On regularity of the logarithmic forward map of electrical impedance tomography

TY - JOUR

T1 - On regularity of the logarithmic forward map of electrical impedance tomography

AU - Garde, Henrik

AU - Hyvönen, Nuutti

AU - Kuutela, Topi

PY - 2020

Y1 - 2020

N2 - 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 Frechet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Frechet 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 Neumannto- 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.

AB - 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 Frechet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Frechet 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 Neumannto- 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.

KW - Electrical impedance tomography

KW - Frechet derivative

KW - Functional calculus

KW - Logarithm

KW - Neumann-to-Dirichlet map

U2 - 10.1137/19M1256476

DO - 10.1137/19M1256476

M3 - Journal article

SN - 0036-1410

VL - 52

SP - 197

EP - 220

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 1

ER -