Section: Application Domains

Inverse problems for elliptic PDE

Participants : Laurent Baratchart, Juliette Leblond, Ana-Maria Nicu, Dmitry Ponomarev.

This work is done in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena project-team.

Solving overdetermined Cauchy problems for the Laplace equation on a spherical layer (in 3-D) in order to extrapolate incomplete data (see section 3.2.1 ) 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 through several layers of different conductivities, from the boundary down to the center of the domain where the singularities (i.e. the sources) lie. Actually, once propagated to the innermost sphere, it turns out that that traces of the boundary data on 2-D cross sections (disks) coincide with analytic functions in the slicing plane, that has branched singularities inside the disk [4] . These singularities are related to the actual location of the sources (namely, they reach in turn a maximum in modulus when the plane contains one of the sources). Hence, we are back to the 2-D framework of section 3.3.3 where approximately recovering these singularities can be performed using best rational approximation.

Numerical experiments gave very good results on simulated data and we are now proceeding with real experimental magneto-encephalographic data, see also sections 5.6 and 6.1 . The PhD thesis of A.-M. Nicu [13] was concerned with these applications, see [16] , in collaboration with the Athena team at Inria Sophia Antipolis, and neuroscience teams in partner-hospitals (hosp. Timone, Marseille).

Similar inverse potential problems appear naturally in magnetic reconstruction. A particular application, which is the object of a joint NSF project with Vanderbilt University and MIT, is to geophysics. There, the remanent magnetization of a rock is to be analysed to draw information on magnetic reversals and to reconstruct the rock history. Recently developed scanning magnetic microscopes measure the magnetic field down to very small scales in a “thin plate” geological sample at the Laboratory of planetary sciences at MIT, and the magnetization has to be recovered from the field measured on a plane located at small distance above the slab.

Mathematically speaking, EEG and magnetization inverse problems both amount to recover the (3-D valued) quantity m (primary current density in case of the brain or magnetization in case of a thin slab of rock) from measurements of the vector potential:

Ω divm(x ' )dx ' |x-x ' |,

outside the volume Ω of the object, from Maxwell's equations.

The team is also getting engaged in problems with variable conductivity governed by a 2-D conjugate-Beltrami equation, see [5] , [58] , [35] . The application we have in mind is to plasma confinement for thermonuclear fusion in a Tokamak, more precisely with the extrapolation of magnetic data on the boundary of the chamber from the outer boundary of the plasma, which is a level curve for the poloidal flux solving the original div-grad equation. Solving this inverse problem of Bernoulli type is of importance to determine the appropriate boundary conditions to be applied to the chamber in order to shape the plasma [54] . These issues are the topics of the PhD theses of S. Chaabi and D. Ponomarev [27] , and of a joint collaboration with the Laboratoire J.-A. Dieudonné at the Univ. of Nice-SA (and the Inria team Castor), and the CMI-LATP at the Univ. of Aix-Marseille I (see section 6.2 ).