Marijn Bartel Schreuders Supervisor Dr Ir M B
Marijn Bartel Schreuders Supervisor: Dr. Ir. M. B. Van Gijzen Date: Monday, 24 February 2014
Overview of this presentation • 2
Iterative methods • 3
Projection methods Subspaces • 4
Projection methods Definition • 5
Projection methods General algorithm • How to choose the subspaces? 7
Krylov subspace methods General • 8
Krylov subspace methods Overview 9
Krylov subspace methods Overview 10
Krylov subspace methods Eigenvalue problems • Computing all eigenvalues can be costly • A is a full matrix • A is large • Idea: find smaller matrix for which it is easy to compute ‘Ritz values’ • Good approximations to some of the eigenvalues of A 11
Krylov subspace methods Overview 12
Krylov subspace methods Overview 13
Krylov subspace methods Symmetric matrices • Conjugate Gradient method (CG) • Optimality condition • Uses short recurrences • Minimises the residual 14
Krylov subspace methods Nonsymmetric matrices • GMRES-type methods • Long recurrences • Minimisation of the residual • Bi-CG – type methods • Short recurrences • No minimisation of the residual • Two matrix-vector operations per iteration • Are their any other possibilities? 15
Induced Dimension Reduction (s) • 16
Induced Dimension Reduction (s) IDR theorem • 17
Induced Dimension Reduction (s) Numerical experiments • 18
Induced Dimension Reduction (s) Numerical experiments • This is an example of a slide 19
Induced Dimension Reduction (s) Numerical experiments • 20
Induced Dimension Reduction (s) Numerical experiments • This is an example of a slide 21
Induced Dimension Reduction (s) Numerical experiments • This is an example of a slide 22
Induced Dimension Reduction (s) Ritz-IDR • 24
Research goals • 25
Marijn Bartel Schreuders Supervisor: Dr. Ir. M. B. Van Gijzen Date: Monday, 24 February 2014
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Research goals • 28
Krylov subspace methods Eigenvalue problems Arnoldi Method Lanczos method & Bi-Lanczos method 29
- Slides: 29