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2.1 Research Themes

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

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

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

• identification and synthesis of analog microwave devices (filters, amplifiers),
• non-destructive control from field measurements in medical engineering (source recovery in magneto/electro encephalography), and paleomagnetism (determining the magnetization of rock samples).

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

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 freezes1 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.

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 a point $i\omega$ of the imaginary axis from the response to a periodic input at frequency $\omega$. Since $f$ is holomorphic in the right half-plane, it satisfies there the Cauchy-Riemann equation $\overline{\partial }f=0$, 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 $\overline{\partial }f={\sum }_1^n{a}_j{\delta }_{z_j}$ where the $z_j$ are its poles and ${\delta }_{z_j}$ 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 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  86 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  39.

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  82, 64. 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 3.2.2 and 7.2.

The previous example of harmonic identification quickly suggests a generalization of itself. Indeed, on identifying $ℂ$ with ${ℝ}^2$, 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 2  37,  90. Such questions are particular instances of the so-called inverse potential problem, where a measure $\mu$ 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 $\mu$. 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 ${ℝ}^n$ 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 ${\sum }_1^N{a}_j{\delta }_{z_j}$). The discreteness assumption also prevails in 3-D inverse source problems, see Section 4.3. 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 2,  46, 50. 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 7.7). Once again, our approach combines steps 1 and 2 above.

3.2 Range of inverse problems

3.3 Approximation

Participants: Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Martine Olivi, Fabien Seyfert.

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

4.1 Introduction

Application domains are naturally linked to the problems described in Sections 3.2.1 and 3.2.2. 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 3.3.

4.2 Inverse magnetization problems

Participants: Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Anass Yousfi.

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 4.3 and inverse magnetization 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:

outside the volume $\Omega$ of the object. Depending on the geometry of models, the magnetization distribution $m$ may lie in a volume or spread out on a surface. This results in quite different identifiability properties, see  42 and Section 7.1.1, 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 7.1.1.

4.3 Inverse source problems in EEG

Participants: Paul Asensio, Laurent Baratchart, Juliette Leblond, Jean-Paul Marmorat, 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 8. 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 6,  47. 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.

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 7.1.2).

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 4.2 and 7.1.1), and it presently focuses on surface-distributed models.

4.4 Identification and design of microwave devices

Participants: Laurent Baratchart, Sylvain Chevillard, Jean-Paul Marmorat, Martine Olivi, 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, 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).

Show image description

Picture of a 6-cavities dual mode filter. Six cylindrical metallic blocks are chained (the cavities). On the five first cavities, three long screws are visible; on the last one, only one screw is visible.

Each cavity (see Figure 1) has three screws, horizontal, vertical and midway (horizontal and vertical are two arbitrary directions, the third direction makes an angle of 45 or 135 degrees, the easy case is when all cavities show the same orientation, and when the directions of the irises are the same, as well as the input and output slits). Since screws are conductors, they behave as capacitors; besides, the electrical field on the surface has to be zero, which modifies the boundary conditions of one of the two modes (for the other mode, the electrical field is zero hence it is not influenced by the screw), the third screw acts as a coupling between the two modes. The effect of an iris is opposite to that of a screw: no condition is imposed on a hole, which results in a coupling between two horizontal (or two vertical) modes of adjacent cavities (in fact the iris is the union of two rectangles, the important parameter being their width). The design of a filter consists in finding the size of each cavity, and the width of each iris. Subsequently, the filter can be constructed and tuned by adjusting the screws. Finally, the screws are glued once a satisfactory response has been obtained. In what follows, we shall consider a typical example, a filter designed by the CNES in Toulouse, with four cavities near 11 GHz.

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 $S$, which may be considered as the transfer function of a stable causal linear dynamical system, with two inputs and two outputs. Its diagonal terms $S_{1,1}$, $S_{2,2}$ correspond to reflections at each port, while $S_{1,2}$, $S_{2,1}$ correspond to transmission. These functions can be measured at certain frequencies (on the imaginary axis). The matrix $S$ is approximately rational of order 4 times the number of cavities (that is 16 in the example on Figure 2), and the key step consists in expressing the components of the equivalent electrical circuit as functions of the $S_{ij}$ (since there are no formulas expressing the lengths of the screws in terms of parameters of this electrical model). This representation is also useful to analyze the numerical simulations of the Maxwell equations, and to check the quality of a design, in particular the absence of higher resonant modes.

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

In short, the strategy for identification is as follows:

• measuring the scattering matrix of the filter near the optimal frequency over twice the pass band (which is 80MHz in the example).
• Solving bounded extremal problems for the transmission and the reflection (the modulus of he response being respectively close to 0 and 1 outside the interval measurement, cf. Section 3.3.1) in order to get a models for the scattering matrix as an analytic matrix-valued function. This provides us with a scattering matrix known to be close to a rational matrix of order roughly 1/4 of the number of data points.
• Approximating this scattering matrix by a true rational transfer-function of appropriate degree (8 in this example) via the Endymion or RARL2 software (cf. Section 3.3.2).
• A state space realization of $S$, viewed as a transfer function, can then be obtained, where additional symmetry constraints coming from the reciprocity law and possibly other physical features of the device have to be imposed.
• Finally one builds a realization of the approximant and looks for a change of variables that eliminates non-physical couplings. This is obtained by using algebraic-solvers and continuation algorithms on the group of orthogonal complex matrices (symmetry forces this type of transformation).

