Uniform stability estimates for the discrete Calderon problems

TL;DR

This paper establishes uniform stability estimates for discrete Calderon problems in dimensions three and higher, adapting continuous techniques like Carleman estimates and CGO solutions to the discrete setting.

Contribution

It introduces a novel approach to obtain stability estimates uniformly with respect to discretization, extending continuous methods to the discrete Calderon problem.

Findings

Discrete Carleman estimates for the Laplace operator

Uniform stability estimates independent of mesh size

Adaptation of CGO solutions to the discrete setting

Abstract

In this article, we focus on the analysis of discrete versions of the Calderon problem in dimension d \geq 3. In particular, our goal is to obtain stability estimates for the discrete Calderon problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we shall prove discrete Carleman estimates for the discrete Laplace operator. A main difference with the continuous ones is that there, the Carleman parameters cannot be taken arbitrarily large, but should be smaller than some frequency scale depending on the mesh size. Following the by-now classical Complex Geometric Optics (CGO) approach, we can thus derive discrete CGO solutions, but with limited range of parameters. As in the continuous case, we then use these solutions to obtain uniform stability estimates for the discrete Calderon problems.