A regularized Newton method for locating thin tubular conductivity inhomogeneities

We consider the inverse problem of determining the position and shape of a thin tubular object, such as for instance a wire, a thin channel or a curve-like crack, embedded in some three-dimensional homogeneous body from a single measurement of electrostatic currents and potentials on the boundary of the body. Using an asymptotic model describing perturbations of electrostatic potentials caused by such thin objects, we reformulate the inverse problem as a nonlinear operator equation. We establish Fréchet differentiability of the corresponding operator, compute its Fréchet derivative and set up a regularized Newton scheme to solve the inverse problem numerically. We discuss our implementation of this method and present numerical results for simulated forward data. (paper)