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

2IDR() as a projection method

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

3IDR() as a projection method

Iterative methods

• Consider a linear system

(1)

with and

• Find an approximate solution to (1), with initial guess

• Residual

4IDR() as a projection method

Projection methods Subspaces

• Define of dimension

• ‘Subspace of candidate approximants’ or ‘Search subspace’

• Define of dimension

• ‘Subspace of constraints’ or ‘Left subspace’

5IDR() as a projection method

Projection methods Definition

Find such that

• Find

• Let form an orthonormal basis for

• Then

How to find this vector?

6IDR() as a projection method

Projection methods How to find

• Let form an orthonormal basis for

• • Hence:

𝑥𝑚=𝑥0+𝑉𝑚 𝑦𝑚

7IDR() as a projection method

Projection methods General algorithm

• How to choose the subspaces?

8IDR() as a projection method

Krylov subspace methods General

•

• Different methods for different choices of

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

9IDR() as a projection method

Krylov subspace methodsOverview

10IDR() as a projection method

Krylov subspace methodsOverview

11IDR() as a projection method

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

12IDR() as a projection method

Krylov subspace methodsOverview

13IDR() as a projection method

Krylov subspace methodsOverview

14IDR() as a projection method

Krylov subspace methods Symmetric matrices

• Conjugate Gradient method (CG)

• Optimality condition• Uses short recurrences• Minimises the residual

15IDR() as a projection method

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?

16IDR() as a projection method

Induced Dimension Reduction (s)

• Residuals are forced to be in certain subspaces

• Compute residuals in each iteration

17IDR() as a projection method

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

18IDR() as a projection method

Induced Dimension Reduction (s)Numerical experiments

• Convection diffusion equation:

• Discretise using finite differences on unit cube; Dirichlet boundary conditions

• internal points equations

• Stopping criterion:

19IDR() as a projection method

Induced Dimension Reduction (s)Numerical experiments

• This is an example of a slide

20IDR() as a projection method

Induced Dimension Reduction (s)Numerical experiments

• Matrix Market: matrix

• Real, nonsymmetric, sparse matrix

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

21IDR() as a projection method

Induced Dimension Reduction (s)Numerical experiments

• This is an example of a slide

22IDR() as a projection method

Induced Dimension Reduction (s)Numerical experiments

• This is an example of a slide

23IDR() as a projection method

Induced Dimension Reduction (s)How to choose

• Recall: )

• Minimisation of the residuals

• Random?

• …… ?

How to choose ?

24IDR() as a projection method

Induced Dimension Reduction (s)Ritz-IDR

• Valeria Simoncini & Daniel Szyld

• Ritz-IDR• Calculates Ritz values

25IDR() as a projection method

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

26IDR() as a projection method

IDR() as a projection methodMarijn Bartel Schreuders

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

27IDR() as a projection method

28IDR() as a projection method

Research goals

• Let

• This is a polynomial in

• To minimise, take derivative w.r.t.

29IDR() as a projection method

Krylov subspace methods Eigenvalue problems

Arnoldi Method

Lanczos method&

Bi-Lanczos method