Alternative Tikhonov-Type Regularization for the Calderón Inverse Conductivity Problem
This paper investigates the application of two regularization methods, classical Tikhonov and Alternative Tikhonov regularization, to the Calderón inverse conductivity problem. We develop an iterative reconstruction scheme based on linearizing the Dirichlet-to-Neumann map and approximating conductivity updates using Gaussian basis functions. The resulting linear systems are solved using both regularization methods, with particular attention given to the derivation and numerical implementation of Alternative Tikhonov regularization. We compare the performance of these methods through numerical experiments on a synthetic conductivity containing two localized regions of differing conductivity. A total of 666 reconstruction tests are performed across varying regularization parameters, damping factors, and iteration counts. The lowest relative reconstruction errors obtained are 0.064730 for Alternative Tikhonov and 0.065173 for classical Tikhonov regularization. Although both methods produce comparable reconstructions, Alternative Tikhonov achieves the lowest observed error. The results demonstrate the effectiveness of both approaches within the proposed reconstruction scheme and motivate further investigation into parameter selection, numerical stability, and reconstruction accuracy.