Introduction to Scientific Computing II

1
Introduction to Scientific Computing II

• Multigrid

2
Multigrid Algorithm

• iterate (GS) on the fine grid
• restrict residual to the coarse grid
• solve coarse grid equation for the error
• interpolate error to the fine grid
• correct fine grid solution

3
Multigrid Things to Choose

• smoother
• relation step sizes coarse fine
• grid transfer operators
• restriction
• interpolation
• processing order of grid levels

4
Multigrid Convergence

• two grid analysis
• h-independent convergence
• for good components

5
Two Grid Multigrid

• Example 2D Poisson 5-point-stencil

h two-grid analysis V-cycle
1/32 0.042 0.042
1/64 0.042 0.044
1/128 0.042 0.044
1/256 0.042 0.043
1/512 0.042 0.043
1/1024 0.042 0.042
1/2048 0.042 0.042
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Multigrid Some Rules

• smoother
• optimal smoothing
• not(!) optimal convergence
• small number of smoothing iterations!

7
Multigrid Some Rules

• grid coarsening
• standard doubling of h
• exceptions
• anisotropic operators
• adaptively refined grids
• unstructured grids/general SLEs

8
Multigrid Some Rules

• restriction/interpolation
• order of restriction order of interpolation
  gt
  order of discretisation

9
Multigrid Some Rules

• V-cycle
• faster
• W-cycle
• more robust

10
Multigrid Parallelisation

• parallel smoothing
• parallel restriction and interpolation
• parallel stopping criteria

11
Multigrid Parallel Smoothing
? Jacobi-like operation!!!
Gauss Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
Gauss Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
Gauss Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
? Jacobi-like operation!!!
Gauss Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
Gauss Seidel ? different result than sequential
GS!!!
processor 2
processor 1
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Multigrid Parallel Smoothing
Alternatives Red-Black Gauss-Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
Alternatives Red-Black Gauss-Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
Alternatives Red-Black Gauss-Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
Alternatives Red-Black Gauss-Seidel
processor 2
processor 1
20
Multigrid Parallel Smoothing
Alternatives Red-Black Gauss-Seidel
processor 2
processor 1
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Multigrid Parallel Smoothing
Alternatives Red-Black Gauss-Seidel
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Multigrid Parallel Smoothing
Alternatives damped Jacobi
processor 2
processor 1
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Multigrid Parallel Smoothing
Alternatives damped Jacobi
processor 2
processor 1
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Multigrid Parallel Smoothing
Alternatives damped Jacobi
processor 2
processor 1
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Multigrid Parallel Smoothing

• Smoothing
• red-black GS robust and fast smoothing
• further reading Irad Yavneh, Multigrid
  smoothing factors for red-black Gauss-Seidel
  relaxation applied to a class of elliptic
  operators, SIAM Journal on Numerical Analysis, 32
  (4), 1995
• Jun Zhang, Acceleration of five-point red-black
  Gauss-Seidel in multigrid for Poisson equation,
  Applied Mathematics and Computation, 80(1), 1996
• damped Jacobi good smoothing

26
Universität Erlangen-Nürnberg
Technische Universität München
Universität Stuttgart
Ferienakademie, Sarntal, Sep 23 Oct 5, 2012
Course 4 Scales and Scalability as Challenges
for CSE (Bader, Schweitzer, Wellein) Deadline
Yesterday