Section: New Results

Source recovery problems

Participants : Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Ana-Maria Nicu.

The works presented here are done in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena EPI, with Doug Hardin and Edward Saff from Vanderbilt University (Nashville, USA), and with Abderrazek Karoui (Univ. Bizerte, Tunisie) and Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris).

This section in dedicated to inverse problems for 3-D Poisson-Laplace equations. Though the geometrical settings differ in the 2 sections below, the characterization of silent sources (that give rise to a vanishing potential at measurement points) is a common problem to both which has been recently achieved, see [37] ,[29] , [39] .These are sums of (distributional) derivatives of Sobolev functions vanishing on the boundary.

Application to EEG

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 [4] 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 milestone in a long-haul research on the behaviour of poles of best rational approximants of fixed degree to functions with branch points has been reached this year [14] , which shows that the technique carries over to finitely many sources (see section 4.2 ). In this connection, a dedicated software “FindSources3D” (see section 5.6 ) has been developed, in collaboration with the team Athena [16] , [26] .

Further, it appears that in the rational approximation step of these schemes, multiple poles possess a nice behaviour with respect to the branched singularities. This is due to the very basic physical assumptions on the model (for EEG data, one should consider triple poles). Though numerically observed in [16] , there is no mathematical justification so far why these multiple poles have such strong accumulation properties, which remains an intriguing observation.

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 [13] , [67] ), while other techniques should be considered as well.

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

It turns out that discretization issues in geophysics can also be approached by these approximation techniques. Namely, in geodesy or for GPS computations, one may need to get a best discrete approximation of the gravitational potential on the Earth's surface, from partial data collected there. This is the topic of a beginning collaboration with a physicist colleague (IGN, LAREG, geodesy). Related geometrical issues (finding out the geoid, level surface of the gravitational potential) are worthy of consideration as well.

Magnetization issues

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 (e.g. meteorites or stalactites) from fine field measurements 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. The problem of recovering a thin plate magnetization distribution from measurements of the field in a plane above the sample lead us to an analysis of the kernel of this operator, which we characterized in various function and distribution spaces (arbitrary compactly supported distributions or derivatives of bounded functions). For this purpose, we introduced a generalization of the Hodge decomposition in terms of Riesz transforms and showed that a thin plate magnetization is “silent” (i.e. in the kernel) if the normal component is zero and the tangential component is divergence free. In particular, we show that a unidirectional non-trivial magnetization with compact support cannot be silent. The same is true for bidirectional magnetizations if at least one of the directions is nontangential. We also proved that any magnetization is equivalent to a unidirectional. We did introduce notions of being silent from above and silent from below, which are in general distinct. These results have been reported in a paper to appear [37] .

We currently work on Fourier based inversion techniques for unidirectional magnetizations, and Figures 5 , 6 , 7 and 8 show an example of reconstruction. A joint paper with our collaborators from VU and MIT is being written on this topic.

Figure 5. Inria's logo were printed on a piece of paper. The ink of the letters “In” were magnetized along a direction $D_1$. The ink of the letters “ria” were magnetized along another direction $D_2$ (almost orthogonal to $D_1$).

Figure 6. The Z-component of the magnetic field generated by the sample is measured by a SQUID microscope. The measure is performed 200µm above the sample.

Figure 7. The field measured in Figure 6 is inversed, assuming that the sample is unidimensionally magnetized along the direction $D_1$. The letters “In” are fairly well recovered while the rest of the letters is blurred (because the hypothesis about the direction of magnetization is false for “ria”).

Figure 8. The field measured in Figure 6 is inversed, assuming that the sample is unidimensionally magnetized along the direction $D_2$. The letters “ria” are fairly well recovered while the rest of the letters is blured (because the hypothesis about the direction of magnetization is false for “In”).

For more general magnetizations, the severe ill-posedness of reconstruction challenges discrete Fourier methods, one of the main problems being the truncation of the observations outside the range of the SQUID measurements. We look forward to develop extrapolation techniques in the spirit of step 1 in section 3.1 .