Stability estimate for the discrete Calderon problem from partial data

TL;DR

This paper establishes logarithmic stability estimates for the discrete Calderon problem with partial boundary data in higher dimensions, using novel discrete Carleman estimates and unique continuation techniques.

Contribution

It introduces a new strategy for discrete unique continuation estimates based on a novel Carleman estimate, advancing the understanding of the discrete Calderon problem.

Findings

Logarithmic stability estimates for discrete Calderon problem

Development of a new discrete Carleman estimate

A novel approach for discrete unique continuation

Abstract

In this paper, we focus on the analysis of discrete versions of the Calderon problem with partial boundary data in dimension d >= 3. In particular, we establish logarithmic stability estimates for the discrete Calderon problem on an arbitrarily small portion of the boundary under suitable a priori bounds. For this end, we will use CGO solutions and derive a new discrete Carleman estimate and a key unique continuation estimate. Unlike the continuous case, we use a new strategy inspired by [32] to prove the key discrete unique continuation estimate by utilizing the new Carleman estimate with boundary observations for a discrete Laplace operator.