The point source method for reconstructing an inclusion from boundary measurements in electrical impedance tomography and acoustic scattering

Abstract

We employ the point source method (PSM) for the reconstruction of some field u on parts of a domain Ω from the Cauchy data for the field on the boundary ∂Ω of the domain. Then, the boundary condition for a perfectly conducting inclusion or a sound-soft object in Ω can be used to find the location and shape of the inhomogeneity. The results show that we can detect perfectly conducting inclusions in impedance tomography from the voltages for one injected current. For acoustic scattering a sound-soft object is found from the knowledge of one (total) field and its normal derivative on ∂Ω. The work redesigns the PSM, which was first proposed in the framework of inverse scattering, to solve inverse boundary value problems. Numerical examples are provided for impedance tomography and the sound-soft acoustic boundary value problem.

Probing for electrical inclusions with complex spherical waves

“…For instance, if γ 1 (x)−γ 0 (x) has a constant sign on each connected component of 1 , and if we know that B(x 0 , R) touches only one of these components, our algorithm works. In dimension 2, numerical detection of inclusions and other anomalies from measurements on the whole boundary has been discussed in [3,5,8,9,10,12,14,16,17,18,27,28,29,33,37,42,43,44]. While some of these algorithms may be modified to accept localized data, the present paper is the first to numerically demonstrate the recovery of two-dimensional electrical inclusions from localized measurements.…”

The method of fundamental solutions for detection of cavities in EIT

“…Multiple star-shaped cavities can also be located in principle by applying the MFS to each cavity as described in Section 3. The MFS technique described in this paper can be extended to solving numerically the inverse cavity problem in the acoustic field [Erhard and Potthast, 2003], the inverse acoustic scattering problem [Colton andKress 1998, Johanasson andSleeman 2007] and the inverse electromagnetic scattering problem [Angell et al 2003], but these investigations are deferred to a future work.…”

Optimal Depth-Dependent Distinguishability Bounds for Electrical Impedance Tomography in Arbitrary Dimension

“…Since we only have a single connected inclusion, 2Λ C,R = Λ C,R | Y C,R , where Y C,R is a linear subspace of H 1/2 (∂B), which again may depend on the inclusion. Due to this inconvenience -in particular, for the inverse conductivity problem where the inclusion is not known a priori -the DN operator with a larger domain Λ C,R is often investigated instead ofΛ C,R (cf., e.g., [7,11,16,26,25,35]). This is also the choice in this work.…”