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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 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 [72] . 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  3.3.2.2 . 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 [78] 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 [36] .

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 [74] , [59] . Apics contributed to this area 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, see Section  3.2.2 .

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 [64] 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 non-tangential limit on a boundary subset of positive linear measure, but there are non-constant harmonic functions in the 3-D 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.) [25] , [83] . 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 produce the same field outside this set; this phenomenon is called balayage [71] . 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. The discreteness assumption also prevails in 3-D inverse source problems, see Section  4.2 . 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. Note that it is different 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 2-D and 3-D [43] [5] , [2] . The team is currently engaged in two kinds of generalizations, to be described further in Section  3.2.1 . The first deals with non-constant conductivities in 2-D, where Cauchy-Riemann equations characterizing holomorphic functions are replaced by conjugate Beltrami equations characterizing pseudo-holomorphic functions; next in line are 3-D situations that we begin to consider, see Sections 6.2 and 4.4 . There, we seek applications to inverse free boundary problems such as plasma confinement in the vessel of a tokamak, or inverse conductivity problems like those arising in impedance tomography. The second generalization lies with inverse source problems for the Laplace equation in 3-D, where holomorphic functions are replaced by harmonic gradients; applications are to EEG/MEG and inverse magnetization problems in paleomagnetism, see Section  4.2 .

The approximation-theoretic 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.