Marijn Bartel Schreuders Supervisor Date Dr Ir M

Marijn Bartel Schreuders Supervisor: Date: Dr. Ir. M. B. Van Gijzen 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: Date: Dr. Ir. M. B. Van Gijzen Monday, 24 February 2014

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Research goals • 28

Krylov subspace methods Eigenvalue problems Arnoldi Method Lanczos method & Bi-Lanczos method 29