Regularized D-bar method for the inverse conductivity problem
Kim Knudsen - Department of Mathematics, Technical University of Denmark, Denmark (email)
Abstract: A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.
Keywords: inverse problem, ill-posed problem,
electrical impedance tomography, inverse conductivity problem,
Received: January 2009; Revised: September 2009; Published: October 2009.
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