Commun. Comput. Phys., 9 (2011), pp. 878-896.


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

Houde Han 1, Leevan Ling 2*, Tomoya Takeuchi 3

1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.
2 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
3 Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212, USA.

Received 20 January 2010; Accepted (in revised version) 6 September 2010
Available online 13 October 2010
doi:10.4208/cicp.200110.060910a

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.

AMS subject classifications: 65N12, 65N15, 65N21

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Key words: Inverse problem, stability, error bounds, Tikhonov regularization, method of fundamental solution.

*Corresponding author.
Email: hhan@math.tsinghua.edu.cn (H. Han), lling@hkbu.edu.hk (L. Ling), tntakeuc@ncsu.edu (T. Takeuchi)
 

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