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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

IDR() as a projection method

MarijnBartelSchreuders

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

overview of this presentation
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
Iterative methods
  • Consider a linear system

(1)

with and

  • Find an approximate solution to (1), with initial guess
  • Residual
projection methods subspaces
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
Projection methodsDefinition

Find such that

  • Find
  • Let form an orthonormal basis for
  • Then

How to find this vector?

projection methods how to find
Projection methods How to find
  • Let form an orthonormal basis for
  • Hence:
projection methods general algorithm
Projection methods General algorithm
  • How to choose the subspaces?
krylov subspace methods g eneral
Krylov subspace methodsGeneral
  • Different methods for different choices of
  • Can be used for
    • eigenvalue problems
    • linear systems of equations
krylov subspace methods eigenvalue problems
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
Krylov subspace methods Symmetric matrices
  • Conjugate Gradient method (CG)
  • Optimality condition
    • Uses short recurrences
    • Minimises the residual
krylov subspace methods nonsymmetric matrices
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
Induced Dimension Reduction (s)
  • Residuals are forced to be in certainsubspaces
  • Compute residuals in each iteration
induced dimension reduction s idr theorem
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
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 experiments2
Induced Dimension Reduction (s)Numerical experiments
  • Matrix Market: matrix
  • Real, nonsymmetric, sparse matrix

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

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

How to choose ?

induced dimension reduction s ritz idr
Induced Dimension Reduction (s)Ritz-IDR
  • Valeria Simoncini & Daniel Szyld
  • Ritz-IDR
    • Calculates Ritz values
research goals
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 projection method1

IDR() as a projectionmethod

MarijnBartelSchreuders

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

research goals1
Research goals
  • Let
  • This is a polynomial in
  • To minimise, take derivative w.r.t.
krylov subspace methods eigenvalue problems1
Krylov subspace methods Eigenvalue problems

Arnoldi Method

Lanczos method

&

Bi-Lanczos method