A coupled complex boundary method for an inverse conductivity problem with one measurement

Abstract

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov regularization method can reconstruct the diffusion parameters stably and effectively.

Keywords:

• Coupled complex boundary method
• inverse conductivity problems
• Tikhonov regularization
• layer potential methods

AMS Subject Classifications:

• 65N21
• 65F22
• 47G40

Acknowledgements

We thank the two anonymous referees for their careful review on our manuscript and for their constructive comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the first author was supported partly by the Natural Science Foundation of China [grant number 11401304]; the Natural Science Foundation of Jiangsu Province [grant number BK20130780]; the Fundamental Research Funds for the Central Universities [grant number NS2014078]. The work of the second author was supported partly by the Natural Science Foundation of China [grant number 11571311]. The work of the third author was partly supported by Simons Foundation [grant number 207052], [grant number 228187]; the National Science Foundation [grant number DMS-1521684].