Section: New Results

Inverse Problems for 2-D and 3-D elliptic operators

Participants : Laurent Baratchart, Aline Bonami [ Univ. Orléans ] , Maureen Clerc [ EPI Odyssée ] , Yannick Fischer, Sandrine Grellier [ Univ. Orléans ] , Mohamed Jaoua, Juliette Leblond, Jean-Paul Marmorat, Ana-Maria Nicu, Théo Papadopoulo [ EPI Odyssée ] , Jonathan R. Partington, Stéphane Rigat [ Univ. Aix-Marseille I ] , Emmanuel Russ [ Univ. Aix-Marseille III ] , Edward Saff, Meriem Zghal.

3-D boundary value problems for Laplace equation

Solving overdetermined Cauchy problems for the Laplace equation on a spherical layer (in 3-D) in order to treat 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. 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 [43] has been obtained. Still, a best approximation on the subset of a general vector field by a harmonic gradient under a L² 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[13] . Issues of robust interpolation on the sphere from incomplete pointwise data are also under study (splines, spherical harmonics, spherical wavelets, spherical Laplace operator, ...), in order to improve numerical accuracy of our reconstruction schemes.

The analogous problem in L^p , p2 , is considerably more difficult. A collaborative work is going on, in the framework of the ANR project AHPI, aiming mainly at the case p = . It was obtained that the BMO distance between a bounded vector field on the sphere and a bounded harmonic gradient is within a constant of the norm of a Hankel-like operator, acting on L² divergence-free vector fields with values in L² gradients. Estimating the constant requires solving further extremal problems in L¹ on the best approximation of a gradient by a divergence free vector field. This issue is currently being studied in L^p where it leads to analyze particular solutions to the the p -Laplacian on the sphere.

Sources recovery in 3-D domains, application to MEEG inverse problems

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 4.2 ).

The team started to consider more realistic geometries for the 3-D domain under consideration. A possibility is to parametrize it in such a way that its planar cross-sections are quadrature domains or R-domains. In this framework, best rational approximation can still be performed in order to recover the singularities of solutions to Laplace equations, but complexity issues are delicate. The preliminary case of an ellipsoid, which requires the preliminary computation of an explicit basis of ellipsoidal harmonics, has been studied in [78] and is one of the topics of the PhD thesis of M. Zghal.

In 3-D, epileptic 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). A breakthrough was made which makes it possible now to proceed via best rational approximation on a sequence of 2-D disks along the inner sphere [5] .

A dedicated numerical software “FindSources3D” (see section 5.7 ) has been developed, in collaboration with the team Odyssée.

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 figures 4 , 5 ). 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 why these multiple poles have such strong accumulation properties, which remains an intriguing observation. This is the topic of [63] .

Figure 5. 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 sources recovery.

This approach should also become interesting for geophysical issues, concerning the discretization of the gravitational potential by means of pointwise masses. This is another recent topic of A.-M. Nicu's PhD thesis and of our present collaboration with LAMSIN-ENIT, hence the reason why she also made a long working stay there (Univ. El Manar, Tunis, Tunisia, April-June).

Magnetic sources localization by analytic and rational approximation on plane sections is currently analyzed from experimental SQUID data, from Vanderbilt University Physics Dept. We have started analyzing the kernel of the magnetization operator, which is the Riesz potential of the divergence. The natural assumptions to handle magnetizations that are piecewise constant (allowing for characteristic elements embedded in the slab to be analyzed) has led us to study Riesz transforms and Hodge decompositions of functions of bounded variation, namely functions whose distributional derivatives are signed measures. The kernel, it has been found, can be described in terms of measures whose balayage on the boundary of the object vanishes. The constructive characterization of those is a rather difficult problem, but the restriction to more specific classes, like piecewise constant or unidirectional magnetizations that are of common use in the field, seems better suited to the purpose of algorithmically recovering m , up to a divergence-free term. The role of the extrapolation techniques initiated by the project team, using bounded extremal problems, should be important in this connection. This research sheds light on the connections between inverse current problems (aiming at the inversion of the Biot-Savart operator) and inverse magnetization problems (aiming at the inversion of the potential of a divergence).

2-D boundary value problems for 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 real Beltrami equation , was investigated. In the case of a smooth domain, and for a smooth conductivity, we analyzed the Dirichlet problem for solutions in Sobolev and then in Hardy classes  [46] .

Their traces merely lie in L^p (1<p< ) 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 . Recently, dual formulations were obtained and some multiplicative (fibered) structure for the solution was obtained based on old work by Bers and Nirenberg on pseudo-analytic functions. An article is being written on these topics.

The case of a conductivity that is merely in , which is important for inverse conductivity problems, is under examination (PhD thesis of S. Chaabi). There, it is still unknown whether solutions exist for all p .

The application that initially motivated this work comes from free boundary problems in plasma confinement (in tokamaks) for thermonuclear fusion. This work was started 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 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, it is possible to explicitly compute a basis of solutions (Bessel functions) that help the computations, see [64] , [65] . However, many other choices are possible, which are under study. This is the topic of the PhD thesis of Y. Fischer.

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 relating to the efficiency of the energy transfer. This will be the topic of future studies.