An order optimal regularization method for the Cauchy problem of a Laplace equation in an annulus domain

The detection of defects due to corrosion in oil pipeline industry is very important. Detecting corrosion by electrical field can be modeled as a Cauchy problem for a Laplace equation in an annulus domain, which is well known to be severely ill-posed. In this paper, a modified Tikhonov regularization method is proposed. And a Hölder-type error estimate is achieved, which is order optimal according to the general regularization theory. Moreover, a Fast Fourier Transform (FFT) is used in the numerical implementation. Finally, numerical results show that the regularization method and FFT work well.