Krylov Subspace Methods - Analysis and Application
| Thesis title in Czech: | Metody krylovovských podprostorů - Analýza a aplikace |
|---|---|
| Thesis title in English: | Krylov Subspace Methods - Analysis and Application |
| Key words: | parciální diferenciální rovnice, diskretizovaná úloha, předpodmínění, spektrální informace, Krylovovské metody, konvergence, zaokrouhlovací chyby |
| English key words: | partial differential equations, discretised problem, preconditioning, spectral information, Krylov subspace methods, convergence, rounding errors |
| Academic year of topic announcement: | 2012/2013 |
| Thesis type: | dissertation |
| Thesis language: | angličtina |
| Department: | Department of Numerical Mathematics (32-KNM) |
| Supervisor: | prof. Ing. Zdeněk Strakoš, DrSc. |
| 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: | 21.09.2020 10:00 |
| Date of electronic submission: | 02.06.2020 |
| Date of submission of printed version: | 22.06.2020 |
| Date of proceeded defence: | 21.09.2020 |
| Opponents: | Patrick Farrell |
| Roland Herzog | |
| Advisors: | prof. Ing. Miroslav Tůma, CSc. |
| Guidelines |
| Krylov subspace methods are being used for decades in problems coming from numerical approximation of partial differential equations but also from other scientific and engineering disciplines like, e.g., image or signal processing. This variety of applications has naturally led to the explosive development of many different algorithms. The tools and strategies used to analyze their behaviour are also often very different. Consequently, because of lack of communication among the various disciplines, the individual approaches are often studied separately which is accompanied by possible loss of useful relationships and which results in quite scattered knowledge.
It is suggested to combine several views of the analysis. In particular, the operator view natural for problem arising in partial differential equations and a matrix view typical for many other applications. We believe this combined view to be very useful and needed for instance in designing efficient preconditioners or in reduction of the studied mathematical model. It is assumed that the analysis will be driven by particular applications. |
| References |
| Benzi, M.: Preconditioning techniques for large linear systems: A survey. Journal of Computational Physics 182 (2002), 418–477.
Mardal, K.-A., and Winther, R. Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18 (2011), 1–40. Günnel, A., Herzog, R. and Sachs, E. A Note on Preconditioners and Scalar Products for Krylov Methods in Hilbert Space. (preprint). Antoulas, A.: Approximations of Large-Scale Dynamical Systems, SIAM, Philadelhia (2005). Elman, H., Silvester, D. and Wathen, A.: Finite elements and fast iterative solvers with applications in incompressible fluid dynamics. Oxford University Press. New York (2005) Liesen, J. and Strakoš, Z.: Krylov Subspace Methods: Principles and Analysis. Oxford University Press. Oxford (2012) Strakoš, Z.: Theory of convegence and effects of finite precision arithmetic in Krylov Subspace Methods. DrSc. Thesis, CAS (2001). Kuijlaars, A.: Convergence analysis of Krylov subspace iterations with methods from potential theory, SIAM Review 48, 3-40 (1996). M. Vohralík: A posteriori error estimates, stopping criteria and inexpensive implementations, Habilitation Thesis, Universite Pierre et Marie Curie - Paris 6, 2010. |