### Abstract

Original language | English |
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Publisher | arXiv.org |

Number of pages | 19 |

Publication status | Published - 2019 |

### Fingerprint

### Keywords

- Neumann-to-Dirichlet map
- Fréchet derivative
- logarithm
- functional calculus
- electrical impedance tomography

### Cite this

*On regularity of the logarithmic forward map of electrical impedance tomography*. arXiv.org.

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**On regularity of the logarithmic forward map of electrical impedance tomography.** / Garde, Henrik; Hyvönen, Nuutti; Kuutela, Topi.

Research output: Working paper › Research

TY - UNPB

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

AU - Garde, Henrik

AU - Hyvönen, Nuutti

AU - Kuutela, Topi

PY - 2019

Y1 - 2019

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 Fréchet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fréchet 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.

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 Fréchet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fréchet 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.

KW - Neumann-to-Dirichlet map

KW - Fréchet derivative

KW - logarithm

KW - functional calculus

KW - electrical impedance tomography

M3 - Working paper

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

PB - arXiv.org

ER -