Volume 9, Issue 4
An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

Houde Han, Leevan Ling & Tomoya Takeuchi

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

Published online: 2011-09

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  • 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.

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@Article{CiCP-9-878, author = {}, title = {An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain}, journal = {Communications in Computational Physics}, year = {2011}, volume = {9}, number = {4}, pages = {878--896}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.200110.060910a}, url = {http://global-sci.org/intro/article_detail/cicp/7526.html} }
TY - JOUR T1 - An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain JO - Communications in Computational Physics VL - 4 SP - 878 EP - 896 PY - 2011 DA - 2011/09 SN - 9 DO - http://doi.org/10.4208/cicp.200110.060910a UR - https://global-sci.org/intro/article_detail/cicp/7526.html KW - AB -

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.

Houde Han, Leevan Ling & Tomoya Takeuchi. (2020). An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain. Communications in Computational Physics. 9 (4). 878-896. doi:10.4208/cicp.200110.060910a
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