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Section: Scientific Foundations

Range of inverse problems

Elliptic partial differential equations (PDE)

Participants : Laurent Baratchart, Slah Chaabi, Juliette Leblond, Ana-Maria Nicu, Dmitry Ponomarev, Elodie Pozzi.

This work is done in collaboration with Alexander Borichev (Univ. Provence).

Reconstructing Dirichlet-Neumann boundary conditions for a function harmonic in a plane domain when these are known on a strict subset $E$ of the boundary, is equivalent to recover a holomorphic function in the domain from its boundary values on $E$. This is the problem raised on the half-plane in step 1 of section 3.1 . It makes good sense in holomorphic Hardy spaces where functions are determined by their values on boundary subsets of positive linear measure, which is the framework for problem $\left(P\right)$ in section 3.3.1 . Such problems naturally arise in nondestructive testing of 2-D (or cylindical) materials from partial electrical measurements on the boundary. Indeed, the ratio between tangential and normal currents (so-called Robin coefficient) tells about corrosion of the material. Solving problem $\left(P\right)$ where $\psi$ is chosen to be the response of some uncorroded piece with identical shape allows one to approach such questions, and this was an initial application of holomorphic extremal problems to non-destructive control [56] , [52] .

A recent application by the team deals with non-constant conductivity over a doubly connected domain, $E$ being the outer boundary. Measuring Dirichlet-Neumann data on $E$, we wanted to check whether the solution is constant on the inner boundary. We first had to define and study Hardy spaces of the conjugate Beltrami equation, of which the conductivity equation is the compatibility condition (just like Laplace's equation is the compatibility condition of the Cauchy-Riemann system). This was done in references [5] and [35] . Then, solving an obvious modification of problem $\left(P\right)$ allows one to numerically check what we want. Further, the value of this extremal problem defines a criterion on inner boundaries, and subsequently a descent algorithm was set up to improve the initial boundary into one where the solution is closer to being constant, thereby trying to solve a free boundary problem..

When the domain is regarded as separating the edge of a tokamak's vessel from the plasma (rotational symmetry makes this a 2-D problem), the procedure just described suits plasma control from magnetic confinement. It was successfully applied in collaboration with CEA (the French nuclear agency) and the University of Nice (JAD Lab.) to data from Tore Supra [58] , see section 6.2 . This procedure is fast because no numerical integration of the underlying PDE is needed, as an explicit basis of solutions to the conjugate Beltrami equation was found in this case.

Three-dimensional versions of step 1 in section 3.1 are also considered, namely to recover a harmonic function (up to a constant) in a ball or a half-space from partial knowledge of its gradient on the boundary. Such questions arise naturally in connection with neurosciences and medical imaging (electroencephalography, EEG) or in paleomagnetism (analysis of rocks magnetization) [2] [37] , see section 6.1 . They are not yet as developed as the 2-D case where the power of complex analysis is at work, but considerable progress was made over the last years through methods of harmonic analysis and operator theory.

The team is also concerned with non-destructive control problems of localizing defaults such as cracks, sources or occlusions in a planar or 3-dimensional domain, from boundary data (which may correspond to thermal, electrical, or magnetic measurements). These defaults can be expressed as a lack of analyticity of the solution of the associated Dirichlet-Neumann problem and we approach them using techniques of best rational or meromorphic approximation on the boundary of the object [4] [16] , see sections 3.3.2 and 4.2 . In fact, the way singularities of the approximant relate to the singularities of the approximated function is an all-pervasive theme in approximation theory, and for appropriate classes of functions the location of the poles of a best rational approximant can be used as an estimator of the singularities of the approximated function (see section 6.1 ). This circle of ideas is much in the spirit of step 2 in section 3.1 .

A genuine 3-dimensional theory of approximation by discrete potentials, though, is still in its infancy.

Systems, transfer and scattering

Participants : Laurent Baratchart, Sylvain Chevillard, Sanda Lefteriu, Martine Olivi, Fabien Seyfert.

Through initial contacts with CNES, the French space agency, the team came to work on identification-for-tuning of microwave electromagnetic filters used in space telecommunications (see section 4.3 ). The problem was to recover, from band-limited frequency measurements, the physical parameters of the device under examination. The latter consists of interconnected dual-mode resonant cavities with negligible loss, hence its scattering matrix is modelled by a $2\times 2$ unitary-valued matrix function on the frequency line, say the imaginary axis to fix ideas. In the bandwidth around the resonant frequency, a modal approximation of the Helmholtz equation in the cavities shows that this matrix is approximately rational, of Mc-Millan degree twice the number of cavities.

