Numerical solution of the real-linear equations of electrical impedance tomography for nonsmooth conductivities

Allan Perämäki

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

This thesis studies the numerical solution and convergence of a certain discretized real-linear Beltrami equation. This equation arises in the uniqueness proof by Astala and Päivärinta for the two-dimensional electrical impedance tomography problem with nonsmooth conductivities. The real-linear matrix equation appearing after discretizing the Beltrami equation is found to have the form appropriate for the application of the real-linear Generalized Minimal Residual (GMRES) method published by Eirola, Huhtanen and von Pfaler. The findings include a fast numerical solution method for the discretized real-linear Beltrami equation, and an implementation of a reconstruction method based on the Astala-Päivärinta uniqueness proof. The solution of the discretized Beltrami equation is shown to converge to the correct solution as the grid is refined, including a convergence rate estimate. For the real-linear GMRES method, the norms of the residuals are bounded in terms of a polynomial approximation problem on the complex plane resembling the situation of classical GMRES. Moreover, complex symmetric matrices are shown to possess a mathematical framework analogous to the classical Hermitian Lanczos framework.
Translated title of the contributionEpäsileiden johtavuuksien sähköisen impedanssitomografian reaalilineaaristen yhtälöiden numeerinen ratkaisu
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Nevanlinna, Olavi, Supervising Professor
  • Huhtanen, Marko, Thesis Advisor
Publisher
Print ISBNs978-952-60-4723-2
Electronic ISBNs978-952-60-4724-9
Publication statusPublished - 2012
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • inverse problem
  • Beltrami equation
  • iterative methods
  • discrete convergence
  • orthogonal polynomials
  • Jacobi matrix
  • condiagonalizable

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