A Newton method for reconstructing non star-shaped domains in electrical impedance tomography
Helmut Harbrecht Thorsten Hohage
Inverse Problems & Imaging 2009, 3(2): 353-371 doi: 10.3934/ipi.2009.3.353
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.
keywords: nonlinear inverse problems regularization inverse obstacle problems. electrical impedance tomography
Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise
Thorsten Hohage Mihaela Pricop
Inverse Problems & Imaging 2008, 2(2): 271-290 doi: 10.3934/ipi.2008.2.271
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.
keywords: parameter identification problems inverse boundary value problems. nonlinear Tikhonov regularization statistical inverse problem Hilbert scales
Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems
Frederic Weidling Thorsten Hohage
Inverse Problems & Imaging 2017, 11(1): 203-220 doi: 10.3934/ipi.2017010

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.

keywords: Inverse scattering Maxwell equations stability regularization source condition

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