Section: New Results

Inverse problems for elliptic operators

Participants : Laurent Baratchart, Aline Bonami [Univ. Orléans] , Slah Chaabi, Sylvain Chevillard, Maureen Clerc [EPI Athena] , Yannick Fischer, Sandrine Grellier [Univ. Orléans] , Doug Hardin [Vanderbilt Univ.] , Abderrazek Karoui [Univ. Bizerte, Tunisie] , Juliette Leblond, Jean-Paul Marmorat, Ana-Maria Nicu, Théo Papadopoulo [EPI Athena] , Jonathan Partington, Elodie Pozzi, Edward Saff.

Boundary value problems for Laplace equation in 3-D

Solving overdetermined Cauchy problems for the Laplace equation on a spherical layer (in 3-D) in order to process incomplete experimental data is a necessary ingredient of the team's approach to inverse source problems, in particular for applications to EEG since the latter involves propagating the initial conditions from the boundary to the center of the domain where the singularities (i.e., the sources) are sought after. Here, the domain is typically made of several homogeneous layers of different conductivities.

Such problems offer an opportunity to state and solve extremal problems for harmonic fields for which an analog of the Toeplitz operator approach to bounded extremal problems [45] has been obtained in [2] . Still, a best approximation on the subset of a general vector field generated by a harmonic gradient under a $L^2$ norm constraint on the complementary subset can be computed by an inverse spectral equation for some Toeplitz operator. Constructive and numerical aspects of the procedure (harmonic 3-D projection, Kelvin and Riesz transformation, spherical harmonics) and encouraging results have been obtained on numerically simulated data.

Issues of robust interpolation on the sphere from incomplete pointwise data are also under study in order to improve numerical accuracy of our reconstruction schemes. Spherical harmonics, Slepian bases and related special functions are of special interest (thesis of A.-M. Nicu), while splines, spherical wavelets, cubature techniques should be considered as well.

It turns out that Slepian functions are eigenfunctions of truncated Toeplitz operators in the complex plane (the framework of 2-D problems). These properties will be used in order to quantify the robustness properties of our resolution schemes for $L^2$ bounded extremal problems [45] , and to establish error estimates.

The analogous problem in $L^p$, $p\ne 2$, is quite important to get tighter control on pointwise approximation. However, it is considerably more difficult. In a collaborative effort with the university of Orléans, within the framework of the ANR project AHPI, we set ourselves the goal of understanding the case $p=\infty$ better. Namely, connections between the BMO (bounded mean oscillation) distance of a bounded vector field on the sphere to a BMO harmonic gradient and the spectral properties of a “big” Hankel-like operator acting on $L^2$ harmonic gradients and valued in its orthogonal space are currently being investigated. We obtained a generalization of the Hodge decomposition (we call it the Hardy-Hodge decomposition) for ${ℝ}^n$-valued vector fields on ${ℝ}^{n-1}$ (resp. ${𝕊}^{n-1}$) which stands analog to the Hardy direct sum decomposition in dimension 1. In this decomposition, the analytic part becomes the trace of the gradient of a harmonic function in the half-space (resp. the ball) whose BMO-norm on parallel hyperplanes (resp. concentric spheres) is uniformly bounded. The two main difficulties facing a generalization of Nehari's theorem are the absence of a constructive derivation of Coifman-Rochberg weak factorizations and Wolff's phenomenon that a harmonic gradient (even $C^1$-smooth) is not determined by its values on a set of positive measure on ${ℝ}^n$ (resp. ${𝕊}^n$). This last point is the only obstacle to establish uniqueness and constancy-of-modulus properties. We shall concentrate on these two items in the future.

The above issue is also interesting in $L^p$, $1<p<\infty$, where it leads to analyze particular solutions to the the $p$-Laplacian on the sphere. This aspect is not pursued in depth at the moment.

Sources recovery in 3-D domains, application to MEEG and geophysics

The problem of sources recovery can be handled in 3-D balls by using best rational approximation on 2-D cross sections (disks) from traces of the boundary data on the corresponding circles (see section 4.1 ).

In 3-D, functional or clinical active regions in the cortex are often represented by pointwise sources that have to be localized from measurements on the scalp of a potential satisfying a Laplace equation (EEG, electroencephalography). In the work [7] it was shown how to proceed via best rational approximation on a sequence of 2-D disks cut along the inner sphere, for the case where there are at most 2 sources. A long-haul research on the behaviour of poles of best rational approximants of fixed degree to functions with branch points was completed this year [19] , which shows that the technique carries over to finitely many sources.

In this connection, a dedicated software “FindSources3D” (see section 5.7 ) has been developed, in collaboration with the team Athena.

Further, it appears that in the rational approximation step of these schemes, multiple poles possess a nice behaviour with respect to the branched singularities (see figure 4 ). This is due to the very basic physical assumptions on the model (for EEG data, one should consider triple poles). Though numerically observed, there is no mathematical justification so far why these multiple poles have such strong accumulation properties, which remains an intriguing observation. This is the topic of [30] .

Figure 4. Localization of 2 sources (red circles) by a single triple pole in plane sections (blue points).

Also, magnetic data from MEG (magneto-encephalography) will soon become available, which should enhance the accuracy of source recovery algorithms.

This approach also appears to be interesting for geophysical issues, namely discretizing the gravitational potential by means of pointwise masses. This is one topic of A.-M. Nicu's PhD thesis.

