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Abstract
This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of the Neumann-to-Dirichlet map is continuously Fr\'echet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fr\'echet derivative defines a bounded linear operator on the space of square integrable functions living on the domain boundary, although the logarithm of the Neumann-to-Dirichlet map itself is unbounded in that topology. In particular, it follows from the fundamental theorem of calculus that the difference between the logarithms of any two Neumann-to-Dirichlet maps is always bounded on the space of square integrable functions. All aforementioned results also hold if the Neumann-to-Dirichlet boundary map is replaced by its inverse, i.e. the Dirichlet-to-Neumann map.
Highlights
This work is motivated by electrical impedance tomography (EIT), i.e., the imaging modality whose aim is to reconstruct the conductivity inside a physical body from boundary measurements of current and voltage
It is shown that the mapping from the the conductivity, i.e., the the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of the Neumann-to-Dirichlet map is continuously Fr\e'chet differentiable between natural topologies
For any essentially bounded perturbation of the conductivity, the Fr\e'chet derivative defines a bounded linear operator on the space of square integrable functions living on the domain boundary, the logarithm of the Neumann-to-Dirichlet map itself is unbounded in that topology
Summary
This work is motivated by electrical impedance tomography (EIT), i.e., the imaging modality whose aim is to reconstruct (useful information about) the conductivity inside a physical body from boundary measurements of current and voltage. To be slightly more precise, the mean relative linearization errors around the unit conductivity were computed over certain random samples of 50,000 conductivities in the unit disk with different parametrizations for the forward map of EIT, and these mean errors were found to be approximately an order of magnitude smaller for the completely logarithmic forward map than for the standard one This lower degree of nonlinearity was observed with the complete electrode model (see [7, 22]) as well as in the mean L2(\Omega ) reconstruction errors for a simple one-step reconstruction algorithm. Our main theorems could as well be formulated for the DN map because the logarithms of the ND and DN maps for a given conductivity only differ by a change of sign
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