Numerical Methods in Discrete Inverse Problems
| Název práce v češtině: | Numerické metody pro řešení diskrétních inverzních úloh |
|---|---|
| Název v anglickém jazyce: | Numerical Methods in Discrete Inverse Problems |
| Klíčová slova: | diskrétní inverzní úlohy, iterační metody, odhadování šumu, smíšený šum, aritmetika s konečnou přesností |
| Klíčová slova anglicky: | discrete inverse problems, iterative solvers, noise estimation, mixed noise, finite-precision arithmetic |
| Akademický rok vypsání: | 2012/2013 |
| Typ práce: | disertační práce |
| Jazyk práce: | angličtina |
| Ústav: | Katedra numerické matematiky (32-KNM) |
| Vedoucí / školitel: | doc. RNDr. Iveta Hnětynková, Ph.D. |
| Řešitel: | skrytý - zadáno a potvrzeno stud. odd. |
| Datum přihlášení: | 27.09.2013 |
| Datum zadání: | 27.09.2013 |
| Datum potvrzení stud. oddělením: | 27.01.2014 |
| Datum a čas obhajoby: | 26.09.2018 00:00 |
| Datum odevzdání elektronické podoby: | 07.08.2018 |
| Datum odevzdání tištěné podoby: | 13.08.2018 |
| Datum proběhlé obhajoby: | 26.09.2018 |
| Oponenti: | Silvia Gazzola |
| prof. Gerard Meurant | |
| Konzultanti: | prof. Ing. Zdeněk Strakoš, DrSc. |
| Zásady pro vypracování |
| 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 |
| Seznam odborné literatury |
| 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 |