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 . 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 . 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 , .

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 and . 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 , see section . 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 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) , see section . 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 , see sections and . 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 ). This circle of ideas is much in the spirit of step 2 in section .

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

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 ). 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 now goes as follows:

Let us also mention that extensions of classical coupling matrix theory to frequency-dependent (reactive) couplings have lately been carried-out 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 to rational functions . 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 ).

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 , 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) . 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 , . They are of fundamental use in our rational approximation software RARL2 (see section ). Schur analysis is also instrumental to approach de-embedding issues considered in section , 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 .

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 , .

Recently, in collaboration with UPV (Bilbao), our attention was driven by CNES, to questions of stability relative to high-frequency amplifiers, see section . 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 ). 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 .

In dimension 2, the prototypical problem to be solved in step 1 of section may be described as: given a domain $D\subset {ℝ}^2$, we want to recover a holomorphic function from its values on a subset of the boundary of $D$. Using conformal mapping, it is convenient for the discussion to normalize $D$. So, in the simply connected case, we fix $D$ to be the unit disk with boundary the unit circle $T$. We denote by $H^p$ the Hardy space of exponent $p$ which is the closure of polynomials in the $L^p$-norm on the circle if $1\le p<\infty$ and the space of bounded holomorphic functions in $D$ if $p=\infty$. Functions in $H^p$ have well-defined boundary values in $L^p\left(T\right)$, which makes it possible to speak of (traces of) analytic functions on the boundary.

To find an analytic function in $D$ approximately matching measured values $f$ on a sub-arc $K$ of $T$, we formulate a constrained best approximation problem as follows.

($P$)  Let $1\le p\le \infty$, $K$ a sub-arc of $T$, $f\in L^p\left(K\right)$, $\psi \in L^p(T\setminus K)$ and $M>0$; find a function $g\in H^p$ such that ${\parallel g-\psi \parallel }_{L^p(T\setminus K)}\le M$ and $g-f$ is of minimal norm in $L^p\left(K\right)$ under this constraint.

Here $\psi$ is a reference behaviour capturing a priori assumptions on the behaviour of the model off $K$, while $M$ is some admissible deviation from them. The value of $p$ reflects the type of stability which is sought and how much one wants to smoothen the data. The choice of $L^p$ classes is well-adapted to handling pointwise measurements.

To fix terminology we refer to ($P$) as a bounded extremal problem. As shown in , , , for $1<p\le \infty$, the solution to this convex infinite-dimensional optimization problem can be obtained upon iterating with respect to a Lagrange parameter the solution to spectral equations for some appropriate Hankel and Toeplitz operators. These equations in turn involve the solution to the standard extremal problem below best approximation problem ($P_0$) below :

($P_0$)  Let $1\le p\le \infty$ and $\varphi \in L^p\left(T\right)$; find a function $g\in H^p$ such that $g-\varphi$ is of minimal norm in $L^p\left(T\right)$.

The case $p=1$ of $\left(P\right)$ is essentially open.

Various modifications of $\left(P\right)$ have been studied in order to meet specific needs. For instance when dealing with loss-less transfer functions (see section ), one may want to express the constraint on $T\setminus K$ in a pointwise manner: $|g-\psi |\le M$ a.e. on $T\setminus K$, see . In this form, it comes close to (but still is different from) $H^{\infty }$ frequency optimization methods for control , .

The analog of problem $\left(P\right)$ on an annulus, $K$ being now the outer boundary, can be seen as a means to regularize a classical inverse problem occurring in nondestructive control, namely recovering a harmonic function on the inner boundary from Dirichlet-Neumann data on the outer boundary (see sections , , , ). For $p=2$ the solution is analysed in . It may serve as a tool to approach Bernoulli type problems where we are given data on the outer boundary and we seek the inner boundary, knowing it is a level curve of the flux. Then, the Lagrange parameter indicates which deformation should be applied on the inner contour in order to improve data fitting.

This is discussed in sections and for more general equations than the Laplacian, namely isotropic conductivity equations of the form $\mathrm{div}\left(\sigma \nabla u\right)=0$ where $\sigma$ is non constant. In this case Hardy spaces in problem $\left(P\right)$ become those of a so-called conjugate or real Beltrami equation , which are studied for $1<p<\infty$ in , . Expansions of solutions needed to constructively handle such issues have been carried out in  .

Though originally considered in dimension 2, problem $\left(P\right)$ carries over naturally to higher dimensions where analytic functions get replaced by gradients of harmonic functions. Namely, given some open set $\Omega \subset {ℝ}^n$ and a ${ℝ}^n$-valued vector $V$ field on an open subset $O$ of the boundary of $\Omega$, we seek a harmonic function in $\Omega$ whose gradient is close to $V$ on $O$.

