Inverse Problems and Imaging (IPI)

Recovering the boundary corrosion from electrical potential distribution using partial boundary data

Pages: 521 - 538, Volume 11, Issue 3, June 2017      doi:10.3934/ipi.2017024

       Abstract        References        Full Text (378.7K)       Related Articles       

Jijun Liu - School of Mathematics, Southeast University, Nanjing 210096, China (email)
Gen Nakamura - Department of Mathematics, Inha University, Incheon 22212, Republic of Korea (email)

Abstract: We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specified on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Hölder type based on some a priori assumptions on the unknown impedance and the electrical current input specified in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data.

Keywords:  Electrical potential, Laplace equation, boundary impedance, uniqueness, stability, regularization, convergence.
Mathematics Subject Classification:  35J05, 35J25, 35R25, 35R30.

Received: January 2016;      Revised: February 2017;      Available Online: April 2017.