Introduction to Scientific Computing II

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Introduction to Scientific Computing II

• Overview

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Recall Scientific Computing Pipeline
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Topic 1 SLE(numerical treatment,
implementation)
???
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Topic 2 Molecular Dynamics(entire pipeline
for one application)
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Prerequisites

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

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Organization

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

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What Determines the Grading?

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

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Course Material

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

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Contact

• for questions contact us after the lectures
• or fix a date per emailMichael Bader
  bader_at_in.tum.de Wolfgang Eckhardteckhardw_at_in.
  tum.de

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Introduction to Scientific Computing II

• From Gaussian Elimination to Multigrid A
  Recapitulation

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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!!!
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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

• Storage (for an n-by-n grid)
• matrix has N n2 rows
• in L and U n new non-zeros per row
• therefore O(Nn) O(n3) bytes
• In 3D
• N n3 rows, n2 new non-zeros
• therefore O(Nn2) O(n5) bytes

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Gaussian Elimination Costs

• Operations
• matrix has N n2 rows
• for each row, eliminate n non-zeros in column
  below
• addition of rows requ. O(n) operations
• therefore O(Nn2) O(n4) operations
• In 3D
• N n3 rows, n2 new non-zeros
• therefore O(Nn4) O(n7) operations

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Gaussian Elimination Costs

• Storage (for an n-by-n grid)
• 2D O(Nn) O(n3) bytes
• 3D O(Nn2) O(n5) bytes
• Computation
• 2D O(Nn2) O(n4) operations
• 3D O(Nn4) O(n7) operations
• Even for problems of modest size (n 100-1000) ?
  Gaussian Elimination is unfeasible

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Iterative Solvers Principle

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

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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
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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
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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
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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
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
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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
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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

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

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Relaxation Methods
problem order an amount of peas on a straight
line (corresponds to solving uxx0)
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Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
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Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
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Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
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Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
61
Relaxation Methods SOR
sequentially correct location of peas a little
more than to the line between two neighbours
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Relaxation Methods SOR
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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
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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
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Relaxation Methods
problem order an amount of peas on a straight
line (corresponds to solving uxx0)
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Relaxation Methods Hierarchical
place peas on the line between two neighbours in
parallel, but in a hierarchical way from coarse
to smooth
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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
84
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