Uniform stability estimates for the discrete Calderon problems

Sylvain Ervedoza (1) , Frédéric de Gournay (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMV - Laboratoire de Mathématiques de Versailles

In this article, we focus on the analysis of discrete versions of the Calderon 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 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.

Mots clés: Calderón problem; Stability estimate; Carleman estimate; CGO solution; Discrete stability estimate

Domaines: Analyse numérique [math.NA]

Sylvain Ervedoza, Frédéric de Gournay. Uniform stability estimates for the discrete Calderon problems. Inverse Problems, 2011, 27 (12), pp.125012, 37. ⟨10.1088/0266-5611/27/12/125012⟩. ⟨hal-00588699⟩