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### Open Access Journals

IPI

We study the reconstruction of the shape of a perfectly conducting inclusion
in three dimensional electrical impedance tomography (EIT) using a regularized
Newton method. This method involves a least squares penalty in the
form of an additional nonlinear operator to cope with the non-uniqueness
of general parametrizations of the unknown boundary.
We provide a convergence result for this
method in the general framework of nonlinear ill-posed operator equations.
Moreover, we discuss the evaluation of the forward operator in EIT,
its derivative, and the adjoint of the derivative using a wavelet based
boundary element method.
Numerical examples illustrate the performance of our method.

IPI

We consider the inverse problem to identify coefficient functions
in boundary value problems from noisy measurements of the solutions. Our
estimators are defined as minimizers of a Tikhonov functional, which is the
sum of a nonlinear data misfit term and a quadratic penalty term involving
a Hilbert scale norm. In this abstract framework we derive estimates of the
expected squared error under certain assumptions on the forward operator.
These assumptions are shown to be satisfied for two classes of inverse elliptic
boundary value problems. The theoretical results are confirmed by Monte
Carlo simulations.

IPI

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to 0. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.

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