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 contribution | Epäsileiden johtavuuksien sähköisen impedanssitomografian reaalilineaaristen yhtälöiden numeerinen ratkaisu |
---|---|
Original language | English |
Qualification | Doctor's degree |
Awarding Institution |
|
Supervisors/Advisors |
|
Publisher | |
Print ISBNs | 978-952-60-4723-2 |
Electronic ISBNs | 978-952-60-4724-9 |
Publication status | Published - 2012 |
MoE publication type | G5 Doctoral dissertation (article) |
Keywords
- inverse problem
- Beltrami equation
- iterative methods
- discrete convergence
- orthogonal polynomials
- Jacobi matrix
- condiagonalizable