Abstract not found

. Every sufficiently regular non-negative function fl (conductivity) on the closed unit disk D induces the Dirichlet-to-Neumann map fl on functions on @D . The main forward problem is to give a characterization of the maps fl . The main inverse problem is to find out when fl uniquely determines fl. In this paper we consider the case of conductivities that are constant on circles centered at the origin, and a discrete analog of this, so called, layered case. We characterize the set of the layered Dirichlet-to-Neumann maps in terms of their kernels and spectra. We also give sharp conditions on fl for the uniqueness in the inverse problems. The characterization in terms of the spectra shows that continuous Dirichlet-to-Neumann maps can be viewed as limits of the discrete Dirichlet-to-Neumann maps. The characterization in terms of the kernels supports the conjecture in [8] that the alternating property essentially characterizes continuous Dirichlet-to-Neumann maps. The chracterization...

Abstract not found

Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on so-called optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting trade-off between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary. 1.

network approaches to electrical impedance tomography