Introduction to Scientific Computing II

1
Introduction to Scientific Computing II

• From Gaussian Elimination to Multigrid A
  Recapitulation

2
Tasks SLE
???
3
Tasks Molecular Dynamics
4
Prerequisites

• discretisation of PDEs
• linear algebra
• Gaussian elimination
• basics on iterative solvers
• Jacobi, Gauss-Seidel, SOR, MG
• matlab

5
Organization

• lecture (90 min/week)
• theory
• methods
• simple examples
• tutorials (45 min/week)
• more examples
• make your own experiences

6
What Determines the Grading?

• written exam at the end of the semester
• no weighting of tutorials
• !!!! solving tutorials is essential !!!!
• for understanding and remembering subjects
• for your success in the exam

7
Materials

• slides (short, only headwords)
• exercise sheets
• make your own lecture notes!
• find your own solutions!
• solutions presented in the tutorials

8
Rules

• for questions ask or fix a date per emailDr.
  Miriam Mehl mehl_at_in.tum.de Martin
  Buchholzbuchholm_at_in.tum.de

9
Introduction to Scientific Computing II

• From Gaussian Elimination to Multigrid A
  Recapitulation

10
Whats the Problem to be Solved?
Application Scenario
Partial Differential Equations
Modelling Scientific Computing I
Finite Elements Finite Differences (Finite
Volumes) Scientific Computing I Numerical
Programming II
Systems of linear equations
LU, Richardson, Jacobi, Gauss-Seidel, SOR,
MG Scientific Computing I, Scientific Computing
Lab, Numerical Programming I
More on this!!!
11
Example Equation

• two-dimensional Poisson equation
• heat equation
• diffusion
• membranes

grid finite differences
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Typical SLE

• sparse
• band structure

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Example
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination (LU)
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Gaussian Elimination Costs
2D O(N4) 3D O(N7)
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Gaussian Elimination Costs 2D
h runtime (HLRB2, 62 TFlop/s)
2-7 0.02 sec
2-8 0.27 sec
2-9 4.4 sec
2-10 1 min 16 sec
2-11 18 min 55 sec
2-12 5 h 02 min 40 sec
2-13 3 d 8 h 37 min 15 sec
hallo
25
Gaussian Elimination Costs 3D
hallo
h runtime (HLRB2, 62 TFlop/s)
2-6 4 min 44 sec
2-7 10 h 05 min 24 sec
2-8 53 d 19 h 21 min 17 sec
2-9 18 a 313 d 21 h 54 min 22 sec
26
Iterative Solvers Principle

• series of approximations
• costs per iteration?
• convergence?
• stopping criterion?

27
Relaxation Methods
problem order an amount of peas on a straight
line (corresponds to solving uxx0)
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Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
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Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
30
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
31
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
32
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
33
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
34
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
35
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
36
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
37
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
38
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
39
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
40
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
41
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
42
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
43
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
44
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
45
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
46
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
47
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
48
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours
49
Relaxation Methods Gauss-Seidel
sequentially place peas on the line between two
neighbours

• we get a smooth curve instead of a straight line
• global error is locally (almost) invisible

50
Relaxation Methods
problem order an amount of peas on a straight
line (corresponds to solving uxx0)
51
Relaxation Methods Jacobi
place peas on the line between two neighbours in
parallel
52
Relaxation Methods Jacobi
place peas on the line between two neighbours in
parallel
53
Relaxation Methods Jacobi
place peas on the line between two neighbours in
parallel
54
Relaxation Methods Jacobi
place peas on the line between two neighbours in
parallel

• we get a high plus a low frequency oscillation
• these fequencies are locally (almost) invisible

55
Relaxation Methods
problem order an amount of peas on a straight
line (corresponds to solving uxx0)
56
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
57
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
58
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
59
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
60
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
61
Relaxation Methods SOR
62
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
63
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
64
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
65
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
66
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
67
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
68
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
69
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
70
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
71
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
72
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
73
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
74
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
75
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
76
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
77
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
78
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
better than GS and J, but still not optimal
79
Relaxation Methods
problem order an amount of peas on a straight
line (corresponds to solving uxx0)
80
Relaxation Methods Hierarchical
place peas on the line between two neighbours in
parallel, but in a hierarchical way from coarse
to smooth
81
Relaxation Methods Hierarchical
place peas on the line between two neighbours in
parallel, but in a hierarchical way from coarse
to smooth
82
Relaxation Methods Hierarchical
place peas on the line between two neighbours in
parallel, but in a hierarchical way from coarse
to smooth
83
Relaxation Methods Hierarchical
place peas on the line between two neighbours in
parallel, but in a hierarchical way from coarse
to smooth

• exact solution in one step
• unfortunately only in 1D, 2D and 3D multigrid