Show image description

Nyquist Diagram of S22S_{22}. A red curve entitled “rational approximation” fits very well sampling points entitled “scattering data of filter” showed in blue.

The final approximation is of high quality. This can be interpreted as a confirmation of the linearity assumption on the system: the relative $L^2$ error is less than ${10}^{-3}$. This is illustrated by a reflection diagram (Figure 2).

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  80. 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 $|S_{1,2}|$ admits a simple expression in terms of the filtering function $D=|S_{1,1}|/|S_{1,2}|$, namely

The filtering function appears to be the ratio of two polynomials $p_1/p_2$, the numerator of the reflection and transmission scattering factors, that may be chosen freely. The denominator $q$ is then obtained as the unique stable unitary polynomial solving the classical Feldtkeller spectral equation:

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 $N$-port devices for $N>3$, like multiplexers, routers and power dividers, or when considering the synthesis of filters under matching conditions. The efficient derivation of multiplexers responses is the subject of recent investigation by Factas, using techniques based on constrained Nevanlinna-Pick interpolation (see Section 7.2).

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  89. 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 7.5. The stability check for limit cycles has made important theoretical advances (see 14, 28), and numerical algorithms are now under investigation.

4.5 Imaging and modeling ancient materials

Participants: Vanna Lisa Coli, Juliette Leblond.

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 9.3; 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 ($\mu$CT) has recently been used for exploring ancient materials micro-structure. This non-invasive method provides quantitative data for a big set of proxies and is perfectly adapted to the analysis of Cultural heritage materials.

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 18.

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

5.1 Footprint of research activities

Participants: Sylvain Chevillard, Martine Olivi.

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 29. We improved the software tool that we developed for that purpose (see 6.1.1), 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 GES1p5 methodology. 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 the website of the local “sustainable development” commission (access restricted to the Inria internal network). Overall, more than 80% of the emissions are due to the mobility.

6.1 New software

7.1 Inverse problems for Poisson-Laplace equations

Participants: Paul Asensio, Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Jean-Paul Marmorat, Masimba Nemaire, Cristóbal Villalobos Guillén, Anass Yousfi.

7.1.1 Inverse magnetization issues for planar samples

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 3 sketches the corresponding experimental set up, brought up to our knowledge by collaborators from the Earth and Planetary Sciences Laboratory at MIT.

Show image description

Schematic view of the experimental setup : the horizontal plane on which the rock sample lies (the sample being a parallelepiped with height rr and a square basis of half-size ss. Its basis is at height 0, and the center of the square basis is at the origin of the system of coordinates). Above, the measurement square lies on a parallel plane at height zz. Its half-size is RR and it it is horizontally centered.

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 ${ℝ}^3$-valued measure supported on a planar set in space). We showed in previous years that consistent recovery is possible, in the Morozov discrepancy limit, by penalizing the total variation, when either the support of the magnetization is purely 1-unrectifiable (which holds in particular for dipolar models) or when the magnetization is unidirectional (an assumption of physical interest: indeed, when igneous rocks cool down in some ambient field, they acquire a magnetization whose intensity may vary internally depending on the composition of the rock, but that is mainly oriented in the same direction as the ambient field), or else when it is supported on sufficiently separated line segments. These notions play a role similar to sparsity in this infinite-dimensional context. We also established that the argument of the minimum of the regularized criterion is unique, by taking advantage of the particular form of the critical point equation for the criterion, together with a loop decomposition of planar divergence free measures which sharpens in the 2-D setting the structure theorem of 87. An article has been published on these results 17.

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.

7.1.2 Inverse problems in medical imaging

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 6, 47 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 4.3).

In this connection, a dedicated software FindSources3D (FS3D, see Section 3.4.3) 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 MNE, that consist in two asynchronous source patches (in the visual cortex), FS3D furnishes the results shown in Figure 4 where they are mapped in a realistic head.

Show image description

A schematic image of the brain inside a head and a figure showing the graph of the potentials generated by two sources over time.

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 8, 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:

$${\displaystyle J(X,p)={∥\frac{p·\left(X-Y\right)}{{|X-Y|}^3}-\frac{p_0·\left(X_0-Y\right)}{|X_0{-Y|}^3}∥}_{L^2(Y,\partial \Omega )}^2,}$$

where $X$ and $X_0$ (actual dipole position) are in a smooth open set $\Omega$ included in ${ℝ}^3$ and $p$, $p_0$ (actual dipole moment) belong to ${ℝ}^3$. In the half-space case where $\Omega =\{(x,y,z)\in {ℝ}^3,z<0\}$, we have proven that there is a unique critical point; i.e., a unique point $X$ where the gradient vanishes, which is but $X_0$ (the absolute minimum). We currently try to obtain an analogous result on the ball, which is considerably more difficult because no translation invariance holds anymore.

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 7.1.1. The approach follows that of 7 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.

Show image description

Three nested ovoids illustrating the three layers of the head (denoted Ωi\Omega _i with i=0,1,2i=0,1,2, 0 being the index of the inner layer). The conductivity of the ii-th layer is denoted with σi\sigma _i and its boundary is denoted with Γi\Gamma _i. Two red straight lines illustrate how deep electrodes take measurements in the inner layer. Little segments perpendicular to each electrode illustrate the different sensors that lie along the electrode. A blue closed curve, denoted SS, in the inner layer illustrates the surface where the sources lie, and little arrows show that their orientation is normal to the curve.

