Conference on Applied Inverse Problems, July 20-24, 2009, Vienna, Austria
Poster Presentation
Cristiana Sebu: A regularized solution for the inverse conductivity problem using mollifiers

Thu, 23 July, 2009, 17:15-18:15, Foyer

The inverse conductivity problem has attracted much attention because although it is relatively easy to state, it turns out to be both nonlinear and, above all, extremely ill-posed. Furthermore it has a practical realisation which is known as Electrical Impedance Tomography (EIT). Substantial progress has been made in designing practical reconstruction algorithms applicable to noisy measurement data. While the iterative methods based on formulating the inverse problem in the framework of nonlinear optimisation techniques are promising for obtaining accurate reconstructed conductivity values, they may be slow to converge and are quite demanding computationally particularly when addressing the three dimensional problem. This concern has encouraged the search for reconstruction algorithms which reduce the computational demands. Some use a
priori information to reconstruct piecewise constant conductivity distributions e.g. [1,2] while others are based on reformulating the inverse problems in terms of integral equations [3-6].

In this paper we describe a reconstruction method suitable for $W^{2,\infty}$ conductivity distributions. It uses a simple transformation to establish a connection between the equations defining the inverse conductivity problem and those used in Inverse Scattering [7]. By combining this process with the powerful concept of mollifiers [8,9] we are able to obtain conductivity reconstructions for arbitrary geometry, extremely rapidly and without relying on accurate a priori information. This is achieved by reformulating the inverse problem in terms of a pair of coupled integral equations, one of which is of the first kind which we solve using mollifier methods. An interesting feature of this method is that the kernel of this integral equation is not given,
but can be modified for the choice of mollifier. We also discuss the choice of optimal boundary data.

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[2] M. Bruhl, M. Hanke and M. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93, 635-54, (2003).
[3] S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem. Inverse Problems 16, 681-99, (2000).
[4] R. Kress and W. Rundell, Nonlinear integral equations and their iterative solution for an inverse boundary value problem. Inverse Problems 21, 1207-23, (2005).
[5] S. Ciulli, M. Pidcock, T.D. Spearman and A. Stroian, Image reconstruction in Electrical Impedance Tomography using an integral equation of the Lippmann-Schwinger type. Phys. Lett. A 271, 377-84,(2000).
[6] S. Ciulli, M. K. Pidcock and C. Sebu, An integral equation method for the inverse conductivity problem. Phys. Lett. A 325, 253-67, (2004).
[7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edition, Springer-Verlag, (1998).
[8] A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind. Inverse Problems 6, 427-40, (1990).
[9] A. K. Louis, Approximate inverse for linear and some nonlinear problems. Inverse Problems 12, 175-90, (1996).

Presentation slides (pdf, 1000 KB)

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