L. Wu and Yan Zhang of the Chinese University of Hong-Kong.

In microwave filter synthesis the dispersive nature of couplings between resonators is usually neglected, due to the narrow band hypothesis. We however showed with S.Amari in 2008 , that a linear dependency in frequency of the couplings could be added to the usual state space model up to the addition of descriptor form, and that filter synthesis was still possible for simple inline topologies. We showed at that time that, if properly controlled, dispersion could be used to enrich the designer possibilities and lead to more selective filter responses by generation of additional transmission zeros.

Last year we developed a completely new filter synthesis for cascaded topologies . The method starts with a factorization of the scattering matrix into blocks, according to a partition of the transmission zeros (see Figure ). In functional analytic terms, this step corresponds to a Potapov factorization. In a second step, each block gets realized as a circuit, and when cascaded these will constitute the whole filter. The procedure stands therefore at the cross-roads between Darlington synthesis and Kalman realization theory, unifying extracted pole techniques and coupling matrix synthesis.

Dwelling on this procedure, we showed that the circuital synthesis of elementary building blocks containing dispersive couplings is equivalent to a generalized Gram-Schmidt orthogonalization process with two scalar products involved. The method allows one for a very flexible, modular synthesis, where each functional block can be implemented according to a different technique. In particular, blocks including dispersive elements (even complex ones ) can be combined with classical triplets, quadruplets as well as with sections involving non-resonating nodes . The manufacturing took place at the Hong-Kong lab (see Figure ).

Recently, we studied in collaboration with P. Macchiarella from Politecnico Milano and S. Tamiazzo from Commscope, coupling topologies with an inline footprint that, unexpectedly, allow one to realize transmission zeros. We proved , , that although these topologies are defective (design parameters are missing to accommodate all possible responses), they can realize Chebyshev characteristics with a particular return loss or variations thereof. Such topologies are particularly suitable for filters to be stacked in recent base stations, that need to line up like sardines in order to save space.

This work was supported by the Contract Inria-Inoveos.

The team has considerable experience in this area, that goes back to the end of the 90's when, in collaboration with CNES and later Xlim, they were first to propose a computer-aided tuning technique for microwave filters (see , ). This year, we participated to a consortium funded by BPI France, led by Inoveos and involving robotic experts (Cisteme) and millimeter wave filter engineers from Xlim. The objective was to built a prototypical robot dedicated to the automatic tuning of microwave devices. Our software was adapted to cope with real time constraints and the massive measurement flow that is possible to perform in such a fully automated scenario. Circuit extractions need now less than 1 second to be performed. This version of Presto-HF (see section ) was transferred and gave rise to the first prototypical tuning robot of Inoveos, see Figure .

The goal is here to help design amplifiers and oscillators, in particular to detect instability at an early stage of the design. This is the subject of a joint research effort with J.-B. Pomet (from the McTao Inria project-team). Application to oscillator design methodologies is also studied in collaboration with Smain Amari from the Royal Military College of Canada (Kingston, Canada).

As opposed to Filters and Antennas, Amplifiers and Oscillators are active components that intrinsically entail non-linear functioning. The latter is due to the use of transistors governed by electric laws exhibiting saturation effects, and therefore inducing input/output characteristics that are no longer proportional to the magnitude of the input signal. Hence, they typically produce non-linear distortions. A central issue arising in the design of amplifiers is to assess stability. The latter may be understood around a functioning point when no input but noise is considered, or else around a periodic trajectory when an input signal at a specific frequency is applied. For oscillators, a precise estimation of their oscillating frequency is crucial during the design process. For devices operating at relatively low frequencies, time domain simulations perform satisfactorily to check stability. For complex microwave amplifiers and oscillators, the situation is however drastically different: the time step necessary to integrate the dynamical equations of the transmission lines (which behave like a simple electrical wire at low frequency) becomes so small that simulations are intractable in reasonable time. Moreover, most linear components of such circuits are known through their frequency response, and prior to any time domain simulation a preliminary, numerically delicate step is needed to obtain their impulse response.

For these reasons, the analysis of such systems is carried out in the frequency domain. To study stability around a functioning point, small input signals are considered and the stability of the linearized system can be
investigated, using a first order approximation of each non-linear component,
via the transfer impedance functions computed at certain ports of the circuit. In recent years, we showed that under realistic
dissipativity assumptions at high frequency for the building blocks of the circuit, these transfer functions are meromorphic in the complex frequency variable

Extending this methodology to the strong signal case, where linearisation is considered around a periodic trajectory, is considerably more difficult and
has received much attention by the team in recent years.
When stability is understood around a periodic trajectory,
computed in practice by Harmonic Balance algorithms, linearization yields a linear time varying dynamical system with periodic coefficients
and a periodic trajectory thereof. While in finite dimension the stability of such systems is well understood
via the Floquet theory, this is no longer the case in the present setting which is
infinite dimensional, due to the presence of delays. Dwelling on known constructions for delay systems
and adapting the resolvent approach for Volterra equations to the context of
functions of bounded variations, it was shown in the PhD thesis
of our former student
S. Fueyo's that, for
general circuits,
the monodromy operator of the linearized system along its periodic trajectory
is a compact perturbation of a “high frequency” circuit where
coils are replaced by short circuits and condensers by wires, so that there is no dynamics anymore but only transmission lines (modeled by delays)
and transistors / resistors. We showed in
that,
under a (realistic) assumption of passivity of components of the circuit at
arbitrary high frequency, such a circuit is exponentially stable,
in any

Altogether, what precedes shows that the system is unstable if and only if the harmonic transfer function has poles in the right half plane, and these must lie equally spaced on finitely many vertical lines in that half plane. In principle, this warrants the approach to instability detection set up in the linear case, by chasing unstable poles of the harmonic transfer function in the right half-plane. It should be remarked that, in general, singularities of the harmonic transfer function need not necessarily be singularities of its Fourier coefficients (the Fourier series can diverge even if all coefficients are very smooth), which is a completely new phenomenon never occurring for finite-dimensional systems. However, in the present case, we know we are dealing with polar singularities (in the right half-plane), and at least one of these Fourier coefficients must have a pole. But since harmonic balance techniques can only estimate finitely many Fourier coefficients, one is not guaranteed to get a singular holomorphic function among them in the neighborhood of the pole under examination. This issue was apparently never considered by practitioners and is now under consideration by the team.

We have also been working on a direct characterization of silent

Besides, an analysis of harmonic equivalent sources, and a uniqueness result (injectivity of the forward operator) for piecewise constant magnetizations on parallelepipeds was obtained in .

In another connection, a version of the Hardy-Hodge decomposition is being developed for the Helmholtz equation, in collaboration with H. Haddar from the team DEFI (Inria Saclay); here, there is an extra 1-dimensional summand whose nonzero elements are non-silent from inside and outside.

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 subscripts

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

where

In order to perform the rational approximation of the function

where

where the coefficients

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

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

We had a contract with the SMB company Inoveos in order to build a prototypical robot dedicated to the automatic tuning of microwave devices. In addition to Inria, this project included the university of Limoges Xlim and the engineering center .

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.

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 supports the PhD of M. Nemaire within Factas, co-advised by IMB partners.

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.

The collaborative project Arch-AI-Story funded by the Idex