1. Mutigrid Methods for Solving Differential Equations Ferien Akademie 05 – Veselin Dikov

2. Multigrid Methods Ferien Akademie 05 Veselin Dikov Agenda Model problem Relaxation. Smoothing property Elements of Multigrid Multigrid schemes

3. Multigrid Methods Model Problem Ferien Akademie 05 Veselin Dikov Discretization in n points, step h = 1/n 1D boundary problem of steady state temperature a long a uniform rod

4. Multigrid Methods Model Problem Ferien Akademie 05 Veselin Dikov Stencil notation Av = f, where and A is Symmetric positive definite

5. Multigrid Methods Ferien Akademie 05 Veselin Dikov Agenda Model problem Relaxation. Smoothing property Elements of Multigrid Multigrid schemes

6. Multigrid Methods Iterative Methods Ferien Akademie 05 Veselin Dikov Jacobi and Gauss-Seidel methods Iterative vs Direct methods Smoothing property More about iterative methods

7. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Error along the domain After 35 sweeps with weighted Jacobi Error was smoothed

8. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes k – wave number

9. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes k = 1k = 2 k = 7 k = 12

10. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes smooth modes - oscillatory modes -

11. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes f = 0, σ = 0 Au = 0 2.Modified model problem exact solution: u = 0 error: e = u – v = -v we can trace the error!

12. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes 2.Modified model problem wJacobi step 3.Weighted Jacobi relaxation error

13. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes 2.Modified model problem we relax with wJacobi with ω = 2/3 on initial guesses respectively: 3.Weighted Jacobi relaxation 4.Three experiments # iterations

14. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes 2.Modified model problem repeat the experiment with: ω = 2/3 and initial guess 3.Weighted Jacobi relaxation 4.Three experiments # iterations

15. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes 2.Modified model problem 3.Weighted Jacobi relaxation 4.Three experiments Explanation R ω has the same eigenvectors as A and they are the same as the wave vectors Recall that for the error e (m) = R m e (0) Eigenvalues of R ω ?

16. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes 2.Modified model problem 3.Weighted Jacobi relaxation 4.Three experiments Explanation wavenumber k Eigenvalue

17. Multigrid Methods Smoothing Property Ferien Akademie 05 Veselin Dikov Smoothing property explained in four steps 1.Fourier modes 2.Modified model problem 3.Weighted Jacobi relaxation 4.Three experiments Explanation Smoothing property Fast damping of oscillatory error modes Common for all iterative methods How to overcome the bad performance effect over smooth error modes?

18. Multigrid Methods Ferien Akademie 05 Veselin Dikov Agenda Model problem Relaxation. Smoothing property Elements of Multigrid Multigrid schemes

19. Multigrid Methods Elements of Multigrid Ferien Akademie 05 Veselin Dikov Element I: A smooth wave looks more oscillatory on a coarser grid Aliasing: k looks like (n-k)

20. Multigrid Methods Elements of Multigrid Ferien Akademie 05 Veselin Dikov Element II: Nested Iterations coarsest grid finest grid Relax transfer the coarse grid result to the finer grid for the initial guess Problems?

21. Multigrid Methods Elements of Multigrid Ferien Akademie 05 Veselin Dikov Element III: Correction scheme Residual equation: Ae = r The scheme: » Relax on Au = f on to obtain an approximation. » Compute. » Relax on Ae = r on to obtain an approximation to the error,. » Correct the approximation.

22. Multigrid Methods Elements of Multigrid Ferien Akademie 05 Veselin Dikov Element IV: Interpolation and restriction Interpolation : Restriction : Variational property: Injection: Full weighting:

23. Multigrid Methods Ferien Akademie 05 Veselin Dikov Agenda Model problem Relaxation. Smoothing property Elements of Multigrid Multigrid schemes

24. Multigrid Methods Two-Grid Ferien Akademie 05 Veselin Dikov Two-Grid = Corr.Scheme+Interpolation+Restriction » Relax times on on with initial guess » Compute and restrict. » Solve on. » Interpolate and correct. » Relax times on on with initial guess

25. Multigrid Methods Two-Grid -> V-Cycle Ferien Akademie 05 Veselin Dikov V-Cycle = Recursive Two-Grid Scheme Two-Grid Scheme V-CycleW-Cycle

26. Multigrid Methods Full Multigrid(FMG) Ferien Akademie 05 Veselin Dikov FMG = V-Cycle + nested iterations FMG

27. Multigrid Methods Costs Ferien Akademie 05 Veselin Dikov V-Cycle costs Computational cost Storage FMG computational costs Speedup because working on smaller domains