By standard properties of conjugate differentials, reconstructing Dirichlet-Neumann boundary conditions for a function harmonic in a plane domain, when these boundary conditions are known already on a subset $E$ of the boundary, is equivalent to recover a holomorphic function in the domain from its boundary values on $E$. This is the problem raised on the half-plane in step 1 of Section . It makes good sense in holomorphic Hardy spaces where functions are entirely determined by their values on boundary subsets of positive linear measure, which is the framework for Problem $\left(P\right)$ that we set up in Section . Such issues naturally arise in nondestructive testing of 2-D (or 3-D cylindrical) materials from partial electrical measurements on the boundary. For instance, the ratio between the tangential and the normal currents (the so-called Robin coefficient) tells one about corrosion of the material. Thus, solving Problem $\left(P\right)$ where $\psi$ is chosen to be the response of some uncorroded piece with identical shape yields non destructive testing of a potentially corroded piece of material, part of which is inaccessible to measurements. This was an initial application of holomorphic extremal problems to non-destructive control , .

Another application by the team deals with non-constant conductivity over a doubly connected domain, the set $E$ being now the outer boundary. Measuring Dirichlet-Neumann data on $E$, one wants to recover level lines of the solution to a conductivity equation, which is a so-called free boundary inverse problem. For this, given a closed curve inside the domain, we first quantify how constant the solution on this curve. To this effect, we state and solve an analog of Problem $\left(P\right)$, where the constraint bears on the real part of the function on the curve (it should be close to a constant there), in a Hardy space of a conjugate Beltrami equation, of which the considered conductivity equation is the compatibility condition (just like the Laplace equation is the compatibility condition of the Cauchy-Riemann system). Subsequently, a descent algorithm on the curve leads one to improve the initial guess. For example, when the domain is regarded as separating the edge of a tokamak's vessel from the plasma (rotational symmetry makes this a 2-D situation), this method can be used to estimate the shape of a plasma subject to magnetic confinement. It was successfully applied, in collaboration with CEA (French nuclear agency) and the University of Nice (JAD Lab.), to data from Tore Supra . The procedure is fast because no numerical integration of the underlying PDE is needed, as an explicit basis of solutions to the conjugate Beltrami equation in terms of Bessel functions was found in this case. Generalizing this approach in a more systematic manner to free boundary problems of Bernoulli type, using descent algorithms based on shape-gradient for such approximation-theoretic criteria, is an interesting prospect, still to be pursued.

The piece of work we just mentioned requires defining and studying Hardy spaces of the conjugate-Beltrami equation, which is an interesting topic by itself. For Sobolev-smooth coefficients of exponent greater than 2, this was done in references and . The case of the critical exponent 2 is treated in , which apparently provides the first example of well-posedness for the Dirichlet problem in the non-strictly elliptic case: the conductivity may be unbounded or zero on sets of zero capacity and, accordingly, solutions need not be locally bounded.

The 3-D version of step 1 in Section  is another subject investigated by Apics: 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. This prototypical inverse problem (i.e. inverse to the Cauchy problem for the Laplace equation) often recurs in electromagnetism. At present, Apics is involved with solving instances of this inverse problem arising in two fields, namely medical imaging e.g. for electroencephalography (EEG) or magneto-encephalography (MEG), and paleomagnetism (recovery of rocks magnetization) , , see Section . In this connection, we collaborate with two groups of partners: Athena Inria project-team, CHU La Timone, and BESA company on the one hand, Geosciences Lab. at MIT and Cerege CNRS Lab.on the other hand. The question is considerably more difficult than its 2-D counterpart, due mainly to the lack of multiplicative structure for harmonic gradients. Still, considerable progress has been made over the last years using methods of harmonic analysis and operator theory.

The team is further concerned with 3-D generalizations and applications to non-destructive control of step 2 in Section . A typical problem is here to localize inhomogeneities or defaults such as cracks, sources or occlusions in a planar or 3-dimensional object, knowing thermal, electrical, or magnetic measurements on the boundary. These defaults can be expressed as a lack of harmonicity of the solution to the associated Dirichlet-Neumann problem, thereby posing an inverse potential problem in order to recover them. In 2-D, finding an optimal discretization of the potential in Sobolev norm amounts to solve a best rational approximation problem, and the question arises as to how the location of the singularities of the approximant (i.e. its poles) reflects the location of the singularities of the potential (i.e. the defaults we seek). This is a fairly deep issue in approximation theory, to which Apics contributed convergence results for certain classes of fields expressed as Cauchy integrals over extremal contours for the logarithmic potential , . Initial schemes to locate cracks or sources via rational approximation on planar domains were obtained this way , , . It is remarkable that finite inverse source problems in 3-D balls, or more general algebraic surfaces, can be approached using these 2-D techniques upon slicing the domain into planar sections , . This bottom line generates a steady research activity within Apics, and again applications are sought to medical imaging and geosciences, see Sections , and .