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 5. Assuming that the current source term $\mu$ is a ${ℝ}^3$-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 $\varphi \left(\mu \right)$ generated by $\mu$ as a double layer potential:

$${\displaystyle \varphi \left(\mu \right)\left(x\right)=-\frac{1}{4\pi }{\int }_S\mu ·\frac{x-y}{{|x-y|}^3}dy,x\notin S.}$$

The associated forward and inverse problems were solved for both an infinite medium conductor and a more realistic single model of the brain ${\Omega }_0$ (as first step towards nested domains for the head). Concerning the latter, it is preliminary assumed that $S\subset \partial {\Omega }_0$, whence $\mu$ consists in patches on the cortex surface.

The numerical implementation was done by approximating the density $\mu$ of the double layer potential with linear shape functions on triangulations of the involved surface $S$. For the discretized version, $\mu$ 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 $\mu$).

The inverse problem for SEEG is ill-posed and a Tychonov regularization is used in order to solve the problem: find a distribution $\mu$ defined on the prescribed surface $S$ such that $\parallel \varphi \left(\mu \right)-{\varphi }^d{\parallel }_{L^2\left({\Omega }_0\right)}+\lambda R\left(\mu \right)$ is minimized, where ${\varphi }^d$ is the given data in ${\Omega }_0$, $\varphi \left(\mu \right)$ is the “forward model” (measurements of the double layer potential), $R\left(\mu \right)$ is an appropriate regularization term that results in $\mu$ having some desired properties (being bounded in $L^2$ norm or $TV$ or $L^1$ on $S$), and $\lambda$ a (Lagrange) parameter. Because ${\varphi }^d$ is only furnished at sensor positions, the $L^2$ term in the above criterion becomes an $l^2$ term. Figures 6 and 7 show the obtained results.

We now consider $S\subset {\Omega }_0$ within the brain, at the gray / white matters interface and we handle outer layers ${\Omega }_1$ and ${\Omega }_2$ for the skull and the scalp, as illustrated in Figure 5. Such configurations involve a data transmission / “cortical mapping” step, that relates the potential and normal current values at the boundaries ${\Gamma }_i$, that the present double layer formulation allows to solve with the source localization issue.

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

For more general source terms (vector valued measures, $L^p$ 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.

7.2 Matching problems and their applications: Finite degree bounds

Participants: Laurent Baratchart, Martine Olivi, Gibin Bose, Fabien Seyfert.

On the topic of uniform matching, Gibin Bose defended his thesis «Approximation $H^{\infty }$, Interpolation Analytique et Optimisation Convexe : Application à l'Adaptation d'Impédance Large Bande» on January 8th 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 $L_{11}$ is the reflection coefficient of the load, and $S_{11}$ the reflection coefficient of the global system (i.e. Matching Circuit + Load), then:

holds. Here $An{g}_{\infty }(.)$ stays for the angular derivative at infinity and $K=20\pi /log\left(10\right)$ is a constant. In other words the log modulus integral of the global system is bounded by the angular derivative of the load. Eventually, if we assume ideally that the global system's response is a step function centered on the pass-band $B$ (which is only possible with an infinite dimensional matching circuit), we get the following bound on the maximal matching log-level $R_L$ of the global system:

where $\left|B\right|$ stands for the length of the pass-band. $F_B$ is called the Fano bound, called after the name of its inventor, Fano, a PhD student of Bode. For a single resonance load, this band is sharp, provided an infinite dimensional matching circuit is at hand. For more complex loads, the Fano bound might be utterly optimistic.

We treated extensively the case of an antenna with a reflection coefficient of degree 3. On Figure 8 an absolute bound for a degree 2 matching circuit has been computed, leading to the depicted reflection coefficient $S_{22}$ yielding a matching level of $9.07\text{dB}$ in the prescribed band. This must be compared here to the $25.92\text{dB}$ of the Fano bound, which appears to be largely over optimistic. This discrepancy between the theoretical Fano bound and the bound computed by our method has two reasons. The antenna is of degree 3 and has in addition to its zero at infinity two additional finite transmission zeros. This deviation from a single resonance behavior is not taken into account by the classical Fano bound. Secondly as already mentioned the Fano bound supposes implicitly a step function shape of the system's response that can hardly be achieved in practice with a finite degree matching circuit. This underlines the necessity to obtain accurate finite degree bounds, as the ones obtained by the Puma software. These render effective a convex relaxation of the finite degree matching problem based on non-linear semi-definite programming and the notion of Pick matrix. A publication on this topic is under way.

Show image description

Graph of the magnitude (in dB) of the three coefficients L11L_{11}, S22S_{22} and F22F_{22} with respect to a normalized frequency ranging from -4-4 to 4. A gray vertical band shows the prescribed band. S22S_{22} is below the level of -9.07dB-9.07\text{dB} in the whole band, where it oscillates and reaches the -9.07dB-9.07\text{dB} level six times. The Fano Bound at -25.92dB-25.92\text{dB} is shown and F22F_{22} has a peak that goes below it around the frequency -0.2-0.2.

7.3 Modular filter synthesis with dispersive elements

Participants: Smain Amari, Fabien Seyfert.

