1. systems of linear equations
St.Petersburg State University
Faculty of Physics
Earth physics department
Ferienakademie 2008
Multigrid methods for
systems of linear equations
Andrey Ponomarenko
Sarntal

2. Iterative methods for linear systems, relaxation schemes
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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3. Efficiency of algorithm, Possibility of parallel computing
The way to multigrid…
Algorithm should not do a lot of empty work
Slow process – a lot of calculations
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
Efficiency of algorithm,
Possibility of parallel computing
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4. Iterative methods for linear systems
1) At first, some basics:
Consider
Let v be an approximation to u
Residual equation
Residual correction
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5. 2) Iteration in the matrix form
Let
D – is diagonal matrix
L and U – are lower and upper parts of A
Then
Now the iteration is
- the iteration error
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6. Now consider the iteration properties on the next equation in 1D:
One-dimensional boundary problem
Introduce the grid:
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7. Then, with the help of Taylor series:
Then our problem can be presented in grid form:
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8. Now we have The discrete model problem:
symmetric, positive defined
So, we can used iteration scheme showed above
Consider relaxation scheme:
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9. Relaxation schemes The equation:
1) Jacobi Method (simultaneous displacements):
2) Weighted Jacobi Method :
Then as we have before
D – is diagonal matrix
L and U – are lower and upper parts of A
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10. Short observation about eigenvalues and eigenvectors:
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)
Now we return to relaxation:
- for one iteration
- we’ll have after n iterations
- initial error
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11. Let’s make convergence analysis on weighted Jakobi on 1D model
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12. 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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13. 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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14. 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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15. 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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16. Coarse Grid Correction
We have now the idea of
The question appears: how to map and
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17. Mapping from the coarse grid to the fine grid:
defined on
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18. Mapping from the fine grid to the coarse grid:
defined on
(there are others methods of mapping, we don’t consider them)
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19. The Coarse Grid Correction scheme:
- is the “coarse-grid version” of operator A
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20. 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 Gh
G2h
So, we have recursive use of CGC sheme.
G4h
G8h
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21. Full-Multigrid method
IDEA: the solution takes fewer iterations if the initial guess is good
How to get a good initial guess:
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
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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22. Some properties of convergence:
Some properties of FMG method:
1) Full Multigtid method computes solution to the error of truncation
The computational cost of FMG is O(Nd), where
N – quantity of fine grid points, d – dimension of the problem
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23. 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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24. 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.
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25. Thank you for attention!
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