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

Through contacts with CNES (French space agency), members of the team became involved in identification and tuning of microwave electromagnetic filters used in space telecommunications, see Section . The initial problem was to recover, from band-limited frequency measurements, physical parameters of the device under examination. The latter consists of interconnected dual-mode resonant cavities with negligible loss, hence its scattering matrix is modeled by a $2\times 2$ 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 comes into play, through the so-called realization process mapping a rational transfer function in the frequency domain to a state-space representation of the underlying system of linear differential equations in the time domain. Specifically, realizing the scattering matrix allows one to construct a virtual electrical network, equivalent to the filter, the parameters of which mediate in between the frequency response and the geometric characteristics of the cavities (i.e. the tuning parameters).

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

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

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

In recent years, our attention was driven by CNES and UPV (Bilbao) to questions about stability of high-frequency amplifiers, see Section . Contrary to previously discussed devices, these are active components. The response of an amplifier can be linearized around a set of primary current and voltages, and then admittances of the corresponding electrical network can be computed at various frequencies, using the so-called harmonic balance method. The initial goal is to check for stability of the linearized model, so as to ascertain existence of a well-defined working state. The network is composed of lumped electrical elements namely inductors, capacitors, negative and positive reactors, transmission lines, and controlled current sources. Our research so far focuses on describing the algebraic structure of admittance functions, so as to set up a function-theoretic framework where the two-steps approach outlined in Section  can be put to work. The main discovery so far is that the unstable part of each partial transfer function is rational, see Section .

In dimension 2, the prototypical problem to be solved in step 1 of Section  may be described as: given a domain $D\subset {ℝ}^2$, to recover a holomorphic function from its values on a subset $K$ of the boundary of $D$. For the discussion it is convenient to normalize $D$, which can be done by conformal mapping. So, in the simply connected case, we fix $D$ to be the unit disk with boundary unit circle $T$. We denote by $H^p$ the Hardy space of exponent $p$, which is the closure of polynomials in $L^p\left(T\right)$-norm 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 $g$ in $D$ matching some measured values $f$ approximately on a sub-arc $K$ of $T$, we formulate a constrained best approximation problem as follows.

$\left(P\right)$  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 behavior capturing a priori assumptions on the behavior of the model off $K$, while $M$ is some admissible deviation thereof. The value of $p$ reflects the type of stability which is sought and how much one wants to smooth out the data. The choice of $L^p$ classes is suited to handle point-wise measurements.

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

($P_0$)  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 $p=1$ is more or less open.

Various modifications of $\left(P\right)$ can be set up in order to meet specific needs. For instance when dealing with lossless transfer functions (see Section ), one may want to express the constraint on $T\setminus K$ in a point-wise manner: $|g-\psi |\le M$ a.e. on $T\setminus K$, see . In this form, the problem comes close to (but still is different from) $H^{\infty }$ frequency optimization used in control , . One can also impose bounds on the real or imaginary part of $g-\psi$ on $T\setminus K$, which is useful when considering Dirichlet-Neuman problems, see .

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 to recover a harmonic function on the inner boundary from Dirichlet-Neumann data on the outer boundary (see Sections , , , ). It may serve as a tool to approach Bernoulli type problems, where we are given data on the outer boundary and we seek the inner boundary, knowing it is a level curve of the solution.. In this case, the Lagrange parameter indicates how to deform the inner contour in order to improve data fitting. Similar topics are discussed in 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 no longer constant. Then, the Hardy spaces in Problem $\left(P\right)$ are those of a so-called conjugate Beltrami equation: $\overline{\partial }f=\nu \overline{\partial f}$ , which are studied for $1<p<\infty$ in , , and . Expansions of solutions needed to constructively handle such issues in the specific case of linear fractional conductivities (these occur in plasma shaping) have been expounded 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 some ${ℝ}^n$-valued vector field $V$ 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 substitute for holomorphic Hardy spaces is provided by the Stein-Weiss Hardy spaces of harmonic gradients . Conformal maps are no longer available when $n>2$, so that $\Omega$ can no longer be normalized. More general geometries than spheres and half-spaces have not been much studied so far.

