Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem
Fabrice Delbary Kim Knudsen
Inverse Problems & Imaging 2014, 8(4): 991-1012 doi: 10.3934/ipi.2014.8.991
The Calderón problem is the mathematical formulation of the inverse problem in Electrical Impedance Tomography and asks for the uniqueness and reconstruction of an electrical conductivity distribution in a bounded domain from the knowledge of the Dirichlet-to-Neumann map associated to the generalized Laplace equation. The 3D problem was solved in theory in late 1980s using complex geometrical optics solutions and a scattering transform. Several approximations to the reconstruction method have been suggested and implemented numerically in the literature, but here, for the first time, a complete computer implementation of the full nonlinear algorithm is given. First a boundary integral equation is solved by a Nyström method for the traces of the complex geometrical optics solutions, second the scattering transform is computed and inverted using fast Fourier transform, and finally a boundary value problem is solved for the conductivity distribution. To test the performance of the algorithm highly accurate data is required, and to this end a boundary element method is developed and implemented for the forward problem. The numerical reconstruction algorithm is tested on simulated data and compared to the simpler approximations. In addition, convergence of the numerical solution towards the exact solution of the boundary integral equation is proved.
keywords: Calderón problem singular boundary integral equation. numerical solution electrical impedance tomography reconstruction algorithm
Electrical impedance tomography using a point electrode inverse scheme for complete electrode data
Fabrice Delbary Rainer Kress
Inverse Problems & Imaging 2011, 5(2): 355-369 doi: 10.3934/ipi.2011.5.355
For the two dimensional inverse electrical impedance problem in the case of piecewise constant conductivities with the currents injected at adjacent point electrodes and the resulting voltages measured between the remaining electrodes, in [3] the authors proposed a nonlinear integral equation approach that extends a method that has been suggested by Kress and Rundell [10] for the case of perfectly conducting inclusions. As the main motivation for using a point electrode method we emphasized on numerical difficulties arising in a corresponding approach by Eckel and Kress [4, 5] for the complete electrode model. Therefore, the purpose of the current paper is to illustrate that the inverse scheme based on point electrodes can be successfully employed when synthetic data from the complete electrode model are used.
keywords: boundary integral equations Newton scheme. Electrical impedance tomography point electrode model complete electrode model

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