Section: Research Program
Introduction
Within the extensive field of inverse problems, much of the research by Apics 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 transferfunction $f$ 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 halfplane, it satisfies there the CauchyRiemann 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 passband of the system which makes the problem illposed [70]. 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) freeboundary inverse problem, this time on the left halfplane. To address such questions, the team has developed a twostep 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 illposed issue requires regularization, for instance constraints on the behavior at nonmeasured frequencies.

Step 2: To compute a reduced order model. This typically consists of rational approximation of the complete model obtained in step 1, or phaseshift 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 McMillan degree when dealing with systems having several inputs and outputs, see Section 3.3.2.2. 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 [76] 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 [33].
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 [72], [56]. Apics contributed to this area the use of Zolotarevlike problems for multiband synthesis, although we presently favor interpolation techniques in which parameters arise in a more transparent manner, see Section 3.2.2.
The previous example of harmonic identification quickly suggests a generalization of itself. Indeed, on identifying $\u2102$ with ${\mathbb{R}}^{2}$, holomorphic functions become conjugategradients 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 DirichletNeumann data; when the portion of boundary where data are not available is itself unknown, we meet a free boundary problem. This framework for 2D nondestructive control was first advocated in [61] 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 (There is a subtle difference here between dimension 2 and higher. Indeed, a function holomorphic on a plane domain is defined by its nontangential limit on a boundary subset of positive linear measure, but there are nonconstant harmonic functions in the 3D ball, ${C}^{1}$ up to the boundary sphere, yet having vanishing gradient on a subset of positive measure of the sphere. Such a “bad” subset, however, cannot have interior points on the sphere.) [13], [79]. Such questions are particular instances of the socalled 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 2D) encompassing the support of $\mu $. For Laplace's operator, potentials are logarithmic in 2D 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 produce the same field outside this set; this phenomenon is called balayage [69]. 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 halfspace), and by requiring in step 2 that the measure be discrete in the left halfplane (in fact: a sum of point masses ${\sum}_{1}^{n}{a}_{j}{\delta}_{{z}_{j}}$). The discreteness assumption also prevails in 3D inverse source problems, see Section 4.3. Conditions that ensure uniqueness of the solution to the inverse potential problem are part of the socalled 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, Apics advocates the use of steps 1 and 2 above, along with some singularity analysis, to approach issues of nondestructive control in 2D and 3D [2], [5], [40]. The team is currently engaged in the generalization to inverse source problems for the Laplace equation in 3D, to be described further in Section 3.2.1. There, holomorphic functions are replaced by harmonic gradients; applications are to EEG/MEG and inverse magnetization problems in geosciences, see Section 4.3.
The approximationtheoretic tools developed by Apics 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.