Reconstructing Discontinuities Using Complex Geometrical Optics Solutions

Abstract

Abstract. In this paper we provide a framework for constructing general complex geometrical optics solutions for several systems of two variables that can be reduced to a system with the Laplacian as the leading order term. We apply these special solutions to the problem of reconstructing inclusions inside a domain filled with known conductivity from local boundary measurements. Computational results demonstrate the versatility of these solutions to determine electrical inclusions. Key words. geometrical optics, inclusions, numerical algorithm AMS subject classifications. 35R30, 65N21DOI. 10.1137/0606763501. Introduction. Inverse boundary value problems are a class of inverse problems where one attempts to determine the internal parameters of body by making measurements only at the surface of the body. A prototypical example that has received a lot of attention is electrical impedance tomography (EIT). In this inverse method one would like to determine the conductivity distribution inside a body by making voltage and current measurements at the boundary.There are many applications of EIT ranging from early breast cancer detection [32] to geophysical sensing for underground objects; see [18], [24], [25], [27]. The article [28] and the ones reviewed in [29] assume that the measurements are made on the whole boundary. However, it is often possible to make the measurements only on part of the boundary; this is the partial data problem. This is the case for the applications in breast cancer detection and geophysical sensing mentioned above.The boundary information is encoded into the Dirichlet-to-Neumann map associated with the conductivity equation. More precisely, let Ω be an open bounded domain with smooth boundary in R d with d = 2 or 3. Assume that γ(x) > 0 in Ω possesses a suitable regularity. The conductivity equation is described by the following elliptic equation:

Complex geometrical optics solutions for anisotropic equations and applications

“…Note that Mittag-Leffler functions can be only applied to the case where the background equation is the Laplacian. Like the results in [21], the method of this paper works for any general second order elliptic equations with coefficients having appropriate finite smoothness.…”

“…With complex spherical waves, one may be able to reconstruct some nonconvex parts of the object [8], [19]. In the two-dimensional case, we are able to get much more information of the object by using Mittag-Leffler functions [14], [15] or the special solutions constructed in [21]. Note that Mittag-Leffler functions can be only applied to the case where the background equation is the Laplacian.…”

“…Those systems include the conductivity equation, the elasticity equations, and the Stokes equations, all with isotropic inhomogeneous coefficients. In this paper we will extend the method in [21] and [22] to scalar equations with anisotropic inhomogeneous coefficients in the plane. We shall focus on the conductivity equation:…”

“…In previous works [21] and [22], special complex geometrical optics (CGO) solutions for certain isotropic systems in the plane were constructed and their applications to the object identification problem were also demonstrated theoretically and numerically. Those systems include the conductivity equation, the elasticity equations, and the Stokes equations, all with isotropic inhomogeneous coefficients.…”

“…This idea is crucial in studying the Calderón problem in two dimensions (see, for example, [2] and [20]). After reducing the anisotropic equation (1.1) to an isotropic one, we can apply the method in [21] to construct CGO solutions for the isotropic equation and then transform the solutions back to the original coordinates.…”