### Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume

Mark Asch, Marion Darbas, Jean-Baptiste Duval
2010 E S A I M: Control, Optimisation and Calculus of Variations
We consider the numerical solution, in two-and three-dimensional bounded domains, of the inverse problem for identifying the location of small-volume, conductivity imperfections in a medium with homogeneous background. A dynamic approach, based on the wave equation, permits us to treat the important case of "limited-view" data. Our numerical algorithm is based on the coupling of a finite element solution of the wave equation, an exact controllability method and finally a Fourier inversion for
more » ... calizing the centers of the imperfections. Numerical results, in 2-and 3-D, show the robustness and accuracy of the approach for retrieving randomly placed imperfections from both complete and partial boundary measurements. Mathematics Subject Classification. 35R30, 35L05, 65M60. the numerical application of the above method to exact boundary control of the wave equation by Glowinski 2 et al. -see ; the theoretical results of the problem of detection for the wave equation by Ammari -see . 3 This paper is structured as follows. We begin, in Section 2, by defining some notation and then formulate the 4 inverse initial boundary value problem. We also address the identification procedure by the Fourier method. In 5 Section 3, we recall the numerical algorithm for the Hilbert Uniqueness Method of Lions . We present the 6 numerical method used for the dynamic detection problem in Section 4. Then numerical results obtained from 7 simulations are shown in Section 5. Finally, some conclusions and perspectives are reported in the last section. 8 2. Dynamic localization theory 9 In this part, we recall the different results contained in the article of Ammari (see  for the full details) on 10 which the numerical algorithm and the subsequent simulations will be based. 11 2.1. Some notation and presentation of the inverse problem 12 2.1.1. Some notation 13 Let Ω be a bounded, smooth subdomain of R d , d = 2, 3, with for simplicity, a smooth boundary ∂Ω, with 14 n denoting the outward unit normal to ∂Ω. We suppose that Ω contains a finite number m of imperfections, 15 each of the form z j + αB j , where B j ⊂ R d is a bounded, smooth domain containing the origin. This gives a 16 collection of imperfections of the form B α = ∪ m j=1 (z j + αB j ). The points z j ∈ Ω, j = 1, ..., m, that define the 17 locations of the imperfections are assumed to satisfy two distance conditions: We also assume that α > 0, the common order of magnitude of the diameters of the imperfections, is sufficiently 19 small so that these are disjoint and their distance to R d \Ω is larger than d 0 /2. 20 2.1.2. Presentation of the inverse problem 21 Let γ 0 denote the conductivity of the background medium which, for simplicity, we shall assume is constant. 22