Multigrid methods for systems of linear equations

Andrey Ponomarenko

Sarntal

St.Petersburg State University

Faculty of Physics

Earth physics department

Ferienakademie 2008

26.09.20081

OUTLINE

The way to multigrid…

Iterative methods for linear systems, relaxation schemes

Coarse grid correction

V-multigrid method

Full-multigrid method

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The way to multigrid…

• Algorithm should not do a lot of empty work

1.Slow process – a lot of calculations2.The convergence - it should be quite quick

• So, multigrid method – for high frequently components we use fine grids, for slow components we use coarse grids

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Iterative methods for linear systems

Consider

Let v be an approximation to u

Residual equation

Residual correction

1) At first, some basics:

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2) Iteration in the matrix form

- the iteration error

Let

D – is diagonal matrixL and U – are lower and upper parts of A

Then

Now the iteration is

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Now consider the iteration properties on the next equation in 1D:

• One-dimensional boundary problem

• Introduce the grid:

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Then our problem can be presented in grid form:

)()(!3

)(!2

)()()( 432

1 hOxuh

xuh

xuhxuxu iiiii

)()(!3

)(!2

)()()( 432

1 hOxuh

xuh

xuhxuxu iiiii

Then, with the help of Taylor series:

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Now we have The discrete model problem:

So, we can used iteration scheme showed above

Consider relaxation scheme:

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symmetric, positive defined

Relaxation schemes

The equation:

1) Jacobi Method (simultaneous displacements):

2) Weighted Jacobi Method :

D – is diagonal matrixL and U – are lower and upper parts of A

Then as we have before

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Now we return to relaxation:

λ – is eigenvalue of matrix B, w – its ascociated eigenvector

Eigenvectors are linearly independent, they form the basis, and for any v it is possible to write (N×N matrix)

Short observation about eigenvalues and eigenvectors:

- for one iteration

- we’ll have after n iterations

- initial error

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Let’s make convergence analysis on weighted Jakobi on 1D model

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Let’s make convergence analysis on weighted Jakobi on 1D model

Expand initial error in the terms of eigenvectors:

So, after M iterations we have: (remember )

The k-th mode of the error is reduced by at each iteration

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Relaxation suppress eigenmodes unevenly

If 0<w<1,

Low frequencies are damped bad

High frequencies:

It can be shown, sm. fact. is the best when , it equals

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So, many relaxation schemes has smoothing property, but smooth modes of error are damped very slowly

For instance (weighted Jakobi method):

• initial error:

• after 35 iterations:

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By using coarse grids we can use the smoothing property in good advantage!

So why??

• Relaxation on the coarse-grid is much cheaper (1/2 – 1D, ¼ - 2D e.t.c )

• Relaxation on the coarse-grid has better convergence ( 1-O(4h2) inst.of 1-O(h2) )

• After relaxing on the fine grid, the error will be smooth. On the coarse grid this error appears more oscillatory, and relaxation will be more effectively

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We have now the idea of

The question appears: how to map and

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defined on

• Mapping from the coarse grid to the fine grid:

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defined on

• Mapping from the fine grid to the coarse grid:

(there are others methods of mapping, we don’t consider them)

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The Coarse Grid Correction scheme:

- is the “coarse-grid version” of operator A

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V-Multigrid method

IDEA: it is unnecessary to

• It is enough to make several relaxation and to obtain approximate solution• We effectively relax high-frequency components• To make slow frequencies smooth we use the next, coarse grid

So, we have recursive use of CGC sheme.

Gh

G2h

G4h

G8h

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Full-Multigrid method

IDEA: the solution takes fewer iterations if the initial guess is good

• Interpolate coarse solution to the fine grid• “Solve” the problem on the coarse grid first• Use interpolated coarse solution as initial guess on fine grid

How to get a good initial guess:

• Let’s use the V-multigrid cycle as the solver on each grid level. This defines the Full Multigrid (FMG) method

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Some properties of convergence:

Some properties of FMG method:

1) Full Multigtid method computes solution to the error of truncation 2) The computational cost of FMG is O(Nd), where N – quantity of fine grid points, d – dimension of the problem

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So, MULTIGRID METHODS - increasingly the right tool:

• Multigrid methods are effective algorithms

• Multigrid method FMG is an optimal (O(N))

• Multigrid algotithms can be parallelized effecintly – it is another real interesting and big topic to present!

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References

Briggs W.M. “Multigrid tutorial”. Presentation by Henson V.E. Center for applied scientific computing, Lawrence livermore National Laboratory.

Stankova E., Zatevahin M. “Multigrid methods. Introduction in the standard methods.” St.Petersburg. In PDF, in russian.

Borzi A. “Introduction to multigrid methods.” In PDF. http://www.uni-graz.at/imawww/borzi/index.html

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Thank you for attention!

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