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  33, 32 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 12. The method starts with a factorization of the scattering matrix into blocks, according to a partition of the transmission zeros (see Figure 9). 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.

 

Hardware realization of an order 6 filter in combline technology

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 26) can be combined with classical triplets, quadruplets as well as with sections involving non-resonating nodes 24. The manufacturing took place at the Hong-Kong lab (see Figure 9).

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 20, 21, 25 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.

7.4 Automatic tuning of microwave devices

Participants: Sylvain Chevillard, Martine Olivi, Fabien Seyfert.

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 86, 63). 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 3.4.4) was transferred and gave rise to the first prototypical tuning robot of Inoveos, see Figure 10.

Show image description

A photography of the tuning robot.

7.5 Stability assessment of microwave circuits

Participants: Laurent Baratchart, Sylvain Chevillard, Martine Olivi, Fabien Seyfert, Sébastien Fueyo, Adam Cooman.

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 $s$, with at most finitely many unstable poles in the right half-plane 13. Dwelling on the unstable/stable decomposition in Hardy Spaces, we developed a procedure to assess the stability or instability of the transfer functions at hand, from their evaluation on a finite frequency grid 10, that was further improved in 9 to address the design of oscillators, in collaboration with Smain Amari. This has resulted in the development of a software library called Pisa (see Section 3.4.1), aiming at making these techniques available to practitioners. Research in this direction now focuses on the links between the width of the measurement band, the density of the measurement points, and the precision with which an unstable pole, located within a certain depth into the complex plane, can be identified.

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 70 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 14 that, under a (realistic) assumption of passivity of components of the circuit at arbitrary high frequency, such a circuit is exponentially stable, in any $L^p$ sense, $1\le p\le \infty$. Consequently, since compact perturbation preserve the essential spectrum, the linearized system along the periodic trajectory has at most finitely many unstable points in the spectrum of its monodromy operator. Now, using tools from function theory (notably on almost periodic analytic functions), realization theory and periodic evolution families, we proved in the recent manuscript 28 that the spectrum of the monodromy operator is the exponential of the singularities of the so-called harmonic transfer function, viewed as a holomorphic function with values in periodic $L^2$ functions. This result may be seen as a generalization to the case of periodic coefficients of the well-known Henry-Hale theorem characterizing the exponential stability of delay systems with constant coefficients.

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.

7.6 Hardy-Hodge decomposition with applications to silent sources

Participants: Laurent Baratchart, Juliette Leblond, Masimba Nemaire.

In a joint work with T. Qian and P. Dang from the university of Macao 54, we proved that on a compact hypersurface $\Sigma$ embedded in ${ℝ}^n$, a ${ℝ}^n$-valued vector field of $L^p$ class decomposes as the sum of a harmonic gradient from inside $\Sigma$, a harmonic gradient from outside $\Sigma$, and a tangent divergence-free field, provided that $2-\epsilon <p<2+{\epsilon }^{\text{'}}$, where $\epsilon$ and ${\epsilon }^{\text{'}}$ depend on the Lipschitz constant of the surface. We also showed that the decomposition is valid for $1<p<\infty$ when $\Sigma$ is $VMO$-smooth (i.e.$\Sigma$ is locally the graph of Lipschitz function with derivatives in $VMO$). 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 15, we have showed that $L^2$-magnetizations (i.e. ${ℝ}^n$-valued vector fields) on a compact connected Lipschitz surface embedded in ${ℝ}^n$ 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 $L^p$-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 has been a collaboration with C. Gerhards and A. Kegeles from TU Freiberg and can be used use to characterize, via balayage, silent magnetizations in a volume rather than on a surface.

We have also been working on a direct characterization of silent ${\left[L^p\left(\Omega \right)\right]}^n$, ($n\ge 3$) vector-fields. We showed that a vector field carried by some finitely connected bounded open set $\Omega$ is silent outside $\Omega$ if and only if, in its Helmholtz decomposition as the sum of a $L^p$ gradient and a divergence-free $L^p$ field over ${ℝ}^n$, both summands vanish a.e. outside of $\Omega$. When $\Omega$ is Lipschitz, this entails that silent vector-fields $M\in {\left[L^p\left(\Omega \right)\right]}^n$ (i.e. vector fields representing magnetizations that produce the zero magnetic field outside $\Omega$) are of the form $M=\nabla \psi +D$ with $\psi \in W_0^{1,p}\left(\Omega \right)$ (the familiar space of Sobolev functions with zero trace) and $D$ divergence-free with zero normal component on the boundary $\partial \Omega$. In fact, the result extends to very general open sets if one uses a potential-theoretic definition of the trace (the mean at Lebesgue points, defined Bessel-$(1,p)$ quasi everywhere). It should be noticed that this holds for $1<p<\infty$, even though the Helmholtz decomposition on $\Omega$ may not hold for every $L^p$-field (in general, for Lipschitz open sets, it only holds for $p$ in a neighborhood of $[2/3,3]$). The previous characterization makes it possible to describe the norm-minimizing equivalent vector-field to a given vector-field $M\in {\left[L^p\left(\Omega \right)\right]}^n$. Here, two vector-fields are equivalent if their magnetic potentials outside $\Omega$ differ by a constant, or equivalently if the vector-fields differ by a silent source. To this effect, we exploited the fact that the norm-minimizing equivalent source can be written as ${\langle M,\nabla \varphi \rangle \left|\nabla \varphi \right|}^{q-2}\nabla \varphi$ with $\varphi$ a harmonic function in $W^{1,q}\left(\Omega \right)$ with $\nabla \varphi$ orthogonal to all silent magnetizations in ${\left[L^p\left(\Omega \right)\right]}^n$. This yields in passing a nonlinear Hodge-like decomposition, valid for all $1<p<\infty$ on arbitrary bounded open sets, containing an extra $p$-divergent-free term. Moreover, $\nabla \varphi$ maximizes $\langle M,\nabla \varphi \rangle$ among all harmonic gradients orthogonal to silent sources in ${\left[L^p\left(\Omega \right)\right]}^n$ with $L^q$-norm 1. In practice, when $\Omega$ is Lipschitz, one can further use that $\varphi$ can be expressed as a single layer potential, say $𝒮f^{*}$ with $f^{*}$ maximizing the duality pairing $\langle P_M\left(y\right),f\left(y\right)\rangle$ on $\partial \Omega$ (here $P_M$ is the Newton potential of the (distributional) divergence of $M$) under appropriate constraints. Then, a Galerkin-type method can be used to approach the problem numerically, using a frame of the space of distributions $W^{\frac{-1}{q},q}\left(\partial \Omega \right)$ that can be constructed as in 77. Alternatively, in the case where $n=3$ and $\Omega$ is a $C^{1,1}$ simply connected domain, a minimization procedure that alternates between approximating $M\in {\left[W^{1,p}\left(\Omega \right)\right]}^3$ 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, it would be very interesting to derive counterparts of these results to the case $p=1$, on smooth domains at least.