Magnetic sources localization from observations of the field away from the support of the magnetization is an issue under investigation in a joint effort with the Math. department of Vanderbilt University and the Earth Sciences department at MIT. The goal is to recover the magnetic properties of rock samples (meteorites) from fine measurements extremely close to the sample that can nowadays be obtained using SQUIDs (supraconducting coil devices).

The magnetization operator is the Riesz potential of the divergence of the magnetization, When the latter has bounded variation, we already described the kernel of this operator (the so-called silent magnetizations or silent source distributions) in terms of measures whose balayage on the boundary of the sample vanishes. This however, is not so very effective, computationally.

The case of a thin slab (the magnetization is then modelled as a vector field on a portion of the plane) has proved more amenable. We have shown that that silent sources from above or below can be characterized via the Hardy-Hodge decomposition mentioned in section 6.2.1 . The smoothness assumptions have been weakened considerably to accomodate magnetizations that may be any distribution with compact support, more generally any finite sum of partial derivatives of any order of $L^p$ or $BMO$ functions. Silent unidirectional and bi-directional magnetizations demonstrably reduce this way to certain divergence free tangential vector fields. In particulait no nonzero compactly supported unidirectional magnetization exists. In the $L^2$ setting, equivalent magnetizations of minimal $L^2$-norm can be computed using the Hardy-Hodge decomposition (which is orthogonal in this case), and an uncertainty principle relating the support of a magnetization and the support of its minimum-norm equivalent magnetization has been obtained. A paper is being written on these results.

Meanwhile, the severe ill-posedness of the reconstruction challenges discrete Fourier methods, one of the main problems being the truncation of the observations outside the range of the SQUID measurements. A next step will be to develop the extrapolation techniques initiated by the project team, using bounded extremal problems, in an attempt to overcome this issue.

Boundary value problems for 2-D conductivity equations, application to plasma control

In collaboration with the CMI-LATP (University Marseille I) and in the framework of the ANR AHPI, the team considers 2-D diffusion processes with variable conductivity. In particular its complexified version, the so-called conjugate or real Beltrami equation, was investigated. In the case of a smooth domain, and for Lipschitz conductivity, we analyzed the Dirichlet problem for solutions in Sobolev and then in Hardy classes [8] .

Their traces merely lie in $L^p$ ($1<p<\infty$) of the boundary, a space which is suitable for identification from pointwise measurements. Again these traces turn out to be dense on strict subsets of the boundary. This allows us to state Cauchy problems as bounded extremal issues in $L^p$ classes of generalized analytic functions, in a reminiscent manner of what was done for analytic functions as discussed in section  3.1.1 .

This year we generalized the construction to finitely connected Dini-smooth domains and $W^{1,q}$-smooth conductivities, with $q>2$ [43] . The case of an annular geometry is the relevant one for the application to plasma shaping mentioned below [17] . The application that initially motivated this work came from free boundary problems in plasma confinement (in tokamaks) for thermonuclear fusion. This work was initiated in collaboration with the Laboratoire J. Dieudonné (University of Nice) and is now the topic of a collaboration with two teams of physicists from the CEA-IRFM (Cadarache).

In the transversal section of a tokamak (which is a disk if the vessel is idealized into a torus), the so-called poloidal flux is subject to some conductivity equation outside the plasma volume for some simple explicit smooth conductivity function, while the boundary of the plasma (in the Tore Supra tokamak) is a level line of this flux [53] . Related magnetic measurements are available on the chamber, which furnish incomplete boundary data from which one wants to recover the inner (plasma) boundary. This free boundary problem (of Bernoulli type) can be handled through the solutions of a family of bounded extremal problems in generalized Hardy classes of solutions to real Beltrami equations, in the annular framework. Such approximation problems also allow us to approach a somewhat dual extrapolation issue, raised by colleagues from the CEA for the purpose of numerical simulation. It consists in recovering magnetic quantities on the outer boundary (the chamber) from an initial guess of what the inner boundary (plasma) is.

In the particular case at hand, the conductivity is $1/x$ and the domain is an annulus embedded in the right half-plane. We obtained a basis of solutions (exponentials times Legendre functions) upon separating variables in toroidal coordinates. This may be viewed as a generalization to the annulus of the Bessel type expansions derived in [21] for simply connected geometries. This provides a computational setting to solve the extremal problems mentioned before, and was the topic of the PhD thesis of Y. Fischer [17] , [22] . In the most recent tokamaks, like Jet or ITER, an interesting feature of the level curves of the poloidal flux is the occurrence of a cusp (a saddle point of the poloidal flux, called an X point), and it is desirable to shape the plasma according to a level line passing through this X point for physical reasons related to the efficiency of the energy transfer. We established well-posedness of the Dirichlet problem in weighted $L^p$ classes for harmonic measure on piecewise smooth domains without cusps, thereby laying ground for such a study. This issue is next in line, now that the present approach has been validated numerically on Tore Supra data.

On the half-plane, the conductivity $1/x$ is severely unbounded but the analysis of this test case is quite important for the convergence of extrapolation algorithms to recover magnetic quantities on the chamber. Additive decompositions into Hardy solutions inside the outer boundary and outside the inner boundary, with controlled vanishing on the imaginary axis, have been obtained as part of the PhD work of S. Chaabi. The latter developed this year a multiplicative parameterization of Hardy-smooth solutions by holomorphic functions for the conjugate Beltrami equation, a result which is both subtler and weaker than the classical Stoilow factorization for solutions to the complex Beltrami equation. This factorization is of considerable numerical interest in situations where conductivity is little or not known.