New global stability estimates for the Calder\'on problem in two dimensions

TL;DR

This paper establishes a new global stability estimate for the two-dimensional Calderón problem, demonstrating that increased potential smoothness leads to exponentially improved stability in reconstruction.

Contribution

It introduces a novel stability estimate linking potential smoothness to reconstruction stability, enhancing understanding of inverse boundary value problems in 2D.

Findings

Stability improves exponentially with potential smoothness.

New estimates apply to electrical impedance tomography.

Enhanced understanding of inverse problem stability.

Abstract

We prove a new global stability estimate for the Gel'fand-Calder\'on inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-\Delta \psi + v\, \psi = 0$ on $D$, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The principal feature of this estimate is that it shows that the more a potential is smooth, the more its reconstruction is stable, and the stability varies exponentially with respect to the smoothness (in a sense to be made precise). As a corollary we obtain a similar estimate for the Calder\'on problem for the electrical impedance tomography.