On the ball, the analog of Problem $\left(P\right)$ is

$\left(P_1\right)$  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. Given $M>0$, 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 $p=2$, Problem $\left(P_1\right)$ was solved in as well as its analog on a shell. The solution extends the one given in for the 2-D case, using a generalization of Toeplitz operators. Thecas of the shell was motivated An important ingredient is a refinement of the Hodge decomposition, that we call the Hardy-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 $H^p\left(B\right)$, a vector field in $H^p({ℝ}^n\setminus \overline{B})$, and a tangential divergence free vector field on $S$; the space of such fields is denoted by $D\left(S\right)$. If $p=1$ or $p=\infty$, $L^p$ must be replaced by the real Hardy space or the space of functions with bounded mean oscillation. More generally this decomposition, which is valid on any sufficiently smooth surface (see Section ), seems to play a fundamental role in inverse potential problems. In fact, it was first introduced formally on the plane to describe silent magnetizations supported in ${ℝ}^2$ (i.e. those generating no field in the upper half space) .

Just like solving problem $\left(P\right)$ appeals to the solution of problem $\left(P_0\right)$, our ability to solve problem $\left(P_1\right)$ will depend on the possibility to tackle the special case where $O=S$:

$\left(P_2\right)$  Let $1\le p\le \infty$ and $V\in L^p\left(S\right)$ be a ${ℝ}^n$-valued vector field. Find a harmonic gradient $G\in H^p\left(B\right)$ such that ${\parallel G-V\parallel }_{L^p\left(S\right)}$ is minimum.

Problem $\left(P_2\right)$ is simple when $p=2$ by virtue of the Hardy Hodge decomposition together with orthogonality of $H^2\left(B\right)$ and $H^2({ℝ}^n\setminus \overline{B})$, which is the reason why we were able to solve $\left(P_1\right)$ in this case. Other values of $p$ cannot be treated as easily and are currently investigated by Apics, especially the case $p=\infty$ which is of particular interest and presents itself as a 3-D analog to the Nehari problem .

Companion to problem $\left(P_2\right)$ is problem $\left(P_3\right)$ below.

$\left(P_3\right)$  Let $1\le p\le \infty$ and $V\in L^p\left(S\right)$ be a ${ℝ}^n$-valued vector field. Find $G\in H^p\left(B\right)$ and $D\in D\left(S\right)$ such that ${\parallel G+D-V\parallel }_{L^p\left(S\right)}$ is minimum.

Note that $\left(P_2\right)$ and $\left(P_3\right)$ are identical in 2-D, since no non-constant tangential divergence-free vector field exists on $T$. It is no longer so in higher dimension, where both $\left(P_2\right)$ and $\left(P_3\right)$ arise in connection with source recovery in electro/magneto encephalography and paleomagnetism, see Sections  and .

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

We refer here to the behavior 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. Normalizing the contour to be the unit circle $T$, we are back to Problem $\left(P_N\right)$ in Section ; invariance of the latter under conformal mapping was established in . Research so far has focused on functions whose singular set inside the contour is zero or one-dimensional.

Generally speaking in approximation theory, assessing the behavior of poles of rational approximants is essential to obtain error rates as the degree goes large, and to tackle constructive issues like uniqueness. However, as explained in Section , Apics considers this issue foremost as a means to extract information on singularities of the solution to a Dirichlet-Neumann problem. The general theme is thus: how do the singularities of the approximant reflect those of the approximated function? This approach to inverse problem for the 2-D Laplacian turns out to be attractive when singularities are zero- or one-dimensional (see Section ). It can be used as a computationally cheap initial condition for more precise but much heavier numerical optimizations which often do not even converge unless properly initialized. As regards crack detection or source recovery, this approach boils down to analyzing the behavior of best meromorphic approximants of 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, when the degree goes large, to some extremal cut of minimum weighted logarithmic capacity connecting the singular points of the crack, or the sources . Moreover, the asymptotic density of the poles turns out to be the Green equilibrium distribution on this cut in $D$, therefore it charges the singular points if one is able to approximate in sufficiently high degree (this is where the method could fail, because high-order approximation requires rather precise data).

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

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

Sylvain Chevillard, joined team in November 2010. His coming resulted in Apics hosting a research activity in certified computing, centered on 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 Sollya is not a long-term objective of Apics.