 
Summary: NUMERICAL SOLUTION OF AN INVERSE INITIAL BOUNDARY VALUE
PROBLEM FOR THE WAVE EQUATION IN THE PRESENCE OF
CONDUCTIVITY IMPERFECTIONS OF SMALL VOLUME
M. ASCH, M. DARBAS, J.B. DUVAL
Abstract. We consider the numerical solution, in two and threedimensional bounded do
mains, of the inverse problem for identifying the location of smallvolume, conductivity imper
fections in a medium with homogeneous background. A dynamic approach, based on the wave
equation, permits us to treat the important case of limitedview data. Our numerical algorithm
is based on the coupling of a nite element solution of the wave equation, an exact controllability
method and nally a Fourier inversion for localizing the centers of the imperfections. Numerical
results, in 2 and 3D, show the robustness and accuracy of the approach for retrieving randomly
placed imperfections from both complete and partial boundary measurements.
1. Introduction
The localization of small imperfections is of great importance since there are numerous practical
applications, particularly in the elds of medical imaging and nondestructive testing of materials.
Generally, when we seek to localize an imperfection contained in a bounded domain, we need to
solve an inverse problem for retrieving the geometry of the imperfection.
The determination of conductivity proles from knowledge of boundary measurements has re
ceived a great deal of attention (see, for example, [1], [2], [7] and [27]). However, the reconstruction
of imperfections within a dynamical (i.e. timedependent) framework has not been widely inves
