@MASTERSTHESIS\{IMM2013-07022,
    author       = "K. R. Jensen",
    title        = "Shape Optimization for Electrical Impedance Tomography",
    year         = "2013",
    school       = "Technical University of Denmark, Department of Applied Mathematics and Computer Science",
    address      = "Richard Petersens Plads, Building 324, {DK-}2800 Kgs. Lyngby, Denmark, compute@compute.dtu.dk",
    type         = "",
    note         = "{DTU} Supervisors: Main supervisor: Kim Knudsen, kiknu@dtu.dk, {DTU} Compute, and Jens Gravesen, and Anton Evgrafov",
    url          = "http://www.compute.dtu.dk/English.aspx",
    abstract     = "This thesis presents a solution of an inverse boundary value problem for harmonic functions arising in Eletrical Impedance Tomography. The concerned problem is regarding shape optimization of perfectly conducting circular inclusions using {B-}splines. In the thesis a representation of the harmonic electrical potential using boundary integrals is set up as to make a Neumann to Dirichlet (current to voltage) map on the outer boundary, represented by the unit circle. The Cauchy data obtained is hereby associated with the shape and location of the perfectly conducting inclusion. The geometry and governing functions have been approximated by {B-}splines and the Boundary Element Method (BEM) has been used to discretize the equations with regards to implementation in matlab.

The forward problem has been set up, as to given the boundary of the perfectly conducting inclusion and an applied current obtain the voltage distribution on the outer boundary. The algorithm has been tested using a solution obtained from separation of variables and seen to approximate the analytical solutions for both a concentric and a non-concentric circular inclusion. The Cauchy data for the non-concentric case have been found using a conformal map from the concentric to the non-concentric case.

The inverse problem of optimizing the shape of the inclusion, that is optimizing the control points for the boundary, represented by {B-}splines, has seen to satisfactory detect and approximate the circular boundary, when applying curve speed regularization. For this purpose the Interior-point algorithm has been used from matlabs optimization framework."
}