Krylov Subspace Methods - Analysis and Application

• Název práce v češtině: Metody krylovovských podprostorů - Analýza a aplikace
• Název v anglickém jazyce: Krylov Subspace Methods - Analysis and Application
• Klíčová slova: parciální diferenciální rovnice, diskretizovaná úloha, předpodmínění, spektrální informace, Krylovovské metody, konvergence, zaokrouhlovací chyby
• Klíčová slova anglicky: partial differential equations, discretised problem, preconditioning, spectral information, Krylov subspace methods, convergence, rounding errors
• Akademický rok vypsání: 2012/2013
• Typ práce: disertační práce
• Jazyk práce: angličtina
• Ústav: Katedra numerické matematiky (32-KNM)
• Vedoucí / školitel: prof. Ing. Zdeněk Strakoš, DrSc.
• 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: 21.09.2020 10:00
• Datum odevzdání elektronické podoby: 02.06.2020
• Datum odevzdání tištěné podoby: 22.06.2020
• Datum proběhlé obhajoby: 21.09.2020
• Oponenti: Patrick Farrell
• Konzultanti: prof. Ing. Miroslav Tůma, CSc.

Zásady pro vypracování

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.

Seznam odborné literatury

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.