Convergence of Derivative Approximations in the Inverse Conductivity Problem

J.G. Wade - K. Senior - S. Seubert

Abstract:

The goal in inverse conductivity problems is to approximately determine the spatially varying conductivity parameter in the interior of some region given certain boundary data. We formulate this problem as a least-squares minimization problem with total variation regularization in order to attenuate the ill-posedness. We outline a numerical approach to this minimization problem, based on the Gauss-Newton idea.

The main results of this paper concern the continuity, regularity and approximability of the forward map underlying the inverse problem in a topology for which total variation regularization induces compact subsets of the parameter space. Specifically, we show that the forward map is Fréchet differentiable in this topology, and we show that standard Galerkin approximations of the Fréchet derivative are convergent. A numerical example is provided.

• Introduction
  • The EIT problem via least-squares
  • Choice of parameter space
  • Regularization
  • Minimization of the objective functional
  • Outline
• Mathematical Framework
  • Variational Formulation of the PDE
  • Continuity of the forward map
  • Additional assumptions and Fréchet differentiability
• Numerical Approximations
  • Definition of the approximations
  • Convergence of the approximations
• A Numerical Example
  • The algorithm
  • Choice of and
  • Synthetic data and results
• References
• About this document ...