When $\Omega$ is a ball or a half-space, a convenient substitute of holomorphic Hardy spaces is provided by Stein-Weiss Hardy spaces of harmonic gradients . Conformal maps are no longer available in ${ℝ}^n$ for $n>2$ and other geometries have not been much studied so far. On the ball, the analog of problem $\left(P\right)$ is

($P1$)  Let $1\le p\le \infty$ and $B\subset {ℝ}^n$ the unit ball. Fix $O$ an open subset of the unit sphere $S\subset {ℝ}^n$. Let further $V\in L^p\left(O\right)$ and $W\in L^p(S\setminus O)$ be ${ℝ}^n$-valued vector fields, and $M>0$; find a harmonic gradient $G\in H^p\left(B\right)$ such that ${\parallel G-W\parallel }_{L^p(S\setminus O)}\le M$ and $G-V$ is of minimal norm in $L^p\left(O\right)$ under this constraint.

When $p=2$, spherical harmonics offer a reasonable substitute to Fourier expansions and problem $\left(P1\right)$ was solved in , together with its natural analog on a shell. The solution generalizes the Toeplitz operator approach to bounded extremal problems , and constructive aspects of the procedure (harmonic 3-D projection, Kelvin and Riesz transformation, spherical harmonics) were derived. An important ingredient is a refinement of the Hodge decomposition allowing us to express a ${ℝ}^n$-valued vector field in $L^p\left(S\right)$, $1<p<\infty$, as the sum of a vector field in $H\left(B\right)$, a vector field in $H^p({ℝ}^n\setminus \overline{B}$, and a tangential divergence free vector field. If $p=1$ or $p=\infty$, $L^p$ must be replaced respectively by the real Hardy space $H^1$ and the bounded mean oscillation space $BMO$, and $H^{\infty }$ should be modified accordingly. This decomposition was fully discussed in (for the case of the half-space) where it plays a fundamental role.

Problem $\left(P1\right)$ is still under investigation in the case $p=\infty$, where even the case where $O=S$ is pending because a substitute of the Adamjan-Arov-Krein theory is still to be built in dimension greater than 2.

Such problems arise in connection with source recovery in electro/mgneto encephalography and paleomagnetism, as discussed in sections and .

The techniques explained in this section are used to solve step 2 in section via conformal mapping and subsequently instrumental to approach inverse boundary value problems for Poisson equation $\Delta u=\mu$, where $\mu$ is some (unknown) distribution.

The following people collaborate with us on this subject: Herbert Stahl (TFH Berlin), Maxim Yattselev (Univ. Oregon at Eugene, USA).

We refer here to the behaviour of poles of best meromorphic approximants, in the $L^p$-sense on a closed curve, to functions $f$ defined as Cauchy integrals of complex measures whose support lies inside the curve. If one normalizes the contour to be the unit circle $T$, we are back to the framework of section and to problem ($P_N$); invariance of the problem under conformal mapping was established in . Research so far has focused on functions whose singular set inside the contour is zero or one-dimensional.

Generally speaking, the behaviour of poles is particularly important in meromorphic approximation to obtain error rates as the degree goes large and to tackle constructive issues like uniqueness. As explained in section , we consider this issue in connection with approximation of the solution to a Dirichlet-Neumann problem, so as to extract information on the singularities. The general theme is thus how do the singularities of the approximant reflect those of the approximated function? This approach to inverse problem for the 2-D Laplacian turns out to be attractive when singularities are zero- or one-dimensional (see section ). It can be used as a computationally cheap initialization of more precise but heavier numerical optimizations.

As regards crack detection or source recovery, the approach in question boils down to analysing the behaviour of best meromorphic approximants of a function with branch points. For piecewise analytic cracks, or in the case of sources, We were able to prove (, , ) that the poles of the approximants accumulate on some extremal contour of minimum weighted energy linkings the singular points of the crack, or the sources . Moreover, the asymptotic density of the poles turns out to be the Green equilibrium distribution of this contour in $D$, hence puts heavy charge around the singular points (in particular at the endpoints) which are therefore well localized if one is able to approximate in sufficiently high degree (this is where the method could fail).

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

It is interesting that inverse source problems inside a sphere or an ellipsoid in 3-D can be attacked with the above 2-D techniques, as applied to planar sections (see section ).

Sylvain Chevillard, joined team in November 2010. His coming resulted in APICS hosting a research activity in certified computing, centered around the software Sollya of which S. Chevillard is a co-author, see section . On the one hand, Sollya is an Inria software which still requires some tuning to a growing community of users. On the other hand, approximation-theoretic methods at work in Sollya are potentially useful for certified solutions to constrained analytic problems described in section . However, developing Solya is not a long-term objective of APICS.