A direct method for reconstructing inclusions and boundary conditions from electrostatic data

Abstract

In this paper, we will discuss the use of the Linear Sampling Method to reconstruct impenetrable inclusions from Electrostatic Cauchy data. We consider the case of a perfectly conducting and impedance inclusion. In either case, we show that the Dirichlet to Neumann mapping can be used to reconstruct impenetrable sub-regions via a sampling method. We also propose a direct method based on boundary integral equations to reconstruct the impedance parameter using the reconstructed boundary of the inclusion from the knowledge of multiple Cauchy pairs which can be computed from Dirichlet to Neumann mapping. Some numerical reconstructions are presented in two space dimensions.

Keywords:

• Inverse boundary value problems
• shape reconstruction
• boundary impedance
• sampling methods
• integral equations

AMS subject classifications:

• 35J05
• 45Q05
• 65R30

Acknowledgements

The research was started while the author was at Texas A&M University. The university's hospitality during the time the author was there is greatly appreciated. The author would also like to thank William Rundell for providing the code to solve the direct problem and for helpful discussions during the author's time at Texas A$\mathrm{\&}$M University and beyond.

Disclosure statement

No potential conflict of interest was reported by the author(s).