Inverse Problems and Imaging (IPI)

Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem

Pages: 991 - 1012, Volume 8, Issue 4, November 2014      doi:10.3934/ipi.2014.8.991

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Fabrice Delbary - University of Genoa, Department of Mathematics, Via Dodecaneso 35, 16146 Genoa, Italy (email)
Kim Knudsen - Danmarks Tekniske Universitet, Department of Applied Mathematics and Computer Science, Matematiktorvet, Building 303 B, DK - 2800 Kgs. Lyngby, Denmark (email)

Abstract: 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, electrical impedance tomography, reconstruction algorithm, numerical solution, singular boundary integral equation.
Mathematics Subject Classification:  Primary: 35R30; Secondary: 35J25, 65N21, 65R32.

Received: May 2014;      Revised: October 2014;      Available Online: November 2014.