CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

INTEGRAL EQUATIONS FOR ELECTROMAGNETISM SCATTERING
[3] A. Bendali, M. Fares, A. Tizaoui, and S. Tordeux, (2012), Matched asymptotic expansions of the eigenvalues of a
3-D boundary-value problem relative to two cavities linked by a hole of small size, ommunications in Computational
Physics, 11 (2), 456–471.
[4] A. Bendali, A. Huard, A. Tizaoui, S. Tordeux, and J. P. Vila, (2009), Asymptotic expansions of the eigenvalues
of a 2-D boundary-value problem relative to two cavities linked by a hole of small size, C. R. Acad. Sci. Paris,
Mathmatiques, 347, 1147–1152.
2.4 Electromagnetic imaging by the Linear Sampling Method
A.-S. Bonnet-Ben Dhia, F. Collino, A. Cossonnière and M. Fares
The theory of inverse scattering for acoustic and electromagnetic waves, is an active area of research
with significant developments in the past few years. Inverse problems consist in getting informations on a
physical object from measurement data. More specifically, the inverse scattering problem is the problem of
finding characteristics of an unknown object referred to as scatterer (location, shape, material properties,...)
from measurement data of acoustic or electromagnetic waves scattered by this object. The question is not
only to detect objects like radar and sonar can do, but also to identify them.
Inverse problems are not easy to solve since they belong to the class of ill-posed problems as defined
by Hadamard. Indeed, a solution may not exist but even if it is the case, the solution does not depend
continuously on the data. Such problems require the use of regularisation schemes to be solved numerically.
The Linear Sampling Method is a technique which aims at reconstructing the shape of a scatterer from
multi-static electromagnetic data at a given frequency : the scatterer, which may be a perfectly conducting
body as well as a penetrable heterogeneity, is illuminated by harmonic plane waves in (almost) all possible
directions and the resulting far-fields are recorded in all directions. These data are used to build the so-called
LSM matrix whose (pseudo) inversion allows to discriminate between sampling points inside or outside the
scatterer.
Our recent contributions on this topic concern on one hand algorithmic aspects : in collaboration with the
ALGO team, a new approach has been developed leading to a significant acceleration of the whole imaging
process. On the other hand, the aim of the thesis of Anne Cossonnière (October 2008 - December 2011),
under the supervision of Houssem Haddar (INRIA project-team DEFI), is to investigate the potential interest
of the so-called interior transmission frequencies in order to image the interior of a penetrable scatterer and
to answer some open problems on this subject.
2.5 Fast solution algorithm : the SVD-tail
In the classical approach, a system involving the LSM matrix has to be inverted for each sampling point
(and in practice, a large number of sampling points is required to get an accurate image). This system being
ill-posed, a Tikhonov-Morozov regularization technique is used, which is quite costly since a full-SVD of
the matrix is achieved and the Tikhonov regularization parameter has to be determined, using the Morozov
discrepancy principle, for each sampling point.
The new approach that we have developed in collaboration with S. Gratton and P. Toint is both simpler and
faster. The main point is that imaging the scatterer does not require the knowledge of the solution of the
LSM system, but only the knowledge of whether this system has or has not a (pseudo) solution. This can
be achieved by a fully iterative algorithm : a small number of left singular vectors associated to the smallest
singular values are first approximated (by the classical power method) ; then the orthogonality of the RHS
to these vectors is simply tested. Let us emphasize that only few left singular vectors are needed thanks to
the presence of noise in the data and in the discretized operator. In a typical application with 2252 incident
directions and 125000 sampling points, a speed-up factor of 50 is obtained [EMA15].
50 Jan. 2010 – Dec. 2011