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Thesis details
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Numerical Methods in Discrete Inverse Problems
Thesis title in Czech: Numerické metody pro řešení diskrétních inverzních úloh
Thesis title in English: Numerical Methods in Discrete Inverse Problems
Key words: diskrétní inverzní úlohy, iterační metody, odhadování šumu, smíšený šum, aritmetika s konečnou přesností
English key words: discrete inverse problems, iterative solvers, noise estimation, mixed noise, finite-precision arithmetic
Academic year of topic announcement: 2012/2013
Thesis type: dissertation
Thesis language: angličtina
Department: Department of Numerical Mathematics (32-KNM)
Supervisor: doc. RNDr. Iveta Hnětynková, Ph.D.
Author: hidden - assigned and confirmed by the Study Dept.
Date of registration: 27.09.2013
Date of assignment: 27.09.2013
Confirmed by Study dept. on: 27.01.2014
Date and time of defence: 26.09.2018 00:00
Date of electronic submission:07.08.2018
Date of submission of printed version:13.08.2018
Date of proceeded defence: 26.09.2018
Opponents: Silvia Gazzola
  prof. Gerard Meurant
 
 
Advisors: prof. Ing. Zdeněk Strakoš, DrSc.
Guidelines
Inverse problems arise in very wide areas of applications. In their solution, numerical methods play a fundamental role. Over the past several decades, a large class of techniques were developed to treat inverse problems resulting from discretization of the model formulated via Fredholm integral equations. Discretized problem inherits undesirable ill-posedness of the integral equation, meaning that a small amount of perturbation (noise) in the data can lead to an enormous errors in the solution of the problem. Here the role of the noise can be played also by (small) inaccuracies in numerical computations. As a consequence, an efficient numerical computation must assure that the computed approximate solution captures enough information from the data while suppressing the possible amplification of the noise (and/or rounding errors).

The investigation will focus on several topics such as, e.g., the numerical interpretation of ill-posedness, theoretical and practical aspects of iterative regularization, an influence of finite precision arithmetic computations, various methods of regularization and discrete inverse problems with noisy model matrix
References
A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996

P.CH.Hansen: Discrete Inverse Problems: Insight and Algorithms, SIAM, 2010

I. Hnětynková, M. Plešinger, Z. Strakoš: The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data, BIT Numerical Mathematics, 2009, 49:669-696

J.Liesen and Z. Strakoš: Krylov Subspace Methods: Principles and Analysis, Oxford University Press, 2012

C.C.Paige and Z. Strakoš: Scaled total least squares fundamentals, Numerische Mathematik, Springer, 2002, 91:1:117-146

C.R.Vogel: Computational Methods for Inverse Problems, SIAM, 2002
 
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