idr( ) as a projection method

1IDR() as a projection method

IDR() as a projection methodMarijn Bartel Schreuders

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 methods Definition

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:

𝑥𝑚=𝑥0+𝑉𝑚 𝑦𝑚


Projection methods General algorithm

• How to choose the subspaces?


Krylov subspace methods General

•

• Different methods for different choices of

• Can be used for• eigenvalue problems• linear systems of equations


Krylov subspace methodsOverview


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

• Conjugate Gradient method (CG)

• 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 certain subspaces

• 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:



• This is an example of a slide



• Matrix Market: matrix

• Real, nonsymmetric, sparse matrix

http://math.nist.gov/MatrixMarket/data/misc/hamm/add20.html





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:

1. Make clear how we can see IDR() in the framework of projection methods

2. Use the IDR(s) algorithm for calculating the


IDR() as a projection methodMarijn Bartel Schreuders

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

idr( ) as a projection method

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