On regularity of the logarithmic forward map of electrical impedance tomography

Henrik Garde, Nuutti Hyvönen, Topi Kuutela

Research output: Working paperResearch

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 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.
Original languageEnglish
PublisherarXiv.org
Number of pages19
Publication statusPublished - 2019

Fingerprint

Electrical Impedance Tomography
Logarithm
Dirichlet-to-Neumann Map
Logarithmic
Regularity
Conductivity
Dirichlet
Fundamental theorem of calculus
Topology
Lipschitz Domains
Continuously differentiable
Elliptic Partial Differential Equations
Bounded Linear Operator
Divergence
Perturbation
Derivative
Coefficient
Term

Keywords

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

Cite this

@techreport{5b469254d0be46a6b03bda88a959584a,
title = "On regularity of the logarithmic forward map of electrical impedance tomography",
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{\'e}chet differentiable between natural topologies. Moreover, 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, 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.",
keywords = "Neumann-to-Dirichlet map, Fr{\'e}chet derivative, logarithm, functional calculus, electrical impedance tomography",
author = "Henrik Garde and Nuutti Hyv{\"o}nen and Topi Kuutela",
year = "2019",
language = "English",
publisher = "arXiv.org",
type = "WorkingPaper",
institution = "arXiv.org",

}

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

arXiv.org, 2019.

Research output: Working paperResearch

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 -