Uniform stability estimates for the discrete Calderón problems

Abstract

In this paper, we focus on the analysis of discrete versions of the Calderón problem in dimension d ⩾ 3. In particular, our goal is to obtain stability estimates for the discrete Calderón problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we prove discrete Carleman estimates for the discrete Laplace operator. A main difference with the continuous ones is that in our case, 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 a limited range of parameters. As in the continuous case, we then use these solutions to obtain uniform stability estimates for the discrete Calderón problems.