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Krylov Subspace Methods - Analysis and Application
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Farrell, Patrick (referee) ; Herzog, Roland (referee)
Title: Krylov Subspace Methods - Analysis and Application Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathematics Abstract: Convergence behavior of Krylov subspace methods is often studied for linear algebraic systems with symmetric positive definite matrices in terms of the condition number of the system matrix. As recalled in the first part of this thesis, their actual convergence behavior (that can be in practice also substantially affected by rounding errors) is however determined by the whole spectrum of the system matrix, and by the projections of the initial residual to the associated invariant subspaces. The core part of this thesis investigates the spectra of infinite dimensional operators −∇ · (k(x)∇) and −∇ · (K(x)∇), where k(x) is a scalar coefficient function and K(x) is a symmetric tensor function, preconditioned by the Laplace operator. Subsequently, the focus is on the eigenvalues of the matrices that arise from the discretization using conforming finite elements. Assuming continuity of K(x), it is proved that the spectrum of the preconditi- oned infinite dimensional operator is equal to the convex hull of the ranges of the diagonal function entries of Λ(x) from the spectral decomposition K(x) =...
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Krylov Subspace Methods - Analysis and Application
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Farrell, Patrick (referee) ; Herzog, Roland (referee)
Title: Krylov Subspace Methods - Analysis and Application Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathematics Abstract: Convergence behavior of Krylov subspace methods is often studied for linear algebraic systems with symmetric positive definite matrices in terms of the condition number of the system matrix. As recalled in the first part of this thesis, their actual convergence behavior (that can be in practice also substantially affected by rounding errors) is however determined by the whole spectrum of the system matrix, and by the projections of the initial residual to the associated invariant subspaces. The core part of this thesis investigates the spectra of infinite dimensional operators −∇ · (k(x)∇) and −∇ · (K(x)∇), where k(x) is a scalar coefficient function and K(x) is a symmetric tensor function, preconditioned by the Laplace operator. Subsequently, the focus is on the eigenvalues of the matrices that arise from the discretization using conforming finite elements. Assuming continuity of K(x), it is proved that the spectrum of the preconditi- oned infinite dimensional operator is equal to the convex hull of the ranges of the diagonal function entries of Λ(x) from the spectral decomposition K(x) =...
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Analysis of Krylov subspace methods
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Hnětynková, Iveta (referee)
Title: Analysis of Krylov subspace methods Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc. Abstract: After the derivation of the Conjugate Gradient method (CG) and the short review of its relationship with other fields of mathematics, this thesis is focused on its convergence behaviour both in exact and finite precision arith- metic. Fundamental difference between the CG and the Chebyshev semi-iterative method is described in detail. Then we investigate the use of the widespread lin- ear convergence bound based on Chebyshev polynomials. Through the example of the composite polynomial convergence bounds it is showed that the effects of rounding errors must be included in any consideration concerning the CG rate of convergence relevant to practical computations. Furthermore, the close corre- spondence between the trajectories of the CG approximations generated in finite precision and exact arithmetic is studied. The thesis is concluded with the discus- sion concerning the sensitivity of the closely related Gauss-Christoffel quadrature. The last two topics may motivate our further research. Keywords: Conjugate Gradient Method, Chebyshev semi-iterative method, fi- nite precision computations, delay of convergence, composite polynomial conver-...
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Teorie a aplikace krylovovských metod v souvislostech
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Hnětynková, Iveta (referee)
Title: Krylov subspace methods: Theory, applications and interconnections Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc. Abstract: After recalling of properties of Chebyshev polynomials and of sta- tionary iterative methods, this thesis is focused on the description of Conjugate Gradient Method (CG), the Krylov method of the choice for symmetric positive definite matrices. Fundamental difference between stationary iterative methods and Krylov subspace methods is emphasized. CG is derived using the minimiza- tion of the quadratic functional and the relationship with several other fields of mathematics (Lanczos method, orthogonal polynomials, quadratic rules, moment problem) is pointed out. Effects of finite precision arithmetic are emphasized. In compliance with the theoretical part, the numerical experiments examine a bound derived assuming exact arithmetic which is often presented in literature. It is shown that this bound inevitably fails in practical computations. The thesis is concluded with description of two open problems which can motivate further research. Keywords: Krylov subspace methods, convergence behaviour, numerical stabil- ity, spectral information, convergence rate bounds
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