IDR( ) as a projection method

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# IDR( ) as a projection method - PowerPoint PPT Presentation

IDR( ) as a projection method. Marijn Bartel Schreuders Supervisor: Dr. Ir. M.B. Van Gijzen Date: Monday, 24 February 2014. Overview of this presentation . Iterative methods Projection methods Krylov subspace methods Eigenvalue problems Linear systems of equations The IDR( ) method

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### IDR() as a projection method

MarijnBartelSchreuders

Supervisor: Dr. Ir. M.B. Van GijzenDate: Monday, 24 February 2014

Overview of this presentation
• Iterative methods
• Projection methods
• Krylov subspace methods
• Eigenvalue problems
• Linear systems of equations
• The IDR() method
• General idea behind the IDR() method
• Numerical examples
• Ritz-IDR
• Research Goals
Iterative methods
• Consider a linear system

(1)

with and

• Find an approximate solution to (1), with initial guess
• Residual
Projection methods Subspaces
• Define of dimension
• ‘Subspace of candidate approximants’ or ‘Search subspace’
• Define of dimension
• ‘Subspace of constraints’ or ‘Left subspace’
Projection methodsDefinition

Find such that

• Find
• Let form an orthonormal basis for
• Then

How to find this vector?

Projection methods How to find
• Let form an orthonormal basis for
• Hence:
Projection methods General algorithm
• How to choose the subspaces?
Krylov subspace methodsGeneral
• Different methods for different choices of
• Can be used for
• eigenvalue problems
• linear systems of equations
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
Krylov subspace methods Symmetric matrices
• Optimality condition
• Uses short recurrences
• Minimises the residual
Krylov subspace methodsNonsymmetric 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?
Induced Dimension Reduction (s)
• Residuals are forced to be in certainsubspaces
• Compute residuals in each iteration
Induced Dimension Reduction (s)IDR theorem

Theorem 1 (IDR theorem):

Let and

Let

Let such that and do not share a subspace of

Define: )

Then the following holds:

(i)

(ii) for some

Induced Dimension Reduction (s)Numerical experiments
• Convection diffusion equation:
• Discretise using finite differences on unit cube; Dirichlet boundary conditions
• internal points equations
• Stopping criterion:
Induced Dimension Reduction (s)Numerical experiments
• Matrix Market: matrix
• Real, nonsymmetric, sparse matrix

Induced Dimension Reduction (s)How to choose
• Recall: )
• Minimisation of the residuals
• Random?
• …… ?

How to choose ?

Induced Dimension Reduction (s)Ritz-IDR
• Valeria Simoncini & Daniel Szyld
• Ritz-IDR
• Calculates Ritz values
Research goals
• Research goals are twofold:
• Make clear how we can see IDR() in the framework of projection methods
• Use the IDR(s) algorithm for calculating the

### IDR() as a projectionmethod

MarijnBartelSchreuders

Supervisor: Dr. Ir. M.B. Van GijzenDate: Monday, 24 February 2014

Research goals
• Let
• This is a polynomial in
• To minimise, take derivative w.r.t.
Krylov subspace methods Eigenvalue problems

Arnoldi Method

Lanczos method

&

Bi-Lanczos method