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 16.

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.

7.7 Identification of resonating frequencies of compact metallic objects in electromagnetic inverse scattering

Participants: Laurent Baratchart, Juliette Leblond, Martine Olivi, Fabien Seyfert.

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.

Show image description

A schematic view of the problem: an antenna (horizontal and on the left) emits a planar wave (propagating horizontally, towards the right and shown in red) with an emitted electric field EE perpendicular to it (represented by a vertical arrow on the figure). On the right, a sphere of radius aa reflects the wave. The reflected wave is radial to the sphere, directed towards its exterior, and is shown in blue. A reception antenna points towards the center of the sphere, and is located at a distance rr of the center of the sphere, while making an angle θ\theta with respect to the horizontal. It measures the reflected electric field, which is perpendicular to it.

The setting is shown on Figure 11. The total field $E_t(r,\theta ,\varphi )$ is the sum of the incident field $E_i$ (here a plane wave) and scattered field $E_s$, that is at every point in space we have $E_t=E_i+E_s$. A harmonic time dependency ($e^{j\omega t}$, 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, following holds,

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 $H(s=j\omega ,{\theta }_o,{\varphi }_o)$. 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 a 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 $s=j{\omega }_i^{\text{'}}s$ 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:

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 $R$ once it has been de-embedded. A preliminary study resulted in a common publication with the Leat team 91.

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:

where $H_O$ and $H_C$ are respectively the optic and creeping wave parts. Their high-frequency behaviors are given by:

where the coefficients $B_0\in ℝ$, $B_1\in ℝ$, $B_2\in {ℝ}^{+}$ and $A_n\in ℝ$ are identified when the observation point $(r_o,{\theta }_o,{\varphi }_o)$ and the emission point $(r_e,{\theta }_e,{\varphi }_e)$ coincide. The above asymptotics of $H_O$ 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 $H_C$ hence that $H_O$ is analytic.

