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Inverse Problems and Imaging (IPI)
 

Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map

Pages: 767 - 789, Volume 9, Issue 3, August 2015      doi:10.3934/ipi.2015.9.767

 
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Daniela Calvetti - Case Western Reserve University, Department of Mathematics, Applied Mathematics, and Statistics, Cleveland, OH 44106, United States (email)
Paul J. Hadwin - University of Auckland, Department of Mathematics, Auckland, New Zealand (email)
Janne M. J. Huttunen - University of Eastern Finland, Department of Applied Physics, Kuopio, Finland (email)
Jari P. Kaipio - University of Auckland, Department of Mathematics, Auckland, New Zealand (email)
Erkki Somersalo - Case Western Reserve University, Department of Mathematics, Applied Mathematics, and Statistics, Cleveland, OH 44106, United States (email)

Abstract: In [3], the authors discussed the electrical impedance tomography (EIT) problem, in which the computational domain with an unknown conductivity distribution comprises only a portion of the whole conducting body, and a boundary condition along the artificial boundary needs to be set so as to minimally disturbs the estimate in the domain of interest. It was shown that a partial Dirichlet-to-Neumann operator, or Steklov-Poincaré map, provides theoretically a perfect boundary condition. However, since the boundary condition depends on the conductivity in the truncated portion of the conductive body, it is itself an unknown that needs to be estimated along with the conductivity of interest. In this article, we develop further the computational methodology, replacing the unknown integral kernel with a low dimensional approximation. The viability of the approach is demonstrated with finite element simulations as well as with real phantom data.

Keywords:  Domain decomposition, Dirichlet-to-Neumann, Steklov-Poincarè, principal component approximation, Bayes' theorem
Mathematics Subject Classification:  Primary: 35J20, 35R30, 65M32, 65M55, 65C60.

Received: April 2014;      Revised: January 2015;      Available Online: July 2015.

 References