An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

Abstract

Abstract. Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

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

“…Assuming that the entire exterior boundary C out is accessible, gðxÞ and hðxÞ are known for allx on C out . Here we use the same assumptions as in [1][2][3].…”

“…For computational aspects, the readers can refer to [5,[21][22][23][24][25][26][27][28]. For the Cauchy problem of the Laplace equation in an annulus domain, two good references are [1,3], which use an energy regularization method and the method of fundamental solutions (MFS) to obtain regularized solutions from the noisy Cauchy data, respectively. However, the error estimates are not order optimal.…”

“…According to the Association of Oil pipe Lines (AOPL), the most recent Crack Detection Tools include ultrasonic crack detection, magnetic flux leakage, and elastic wave tool. Varieties of detects in pipeline can be classified into three categories: defects due to corrosion, defects generated by mechanical damage, and cracks created by stresses in the pipe wall, see [1] for details.…”

“…Once the solution of Cauchy problem (1) and (2) is obtained, one can easily use (3) to find cðxÞ as follows:…”

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

“…Laleg-Kirati is an assistant professor in CEMSE, KAUST, K.S.A. email: taousmeriem.laleg@kaust.edu.sa method is introduced in [10]. Regularization methods in general affect the solution depending on the choice of a particular regularization parameter and it also increase the computation cost.…”