Krylovovské metody v numerickém řešení parciálních diferenciáních rovnic - analytický a algebraický pohled
| Název práce v češtině: | Krylovovské metody v numerickém řešení parciálních diferenciáních rovnic - analytický a algebraický pohled |
|---|---|
| Název v anglickém jazyce: | Krylov subspace methods in numerical solution of partial differential equations - analytic and algebraic view |
| Klíčová slova: | Parciální diferenciální rovnice, operátorový popis úlohy, konvergenční chování iteračních metod, předpodmínění |
| Klíčová slova anglicky: | Partial differential equations, operator formulation, convergence behaviour of iterative methods, preconditioning |
| Akademický rok vypsání: | 2015/2016 |
| Typ práce: | disertační práce |
| Jazyk práce: | |
| Ústav: | Katedra numerické matematiky (32-KNM) |
| Vedoucí / školitel: | prof. Ing. Zdeněk Strakoš, DrSc. |
| Řešitel: | skrytý - zadáno a potvrzeno stud. odd. |
| Datum přihlášení: | 03.10.2016 |
| Datum zadání: | 03.10.2016 |
| Datum potvrzení stud. oddělením: | 03.10.2016 |
| Zásady pro vypracování |
| Krylov subspace methods can be seen as highly nonlinear model reduction that can be very efficient in some cases and not easy to handle in others. Convergence behaviour is well understood for the self-adjoint and normal operators (matrices), where we can conveniently rely on the spectral decomposition. That does not have a parallel in non-normal cases. Theoretical analysis of efficient preconditioners is therefore complicated and it is often based on a simplified view to Krylov subspace methods as linear contractions. In numerical solution of boundary value problems, e.g., the infinite dimensional formulation, discretization, and algebraic iteration (including preconditioning) should be tightly linked to each other. Computational efficiency requires accurate, reliable and cheap a posteriori error estimators that relate the discretization and algebraic errors in order to construct an appropriate (problem dependent) stopping criteria. Understanding numerical stability issues is crucial and this becomes even more urgent with increased parallelism where the communication cost becomes a prohibitive factor.
Operator preconditioning and the closely related concept of the spectrally equivalent operators are general frameworks that link preconditioning in algebraic computations with the infinite dimensional operator description and motivation. Many practically used methods can be put into that framework. Their application is, however, somewhat restricted to bounding the condition number of preconditioned matrix of the algebraic system independently of the discretization parameter (mesh). Link between operators, preconditioning and discretizations can perhaps be exploited further, e.g., by considering the connection between preconditioning and nonlocal discretization basis components, more refined spectral information on the preconditioned matrices etc. The work will benefit from an interplay between the functional (infinite dimensional) and algebraic approaches. |
| Seznam odborné literatury |
| P. G. Ciarlet: Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013
A. Friedman, Foundations of Modern Analysis, Dover, 1982 H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers with Applications to Incompressible Fluid Dynamics, second ed., Oxford University Press, 2014 J. Malek and Z. Strakos, Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs, SIAM, 2015 |