introduction to scientific computing ii from gaussian elimination to multigrid – a recapitulation...
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Introduction to Scientific
Computing II
From Gaussian Elimination to Multigrid – A Recapitulation
Dr. Miriam Mehl
Tasks – SLE
fu
nn
nn
nn
nn
r
r
r
r
u
u
u
u
,
,1
1,2
1,1
,
,1
1,2
1,1
411
1411
11411
1141
114
???
Tasks – Molecular Dynamics
Prerequisites
• discretisation of PDEs
• linear algebra
• Gaussian elimination
• basics on iterative solvers
• Jacobi, Gauss-Seidel, SOR, MG
• matlab
Organization
• lecture (90 min/week)
– theory
– methods
– simple examples
• tutorials (45 min/week)
– more examples
– make your own experiences
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
Materials
• slides (short, only headwords)
• exercise sheets
make your own lecture notes!
find your own solutions!
solutions presented in the tutorials
Rules
• for questions ask or fix a date per email
Dr. Miriam Mehl:
Martin Buchholz:
Introduction to Scientific
Computing IIFrom Gaussian Elimination to Multigrid
– A Recapitulation
Dr. Miriam Mehl
What’s the Problem to be Solved?
0
)(Re1
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puuuu
T
t
Finite Elements
Finite Differences
(Finite Volumes)
Scientific Computing I
Numerical Programming II
Systems of linear equations
Application
ScenarioModelling
Scientific
Computing I
Partial Differential Equations
bAuh LU, Richardson, Jacobi, Gauss-Seidel,
SOR, MG
Scientific Computing I, Scientific Computing Lab,
Numerical Programming IMore on this!!!
two-dimensional Poisson equation
heat equation diffusion membranes …
Example Equation
21;0 in fu
vvvvvvvvvvvvvvv
grid +
finite differences
bAuh
Typical SLE
sparse
band structure
411
1411
11411
1141
114
Example
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]2[
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f
f
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f
f
f
u
u
u
u
u
u
u
u
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Gaussian Elimination (LU)
411
1411
141
1411
11411
1141
141
1141
114
1
1
1
1
1
1
1
1
1
Gaussian Elimination (LU)
411
1411
141
1411
11411
****
141
****
114
1
1
1
1
1
1*
1
1*
1
Gaussian Elimination (LU)
411
1411
141
1411
*****
****
****
****
114
1
1
1
1
1*
1**
1*
1*
1
Gaussian Elimination (LU)
411
1411
141
****
****
****
****
****
114
1
1
1
1*
1**
1***
1*
1*
1
Gaussian Elimination (LU)
411
1411
****
****
****
****
****
****
114
1
1
1*
1**
1***
1***
1*
1*
1
Gaussian Elimination (LU)
411
****
***
****
****
****
****
****
114
1
1*
1**
1***
1***
1***
1*
1*
1
Gaussian Elimination (LU)
***
***
***
****
****
****
****
****
114
1*
1**
1***
1***
1***
1***
1*
1*
1
Gaussian Elimination (LU)
**
**
***
****
****
****
****
****
114
1**
1***
1***
1***
1***
1***
1*
1*
1
Gaussian Elimination (LU)
*
**
***
****
****
****
****
****
114
1***
1***
1***
1***
1***
1***
1*
1*
1
Gaussian Elimination – Costs
2D: O(N4)
3D: O(N7)
Gaussian Elimination – Costs 2D
halloh 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
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
Iterative Solvers – Principle
series of approximations
costs per iteration? convergence? stopping criterion?
hMh
Mhhh uuuuu with10
Relaxation Methods
problem: order an amount of peas on a straight line
(corresponds to solving uxx=0)
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
Relaxation Methods – Gauss-Seidel
sequentially place peas on the line between two neighbours
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
Relaxation Methods – Gauss-Seidel
Relaxation Methods
problem: order an amount of peas on a straight line
(corresponds to solving uxx=0)
Relaxation Methods – Jacobi
place peas on the line between two neighbours in parallel
Relaxation Methods – Jacobi
place peas on the line between two neighbours in parallel
Relaxation Methods – Jacobi
place peas on the line between two neighbours in parallel
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
Relaxation Methods
problem: order an amount of peas on a straight line
(corresponds to solving uxx=0)
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
Relaxation Methods – SOR
sequentially correct location of peas a little more than
to the line between two neighbours
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
Relaxation Methods
problem: order an amount of peas on a straight line
(corresponds to solving uxx=0)
Relaxation Methods – Hierarchical
place peas on the line between two neighbours in parallel,
but in a hierarchical way from coarse to smooth
Relaxation Methods – Hierarchical
place peas on the line between two neighbours in parallel,
but in a hierarchical way from coarse to smooth
Relaxation Methods – Hierarchical
place peas on the line between two neighbours in parallel,
but in a hierarchical way from coarse to smooth
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