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

8.1 Bilateral Contracts with Industry

9.1 European initiatives

9.2 National initiatives

9.3 Regional initiatives

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

10.3 Popularization

11.1 Major publications

• 1 articleS.Smain Amari, F.Fabien Seyfert and M.M. Bekheit. Theory of Coupled Resonator Microwave Bandpass Filters of Arbitrary Bandwidth.Microwave Theory and Techniques, IEEE Transactions on588August 2010, 2188--2203
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• 2 articleB.Bilal Atfeh, L.Laurent Baratchart, J.Juliette Leblond and J. R.Jonathan R. Partington. Bounded extremal and Cauchy-Laplace problems on the sphere and shell.J. Fourier Anal. Appl.162Published online Nov. 20092010, 177--203URL: http://dx.doi.org/10.1007/s00041-009-9110-0
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• 3 articleL.Laurent Baratchart, S.Sylvain Chevillard and T.Tao Qian. Minimax principle and lower bounds in $H^2$-rational approximation.Journal of Approximation Theory2062015, 17--47
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• 4 articleL.Laurent Baratchart, J.Juliette Leblond, S.Stéphane Rigat and E.Emmanuel Russ. Hardy spaces of the conjugate Beltrami equation.Journal of Functional Analysis25922010, 384-427URL: http://dx.doi.org/10.1016/j.jfa.2010.04.004
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• 5 articleL.Laurent Baratchart, M.Martine Olivi and F.Fabien Seyfert. Boundary nevanlinna-pick interpolation with prescribed peak points. Application to impedance matching.SIAM Journal on Mathematical Analysis2017
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• 6 articleL.Laurent Baratchart, H.Herbert Stahl and M.Maxim Yattselev. Weighted Extremal Domains and Best Rational Approximation.Advances in Mathematics2292012, 357-407URL: http://hal.inria.fr/hal-00665834
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• 7 articleL.Laurent Baratchart, C.Cristobal Villalobos Guillén, D.Doug Hardin, M.Michael Northington and E.Edward Saff. Inverse Potential Problems for Divergence of Measures with Total Variation Regularization.Foundations of Computational MathematicsNovember 2019
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• 8 articleM.Maureen Clerc, J.Juliette Leblond, J.-P.Jean-Paul Marmorat and T.Théodore Papadopoulo. Source localization using rational approximation on plane sections.Inverse Problems285May 2012, 24URL: http://hal.inria.fr/inria-00613644
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• 9 inproceedingsA.Adam Cooman, F.Fabien Seyfert and S.Smain Amari. Estimating unstable poles in simulations of microwave circuits.IMS 2018Philadelphia, United StatesJune 2018
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• 10 articleA.Adam Cooman, F.Fabien Seyfert, M.Martine Olivi, S.Sylvain Chevillard and L.Laurent Baratchart. Model-Free Closed-Loop Stability Analysis: A Linear Functional Approach.IEEE Transactions on Microwave Theory and Techniqueshttps://arxiv.org/abs/1610.032352017
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• 11 articleM.Martine Olivi, F.Fabien Seyfert and J.-P.Jean-Paul Marmorat. Identification of microwave filters by analytic and rational H2 approximation.Automatica492January 2013, 317-325URL: http://hal.inria.fr/hal-00753824
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• 12 articleY.Yan Zhang, F.Fabien Seyfert, S.Smain Amari, M.Martin Olivi and K.-L.KE-Li Wu. General Synthesis Method for Dispersively Coupled Resonator Filters With Cascaded Topologies.IEEE Transactions on Microwave Theory and TechniquesDecember 2020, 15
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11.2 Publications of the year

