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## Section: Scientific Foundations

### 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 behaviour. Perhaps the most basic example of such a problem is harmonic identification of a stable linear dynamical system: the transfer-function $f$ is holomorphic in the right half-pane, which means it satisfies there the Cauchy-Riemann equation $\overline{\partial }f=0$, and in principle $f$ can be recovered from its values on the imaginary axis, e.g. by 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 [69] . 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 $n$, it satisfies $\overline{\partial }f={\sum }_{1}^{n}{a}_{j}{\delta }_{{z}_{j}}$ where the ${z}_{j}$ are its poles, and finding the domain of holomorphy (i.e. locating the ${z}_{j}$) amounts to solve a (degenerate) free-boundary inverse problem, this time on the left half-plane. To address these questions, the team has developed a two-step approach as follows.

• Step 1:

To determine a complete model, that is, one which is defined for every frequency, in a sufficiently flexible function class (e.g. Hardy spaces). This ill-posed issue requires regularization, for instance constraints on the behaviour 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. Derivation of the complete model is important to achieve stability of the reduced one.

Step 1 makes connection with 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 output, see section 3.3.2.2 . It also makes contact with the topology of rational functions, to count critical points and to derive bounds, see section 3.3.2 . Moreover, this step raises issues in approximation theory regarding the rate of convergence and whether the singularities of the approximant (i.e. its poles) converge to the singularities of the approximated function; this is where logarithmic potential theory becomes effective, see section 3.3.3 .

Iterating the previous steps coupled with a sensitivity analysis yields a tuning procedure which was first demonstrated in [77] on resonant microwave filters.

Similar steps can be taken to approach design problems in frequency domain, replacing measured behaviour by desired behaviour. However, describing achievable responses from the design parameters at hand is generally cumbersome, and most constructive techniques rely on rather specific criteria adapted to the physics of the problem. This is especailly true of circuits and filters, whose design classically appeals to standard polynomial extremal problems and realization procedures from system theory [70] , [55] . APICS is active in this field, where we introduced the use of Zolotarev-like problems for microwave multiband filter design. We currently favor interpolation techniques because of their transparency with respect to parameter use, see section 3.2.2 .

In another connection, the 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; portion of the boundary where data are not available may be unknown, in which case we meet a free boundary problem. This framework for 2-D non-destructive control was first advocated in [59] and subsequently received considerable attention. This framework makes it 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) [79] . All these questions are particular instances of the so-called inverse potential problem, where a measure $\mu$ has to be recovered from 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. In any case, by construction, the operator applied to the potential yields back the measure.

Inverse potential problems are severely indeterminate because infinitely many measures within an open set produce the same field outside this set [68] . In step 1 above we implicitly removed this indeterminacy by requiring that the measure be supported on the boundary (because we seek a function holomorphic throughout the right half space), and in step 2 by requiring, say, in case of rational approximation that the measure be discrete in the left half-plane. The same discreteness assumption prevails in 3-D inverse source problems. 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 this 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, the team initiated the use of steps 1 and 2 above, along with singularity analysis, to approach issues of nondestructive control in 2 and 3-D [41] [6] , [2] . We are currently engaged in two kinds of generalization, further described in section 3.2.1 . The first one deals with non-constant conductivities, where Cauchy-Riemann equations for holomorphic functions are replaced by conjugate Beltrami equations for pseudo-holomorphic functions; there we seek applications to plasma confinement. The other one lies with inverse source problems for Laplace's equation in 3-D, where holomorphic functions are replaced by harmonic gradients, developing applications to EEG/MEG and inverse magnetization problems in paleomagnetism, see section 4.2

The main approximation-theoretic tools developed by APICS to get to grips with issues mentioned so far are outlined in section 3.3 . In section 3.2 to come, we make more precise which problems are considered and for which applications.