Reconstruction in the Calder\'on problem on a fixed partition from finite and partial boundary data

TL;DR

This paper presents a modified reconstruction method for the Calderón problem that allows for the recovery of piecewise constant conductivities using finite, partial boundary data without requiring bounds or unknown partitions.

Contribution

The author adapts a previous method to work with known partitions, general conductivities, and limited boundary measurements, removing the need for bounds and the local Neumann-to-Dirichlet map.

Findings

Effective reconstruction with finite partial boundary data

No bounds on conductivities needed

Applicable to known partitions and general conductivities

Abstract

This short note modifies a reconstruction method by the author (Comm. PDE, 45(9):1118-1133, 2020), for reconstructing piecewise constant conductivities in the Calder\'on problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.