11.3 Cited publications

• 30 bookN.N.~I. Achieser. Elements of the Theory of Elliptic Functions.AMS1990
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• 31 articleD.D. Alpay, L.Laurent Baratchart and A.Andrea Gombani. On the Differential Structure of Matrix-Valued Rational Inner Functions.Operator Theory~: Advances and Applications731994, 30--66
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• 32 inproceedingsS.S. Amari, M.M. Bekheit and F.F. Seyfert. Notes on Bandpass Filters Whose Inter-Resonator Coupling Coefficients Are Linear Functions of Frequency.2008 IEEE MTT-S International Microwave Symposium DigestJune 2008, 1207-1210
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• 33 articleS.Smain Amari, F.Fabien Seyfert and M.M. Bekheit. Theory of Coupled Resonator Microwave Bandpass Filters of Arbitrary Bandwidth.IEEE Transactions on Microwave Theory and Techniques5882010, 2188 -2203
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• 34 bookJ.J.~A. Ball, I.I. Gohberg and L.L. Rodman. Interpolation of rational matrix functions.Birkhäuser1990
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• 35 articleL.Laurent Baratchart. A remark on uniqueness of best rational approximants of degree 1 in $L^2$ of the circle. Elec. Trans.on Numerical Anal.252006, 54--66
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• 36 articleL.Laurent Baratchart, A.Alexandre Borichev and S.Slah Chaabi. Pseudo-holomorphic functions at the critical exponent.Journal of the European Mathematical Society1892016
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• 37 articleL.Laurent Baratchart, L.Laurent Bourgeois and J.Juliette Leblond. Uniqueness results for inverse Robin problems with bounded coefficient.Journal of Functional Analysis2016
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• 38 articleL.Laurent Baratchart, M.M. Cardelli and M.Martine Olivi. Identification and rational $L^2$ approximation: a gradient algorithm.Automatica271991, 413--418
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• 39 articleL.Laurent Baratchart, M.M. Chyba and J.-B.Jean-Baptiste Pomet. A Grobman-Hartman theorem for control systems.J. Dyn. Differential Eqs.192007, 75-107
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• 40 articleL.Laurent Baratchart, Y.Yannick Fischer and J.Juliette Leblond. Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation.Complex Variables and Elliptic Equations2014, 41
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• 41 inproceedingsL.Laurent Baratchart, L.L. Golinskii and S.Stanislas Kupin. Orthogonal rational functions and nonstationary stochastic processes: a Szegő theory.Proc. 19th Symposium on Mathematical Theory of Networks and SystemsBudapest2010
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• 42 articleL.Laurent Baratchart, D. P.Douglas P. Hardin, E. A.Eduardo Andrade Lima, E.Edward~B. Saff and B.Benjamin Weiss. Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions.Inverse Problems2912013, URL: https://hal.inria.fr/hal-00919261
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• 43 articleL.Laurent Baratchart, R.Reinhold Kuestner and V.Vilmos Totik. Zero distributions via orthogonality.Annales de l'Institut Fourier5552005, 1455--1499
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• 44 articleL.Laurent Baratchart, S.Stanislas Kupin, V.Vincent Lunot and M.Martine Olivi. Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I.Journal d'Analyse1122011, 207-255URL: https://arxiv.org/abs/0812.2050v3
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• 45 articleL.Laurent Baratchart and J.Juliette Leblond. Hardy approximation to $L^p$ functions on subsets of the circle with $1p<$.Constructive Approximation141998, 41--56
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• 46 articleL.Laurent Baratchart, J.Juliette Leblond, F.Frédéric Mandréa and E.Edward~B. Saff. How can meromorphic approximation help to solve some 2D inverse problems for the Laplacian?Inverse Problems1511999, 79--90URL: https://dx.doi.org/10.1088/0266-5611/15/1/012
  back to textback to textback to text
• 47 articleL.Laurent Baratchart, J.Juliette Leblond and J.-P.Jean-Paul Marmorat. Sources identification in 3D balls using meromorphic approximation in 2D disks.Electronic Transactions on Numerical Analysis (ETNA)252006, 41--53
  back to textback to textback to text
• 48 articleL.Laurent Baratchart, J.Juliette Leblond and J.Jonathan~R. Partington. Hardy approximation to $L^{}$ functions on subsets of the circle.Constructive Approximation121996, 423--435
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• 49 articleL.Laurent Baratchart, J.Juliette Leblond and F.Fabien Seyfert. Constrained extremal problems in H2 and Carleman's formulas.Matematicheskii Sbornik20972018, 36
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• 50 articleL.Laurent Baratchart, F.Frédéric Mandréa, E.Edward~B. Saff and F.Franck Wielonsky. 2D inverse problems for the Laplacian: a meromorphic approximation approach.Journal de Math. Pures et Appliquées862008, 1-41
  back to textback to textback to textback to text
• 51 articleL.Laurent Baratchart and M.Martine Olivi. Critical points and error rank in best $H^2$ matrix rational approximation of fixed McMillan degree.Constructive Approximation141998, 273--300
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• 52 articleL.Laurent Baratchart and M.Martine Olivi. Index of critical points in $l^2$-approximation.System and Control Letters101988, 167--174
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• 53 inproceedingsL.Laurent Baratchart. On the $H^2$ Rational Approximation of Markov Matrix-Valued Functions.Proc. 17th Symposium on Mathematical Theory of Networks and Systems (MTNS)Kyoto, Japon2006, 180--182
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• 54 unpublishedL.Laurent Baratchart, D.Dang Pei and T.Tao Qian. Hardy-Hodge decomposition of vector fields on compact Lipschitz hypersurfaces., in preparation
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• 55 articleL.Laurent Baratchart, E.Edward~B. Saff and F.Franck Wielonsky. A criterion for uniqueness of a critical point in $H^2$ rational approximation.Journal d'Analyse701996, 225--266
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• 56 articleL.Laurent Baratchart and F.Fabien Seyfert. An $L^p$ analog to AAK theory for $p2$.Journal of Functional Analysis19112002, 52--122
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• 57 articleL.Laurent Baratchart, H.Herbert Stahl and F.Franck Wielonsky. Asymptotic uniqueness of best rational approximants of given degree to Markov functions in $L^2$ of the circle.Constr. Approx.1712001, 103--138
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• 58 incollectionL.Laurent Baratchart and M.Maxim Yattselev. Asymptotic uniqueness of best rational approximants to complex Cauchy transforms in $L^2$ of the circle.Recent trends in orthogonal polynomials and approximation theory507Contemp. Math.Providence, RIAmer. Math. Soc.2010, 87--111
