Section: Application Domains

Geometric inverse problems for elliptic partial differential equations

Participants : Laurent Baratchart, Sylvain Chevillard, Maureen Clerc [EPI Athena] , Yannick Fischer [Until November] , Juliette Leblond, Ana-Maria Nicu, Théo Papadopoulo [EPI Athena] .

This domain is mostly connected to the techniques described in section  3.1 .

We are mainly concerned with classical inverse problems like the one of localizing defaults (as cracks, pointwise sources or occlusions) in a two or three dimensional domain from boundary data (which may correspond to thermal, electrical, or magnetic measurements), of a solution to Laplace or to some conductivity equation in the domain. These defaults can be expressed as a lack of analyticity of the solution of the associated Dirichlet-Neumann problem that may be approached, in balls, using techniques of best rational or meromorphic approximation on the boundary of the object (see section 3.1 ).

Actually, it turns out 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 [7] . 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 where approximately recovering these singularities can be performed using best rational approximation.

In this connection, the realistic case where data are available on part of the boundary only offers a typical opportunity to apply the analytic extension techniques (see section 3.1.1 ) to Cauchy type issues, a somewhat different kind of inverse problems in which the team is strongly interested.

The approach proposed here consists in recovering, from measured data on a subset $K$ of the boundary $\partial D$ of a domain $D$ of $R^2$ or $R^3$, say the values $F_K$ on $K$ of some function $F$, the subset $\gamma \subset D$ of its singularities (typically, a crack or a discrete set of pointwise sources), provided that $F$ is an analytic function in $D\setminus \gamma$.

• The analytic approximation techniques (section 3.1.1 ) first allow us to extend $F$ from the given data $F_K$ to all of $\partial D$, if $K\ne \partial D$, which is a Cauchy type issue for which our algorithms provide robust solutions, in plane domains (see [2] for 3D spherical situations, also discussed in section 6.2 ).

• From these extended data on the whole boundary, one can obtain information on the presence and location of $\gamma$, using rational or meromorphic approximation on the boundary (section 3.1 ). This may be viewed as a discretization of $\gamma$ by the poles of the approximants [6] .

  This is the case in dimension 2, using classical links between analyticity and harmonicity [4] , but also in dimension 3, at least in spherical or ellipsoidal domains, working on 2-D plane sections, [7] , [73] .

The previous two steps were shown to provide a robust way of locating sources from incomplete boundary data in a 2-D situation with several annular layers. Numerical experiments have already yielded excellent results in 3-D situations and we are now on the way to process real experimental magneto-encephalographic data, see also sections 5.7 and 6.2.2 . The PhD thesis of A.-M. Nicu is concerned with these applications, see [30] , in collaboration with the Athena team of Inria Sophia Antipolis, and with neuroscience teams in partner-hospitals (hosp. Timone, Marseille).

Such methods are currently being generalized to problems with variable conductivity governed by a 2-D conjugate-Beltrami equation, see [8] , [21] . 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 [53] . These issues are the topics of the PhD theses of S. Chaabi and Y. Fischer [17] , and of a joint collaboration with the CEA-IRFM (Cadarache), the Laboratoire J.-A. Dieudonné at the Univ. of Nice-SA, and the CMI-LATP at the Univ. of Marseille I (see section 6.2.3 ).

Inverse potential problems are also naturally encountered in magnetization issues that arise in nondestructive control. A particular application, which is the object of a joint NSF-supported project with Vanderbilt University and MIT, is to geophysics where the remanent magnetization of a rock is to be analyzed using a squid-magnetometer in order to analyze the rock history; specifically, the analysis of Martian rocks is conducted at MIT, for instance to understand if inversions of the magnetic field took place there. Mathematically speaking, the problem is to recover the (3-D valued) magnetization $m$ from measurements of the vector potential:

outside the volume $\Omega$ of the object.

In 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 also the topic of the PhD theses of A.-M. Nicu, and 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.