Iterative Methods and
Iterative
Multigrid
Part 3: Methods based on the
two-sided Lanczos algorithm
Short Recurrence Methods?
Short
GMRES optimal in iterations but expensive in time and
memory if many iterations required. Main cost is keeping all
vectors a
Convergence bounds for CG
Consider case where one eigenvalue n much larger than others.
2 n 1
n 1
Construct better polynomial than T m
information.
/T m
n 1
n 1
using this
For example, polynomial that is zero at extreme eigenvalue and lower
degree Cheb
Introduction Iterative Methods
Eric de Sturler
Department of Mathematics, Virginia Tech
[email protected]
http:/www.math.vt.edu/people/sturler
Iterative Methods for Linear Systems: Basics to Research
Numerical Analysis and Software I
Purpose of this Course
I
Convergence bounds for CG
Consider the CG residual and from that the CG error.
CG: r m = b A(x 0 + z m ) = r 0 Az m , where z m c Km (A, r 0 ).
This gives polynomial r m = r 0 AP m1 (A )r 0 = (I AP m1 (A )r 0
Multiplying by A 1 gives e m = A 1 r m = (I AP
Fixed-Point Iterations,
Krylov Spaces, and Krylov Methods
Fixed-Point Iterations
Solve nonsingular linear system: Ax = b
(solution x = A1b )
Solve an approximate, but simpler system: Mx 0 = b x 0 = M 1b
(iterative refinement)
Improve the solution using th
Iterative Methods for
Iterative
Linear Systems
Comparisons of methods for
model problems
GMRES - Computational Cost
GMRES: Ax = b
choose x 0 (e.g. x 0 = 0 ) and tol
r 0 = b Ax 0 ; k = 0; v 1 = r 0 /r 0 2 ;
while r k 2 > tol
k = k + 1;
w = Av k;
Solve Pvk+
Iterative Methods and
Multigrid
Part 3: Preconditioning
2000 Eric de Sturler
Preconditioning
The general idea behind preconditioning is that convergence of
some method for the linear system Ax = b can be improved by
applying the method to the precondition
Multigrid Methods
Convergence Proof and Analysis
2003 by Eric de Sturler. All rights reserved.
Fixed Point Iterations
First we make some general observations about fixed point iterations.
Consider the equation Ax = b and the splitting A = (A P ) + P .
We
Iterative Methods and
Multigrid
Part 4: Local mode analysis
2000 Eric de Sturler
Local mode analysis
How to analyze smoothing behavior for more general problems and
relaxations: local mode (normal mode/Fourier) analysis.
In general computing eigenvectors/
Iterative Methods and
Multigrid
Part 1: Introduction to Multigrid
2000 Eric de Sturler
1
12/02/09
MG01.prz
Basic Iterative Methods (1)
Nonlinear equation: f(x) = 0
Rewrite as x = F(x) , and iterate xi +1 = F(xi ) (fixed-point iteration)
Converges locally
Department of Computer Science,
University of Illinois at Urbana-Champaign
Probing for Schur Complements and
Preconditioning Generalized
Saddle-Point Problems
Eric de Sturler,
[email protected],
http:/www-faculty.cs.uiuc.edu/~sturler
Optimization Techno
Iterative Methods and
Multigrid
Part 3: Preconditioning: Based on
Domain Decomposition
2000 Eric de Sturler
Domain Decomposition Book
Significant parts of this discussion come from the (excellent) book
Barry Smith, Petter Bjrstad, and William Gropp,
Domai
Synopsis of Numerical Linear Algebra
Eric de Sturler
Department of Mathematics, Virginia Tech
[email protected]
http:/www.math.vt.edu/people/sturler
Iterative Methods for Linear Systems: Basics to Research
Numerical Analysis and Software I
Systems of Linea