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• 59 articleL.Laurent Baratchart and M.Maxim Yattselev. Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights.International Mathematics Research Notices2010222010, 4211--4275URL: https://hal.inria.fr/hal-00508314
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• 60 articleL.Laurent Baratchart and M.Maxim Yattselev. Convergent interpolation to Cauchy integrals over analytic arcs.Found. Comp. Math.962009, 675--715
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• 61 articleL.Laurent Baratchart and M.Maxim Yattselev. Meromorphic approximants for complex Cauchy transforms with polar singularities.Mat. Sbornik20092009, 3-40
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• 62 articleA.Amel Ben~Abda, F.Fehmi Ben~Hassen, J.Juliette Leblond and M.Moncef Mahjoub. Sources recovery from boundary data: a model related to electroencephalography.Mathematical and Computer Modelling4911--122009, 2213--2223URL: https://dx.doi.org/10.1016/j.mcm.2008.07.016
  back to textback to text
• 63 articleR.R.J. Cameron, J.-C.Jean-Charles Faugère, F.Fabrice Rouillier and F.Fabien Seyfert. Exhaustive approach to the coupling matrix synthesis problem and application to the design of high degree asymmetric filters.International Journal of RF and Microwave Computer-Aided Engineering1712007, 4--12URL: https://hal.inria.fr/hal-00663777
  back to textback to text
• 64 articleR.R.J. Cameron, A.A.R. Harish and C.C.J. Radcliffe. Synthesis of advanced microwave filters without diagonal cross-couplings.IEEE Transactions on Microwave Theory and Techniques5012dec 2002, 2862--2872URL: https://dx.doi.org/10.1109/TMTT.2002.805141
  back to text
• 65 articleS.Slim Chaabane, I.I. Fellah, M.Mohamed Jaoua and J.Juliette Leblond. Logarithmic stability estimates for a Robin coefficient in 2D Laplace inverse problems.Inverse Problems2012004, 49--57URL: https://dx.doi.org/10.1088/0266-5611/20/1/003
  back to text
• 66 phdthesisS.Slah Chaabi. Analyse complexe et problèmes de Dirichlet dans le plan : équation de Weinstein et autres conductivités non bornées.Mathématiques et Informatique de Marseille2013
  back to text
• 67 articleR.R.R. Coifman, R.R. Rochberg and G.G. Weiss. Factorization theorems for Hardy spaces in several variables.Ann. Math.1031976, 611--635
  back to text
• 68 phdthesisY.Yannick Fischer. Approximation des des classes de fonctions analytiques généralisées et résolution de problèmes inverses pour les tokamaks.Univ. Nice Sophia Antipolis2011, URL: https://tel.archives-ouvertes.fr/tel-00643239/
  back to textback to text
• 69 articleA.A. Friedman and M.M. Vogelius. Determining cracks by boundary measurements.Indiana Univ. Math. J.3831989, 527--556
  back to text
• 70 phdthesisS.Sébastien Fueyo. Time-varying delay systems and 1-D hyperbolic equations, Harmonic transfer function and nonlinear electric circuits.Université Cote d'AzurOctober 2020, URL: https://hal.archives-ouvertes.fr/tel-03105344
  back to text
• 71 articleP.P. Fulcheri and M.Martine Olivi. Matrix rational $H^2$-approximation: a gradient algorithm based on Schur analysis.SIAM J. on Control & Optim.361998, 2103--2127
  back to text
• 72 bookJ.J.~B. Garnett. Bounded analytic functions.Academic Press1981
  back to text
• 73 articleT.T. Georgiou. A topological approach to Nevanlinna-Pick interpolation.SIAM J. Math. Anal.1851987, 1248--1260
  back to text
• 74 articleB.Bernard Hanzon, M.Martine Olivi and R.Ralf~L.M. Peeters. Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm.Linear Algebra and its Applications4182006, 793-820URL: https://dx.doi.org/10.1016/j.laa.2006.09.029
  back to textback to textback to text
• 75 articleJ.J.W. Helton and D.D.E. Marshall. Frequency domain analysis and analytic selections.Indiana Univ. Math. J.3911990, 157--184
  back to text
• 76 bookT.Tadeusz Iwaniec and G.Gaven Martin. Geometric function theory and non-linear analysis.Oxford Univ. Press2001
  back to text
• 77 articleG.G. Kyriazis and P.P. Petrushev. On the construction of frames for spaces of distributions.Journal of Functional Analysis2572009, 2159--2187
  back to text
• 78 bookN.N.S. Landkhof. Foundations of modern potential theory.Springer-Verlag1972
  back to text
• 79 bookM.M.M. Lavrentiev. Some Improperly Posed Problems of Mathematical Physics.Springer1967
  back to text
• 80 articleV.Vincent Lunot, F.Fabien Seyfert, S.Stéphane Bila and A.A. Nasser. Certified Computation of Optimal Multiband Filtering Functions.IEEE Transactions on Microwave Theory and Techniques5612008, 105-112URL: https://dx.doi.org/10.1109/TMTT.2007.912234
  back to textback to text
• 81 articleJ.-P.Jean-Paul Marmorat and M.Martine Olivi. Nudelman Interpolation, Parametrization of Lossless Functions and balanced realizations.Automatica432007, 1329--1338URL: https://dx.doi.org/10.1016/j.automatica.2007.01.020
  back to textback to textback to text
• 82 articleH.H. Orchard and G.G. Temes. Filter Design Using Transformed Variables.IEEE Transactions on Circuit Theory154dec 1968, 385--408URL: https://dx.doi.org/10.1109/TCT.1968.1082870
  back to text
• 83 bookV.Vladimir Peller. Hankel Operators and their Applications.Springer2003
  back to textback to text
• 84 articleA.A. Schneck. Constrained Hardy space approximation.J. Approx. Theory82010, 1466--1483
  back to text
• 85 articleI.I. Schur. Über Potenzreihen die im innern des einheitskreises beschränkt sind.J. Reine Angew. Math.1471917, 205--232
  back to text
• 86 inproceedingsF.Fabien Seyfert, L.Laurent Baratchart, J.-P.Jean-Paul Marmorat, S.Stéphane Bila and J.Jacques Sombrin. Extraction of coupling parameters for microwave filters: determinati on of a stable rational model from scattering data.2003 IEEE MTT-S International Microwave Symposium Digest1Philadelphie, États-UnisIEEE2003, 25--28URL: https://hal.inria.fr/hal-00663504
  back to textback to text
• 87 articleS.S. Smirnov. Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents.Algebra i Analiz4Transl. St Petersburg Math. Journal, n. 4, pp. 841--867, 19941993, 206-238
  back to text
• 88 bookE.E.~M. Stein. Harmonic Analysis.Princeton University Press1993
  back to text
• 89 bookA.Almudena Suàrez and R.Raymond Quéré. Stability analysis of nonlinear microwave circuits.Artech House2003
  back to text
• 90 inproceedingsT.T. Wolff. Counterexamples with harmonic gradients in ${𝐑}^3$.Essays on Fourier analysis in honor of Elias M. Stein42Math. Ser.Princeton Univ. Press1995, 321--384
  back to text
• 91 inproceedingsY.Yasmina Zaky, N.Nicolas Fortino, J.-Y.Jean-Yves Dauvignac, F.Fabien Seyfert, M.Martine Olivi and L.Laurent Baratchart. Comparison of SEM Methods for Poles Estimation from Scattered Field by Canonical Objects.Renaissance meets advancing technologyFlorence, ItalySeptember 2020, 6URL: https://hal.archives-ouvertes.fr/hal-02941637
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