Reconstruction of cracks in Calderón's inverse conductivity problem using energy comparisons
Henrik Garde, Michael Vogelius
TL;DR
The paper tackles the reconstruction of cracks in Calderón's inverse conductivity problem by introducing two monotonicity-based approaches that rely on local Neumann-to-Dirichlet maps. The first method yields exact crack locations by shrinking test inclusions $C$ and verifying the operator inequality $\Lambda_{C}^{\emptyset} \geq \Lambda_{D_0}^{D_\infty} \geq \Lambda_{\emptyset}^{C}$, accommodating mixtures of insulating and conducting cracks. The second method handles the pure insulating or pure conducting cases to produce inner-approximations $\chi \subseteq D$ using similar operator tests. The authors develop a localisation framework based on localised potentials and range relations of variational operators to prove both directions of the main theorems, enabling rigorous reconstruction in dimensions $d \geq 2$ and for general, non-homogeneous backgrounds $\gamma_0$. These results extend prior inclusion-reconstruction techniques and offer a robust, constructive paradigm for detecting crack geometries from boundary data, with potential numerical applicability in EIT-like settings.
Abstract
We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calderón's inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann-to-Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.