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Inverse Problems and Imaging (IPI)
 

Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results

Pages: 749 - 766, Volume 9, Issue 3, August 2015      doi:10.3934/ipi.2015.9.749

 
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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)
David Isaacson - Rensselaer Polytechnic Institute, Department of Mathematics, Troy, NY 12180, United States (email)
Jari P. Kaipio - University of Auckland, Department of Mathematics, Auckland, New Zealand (email)
Debra McGivney - Case Western Reserve University, Department of Radiology, Cleveland, OH 44106, United States (email)
Erkki Somersalo - Case Western Reserve University, Department of Mathematics, Applied Mathematics, and Statistics, Cleveland, OH 44106, United States (email)
Joseph Volzer - Case Western Reserve University, Department of Mathematics, Applied Mathematics, and Statistics, Cleveland, OH 44106, United States (email)

Abstract: Artificial boundary conditions have long been an active research topic in numerical approximation of scattering waves: The truncation of the computational domain and the assignment of the conditions along the fictitious boundary must be done so that no spurious reflections occur. In inverse boundary value problems, a similar problem appears when the estimation of the unknowns is restricted to a domain that represents the whole domain of the solutions of a partial differential equation with unknown coefficient. This problem is significantly more challenging than general scattering problems, because the coefficients representing the unknown material parameter of interest are not known in the truncated portion and assigning suitable condition on the fictitious boundary is part of the problem also. The problem is addressed by defining a Dirichlet-to-Neumann map, or Steklov-Poincaré map, on the boundary of the domain truncation. In this paper we describe the procedure, provide a theoretical justification and illustrate with computed examples the limitations of imposing fixed boundary condition. Extensions of the proposed approach will be presented in a sequel article.

Keywords:  Domain decomposition, Dirichlet-to-Neumann, Steklov-PoincarĂ©, coercivity, electrode model.
Mathematics Subject Classification:  Primary: 35J20, 35R30, 65M32, 65M55.

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

 References