Regularized D-bar method for the inverse conductivity problem

Pages: 599 - 624, Volume 3, Issue 4, November 2009

doi:10.3934/ipi.2009.3.599       Abstract        Full Text (451.7K)       Related Articles

Kim Knudsen - Department of Mathematics, Technical University of Denmark, Denmark (email)
Matti Lassas - Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hallstromin katu 2b) FI-00014, Finland (email)
Jennifer L. Mueller - Department of Mathematics, Colorado State University, Fort Collins, CO 80523,, United States (email)
Samuli Siltanen - Department of Mathematics and Statistics, University of Helsinki, Finland (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, regularization.
Mathematics Subject Classification:  Primary: 35R30, 65N21; Secondary: 65J20.

Received: January 2009;      Revised: September 2009;      Published: October 2009.