This is where system theory enters the scene, through the so-called realization process mapping a rational transfer function in the frequency domain to a state-space representation of the underlying system as a 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, and in particular the Hilbert space $H^2$, provide a framework to transform this classical ill-posed issue into a series of well-posed analytic and meromorphic approximation problems. The procedure sketched in section 3.1 now goes as follows:

1. infer from the pointwise boundary data in the bandwidth a stable transfer function (i.e. one which is holomorphic in the right half-plane), that may be infinite dimensional (numerically: of high degree). This is done by solving in the Hardy space $H^2$ of the right half-plane a problem analogous to $\left(P\right)$ in section 3.3.1 , taking into account prior knowledge on the decay of the response outside the bandwidth, see [18] for details.

2. From this stable model, a rational stable approximation of appropriate degree is computed. For this a descent method is used on the relatively compact manifold of inner matrices of given size and degree, using a novel parametrization of stable transfer functions [18] .

3. From this rational model, realizations meeting certain constraints imposed by the technology in use are computed (see section  6.3 ). These constraints typically come from the nature and topology of the equivalent electrical network used to model the filter. This network is composed of resonators, coupled to each other by some specific coupling topology. Performing this realization step for given coupling topology can be recast, under appropriate compatibility conditions [8] , as the problem of solving a zero-dimensional multivariate polynomial system. To tackle this problem in practice, we use Groebner basis techniques as well as continuation methods as implemented in the Dedale-HF software ( 5.4 ).

Let us also mention that extensions of classical coupling matrix theory to frequency-dependent (reactive) couplings have lately been carried-out [1] for wide-band design applications, but further study is needed to make them effective.

Subsequently APICS started investigating issues pertaining to filter design rather than identification. Given the topology of the filter, a basic problem is to find the optimal response with respect to amplitude specifications in frequency domain bearing on rejection, transmission and group delay of scattering parameters. Generalizing the approach based on Tchebychev polynomials for single band filters, we recast the problem of multi-band response synthesis in terms of a generalization of classical Zolotarev min-max problem [30] to rational functions [11] . Thanks to quasi-convexity, the latter can be solved efficiently using iterative methods relying on linear programming. These are implemented in the software easy-FF (see section 5.5 ).

Later, investigations by the team extended to design and identification of more complex microwave devices, like multiplexers and routers, which connect several filters through wave guides. Schur analysis plays an important role in such studies, which is no surprise since 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 [76] , who devised a recursive test to check for contractivity of a holomorphic function in the disk. Generalizations thereof turned out to be very efficient to parametrize solutions to contractive interpolation problems subject to a well-known compatibility condition (positive definiteness of the so-called Pick matrix) [32] . Schur analysis became quite popular in electrical engineering, as the Schur recursion precisely describes how to chain two-port circuits.

Dwelling on this, members of the team contributed to differential parametrizations (atlases of charts) of lossless matrix functions to the theory [31] [12] , [10] . They are of fundamental use in our rational approximation software RARL2 (see section 5.1 ). Schur analysis is also instrumental to approach de-embedding issues considered in section 6.4 , and provides further background to current studies by the team of synthesis and adaptation problems for multiplexers. At the heart of the latter lies a variant of contractive interpolation with degree constraint introduced in [62] .

We also mention the role played by multipoint Schur analysis in the team's investigation of spectral representation for certain non-stationary discrete stochastic processes [3] , [36] .

Recently, in collaboration with UPV (Bilbao), our attention was driven by CNES, to questions of stability relative to high-frequency amplifiers, see section 7.2 . Contrary to previously mentioned devices, these are active components. The amplifier can be linearized at a functioning point and admittances of the corresponding electrical network can be computed at various frequencies, using the so-called harmonic balance method. The goal is to check for stability of this linearised model. The latter is composed of lumped electrical elements namely inductors, capacitors, negative and positive reactors, transmission lines, and commanded current sources. Research so far focused on determining the algebraic structure of admittance functions, and setting up a function-theoretic framework to analyse them. In particular, much effort was put on realistic assumptions under which a stable/unstable decomposition can be claimed in $H^2\oplus \overline{H^2}$ (see section 6.5 ). Under them, the unstable part of the elements under examination is rational and we expect to bring valuable estimates of stability to the designer using the general scheme in section 3.1 .