Iterative Methods (pdf) - School of Mathematics

Review Iterative Methods Relaxation Summary

I TERATIVE M ETHOD FOR E LLIPTIC PDE S Dr. Johnson School of Mathematics

Semester 1 2008

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

O UTLINE 1

R EVIEW

2

I TERATIVE M ETHODS Point Iteration Residual Gauss-Siedel iteration

3

R ELAXATION Relaxation and SOR Line Relaxation

4

S UMMARY

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

We have shown how to discretise a model elliptic problem. ∂2 φ ∂2 φ + 2 = f (x, y ), ∂x 2 ∂y The discretised equations can be expressed as the matrix equation. Aw = f Usually solve using iterative techniques. The matrix form helps us think about methods in an abstract way. The matrix form will allow us to analyse the stability and convergence of schemes. Dr. Johnson

MATH65241

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Review Iterative Methods Relaxation Summary

Point Iteration Residual Gauss-Siedel iteration

I TERATIVE M ETHODS

Point relaxation. Jacobi, Gauss-Seidel, SOR Line Relaxation Conjugate gradient and PCG methods Minimal residual methods Multigrid Fast Direct Methods (FFT)

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Point Iteration Residual Gauss-Siedel iteration

J ACOBI M ETHOD Rewrite the discrete equations wi,j +1 − 2wi,j + wi,j −1 wi +1,j − 2wi,j + wi −1,j + = fi,j , 2 ∆x ∆y 2 for 1 ≤ i ≤ N − 1, and 1 ≤ j ≤ M − 1. as

wi,j =

£ ¤ 1 wi +1,j + wi −1,j + β2 (wi,j +1 + wi,j −1 ) − ∆x 2 fi,j 2 2 + 2β

with β = ∆x /∆y .

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Point Iteration Residual Gauss-Siedel iteration

J ACOBI M ETHOD

This suggests the iterative scheme

q +1 wi,j =

h i 1 q q q q 2 2 w + w + β ( w + w ) − ∆x f i,j i +1,j i −1,j i,j +1 i,j −1 2 + 2β2

q here wi,j is the qth guess at the solution.

How do we know when we have the real solution? We must define some convergence criteria.

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Point Iteration Residual Gauss-Siedel iteration

E RRORS IN THE SCHEME Suppose we write the linear system as Av = f where v is the exact solution of the linear system. If w is an approximate solution, the error e is defined by e = v − w. Thus Ae = A(v − w) = f − Aw.

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Point Iteration Residual Gauss-Siedel iteration

D EFINING THE RESIDUAL The residual is defined by r = f − Aw. We don’t know the exact solution... but we can calculate the residual. The residual is related to the error, e = 0 if and only if r = 0 So, we choose the magnitude of the residual as the convergence criteria.

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Point Iteration Residual Gauss-Siedel iteration

J ACOBI R ESIDUAL

For the Jacobi scheme the residual is given by ri,j = wi +1,j + wi −1,j − (2 + 2β2 )wi,j + β2 (wi,j +1 + wi,j −1 ) − ∆x 2 fi,j Suitable stopping conditions might be max |ri,j | < ǫ, i,j

or

r

∑ ri,j2 < ǫ i,j

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Point Iteration Residual Gauss-Siedel iteration

G AUSS -S IEDEL M ETHOD

In this iterative scheme we use the most up-to-date values of w q +1 wi,j =

h i 1 q q +1 q q +1 2 2 w + w + β ( w + w ) − ∆x f i,j i +1,j i −1,j i,j +1 i,j −1 2 + 2β2

q here wi,j is again the qth guess at the solution,

and we have already calculated w at (i − 1, j ) and (i, j − 1) Gauss-Siedel will usually converge faster than Jacobi iteration.

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Relaxation and SOR Line Relaxation

FASTER C ONVERGENCE WITH R ELAXATION

Point iteration can be slow to converge. A faster way to find the solution is overestimate the correction at each point. We call this relaxation. We define q +1 q ∗ wi,j = (1 − ω )wi,j + ωwi,j

∗ is the solution from the iteration scheme. where wi,j

ω is the relaxation factor.

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Relaxation and SOR Line Relaxation

S UCCESSIVE O VER R ELAXATION

If ω = 1 we just have the Jacobi or Gauss-Seidel scheme. If ω > 1 with Gauss-Seidel then it is called successive overrelaxation ot the SOR scheme Theorems can show that 0 < ω < 2 is a condition for a scheme to converge. If we choose ω < 1 then we under-relax, and the convergence will be slower. This can be used to bound errors in unstable schemes.

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Relaxation and SOR Line Relaxation

M ORE D IRECT M ETHODS The Jacobi, Gauss-Seidel and SOR schemes are called point relaxation methods. We can compute a whole line of new values using a direct method, this leads to line-relaxation methods. y

x

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

Relaxation and SOR Line Relaxation

L INE R ELAXATION

Keeping i constant, we may write the system of equations q +1 q +1 q q +1 2 2 q +1 2 β2 wi,j +1 − (2 + 2β )wi,j + β wi,j −1 = ∆x fi,j − (wi +1,j + wi −1,j )

for 1 ≤ j ≤ M − 1. We can solve this system using a direct-solver. We can also over-relax wqi +1 = (1 − ω )wqi + ωwi∗

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Dr. Johnson

MATH65241

Review Iterative Methods Relaxation Summary

R ELAXATION M ETHODS The model equations can be rewritten wi,j +1 − 2wi,j + wi,j −1 wi +1,j − 2wi,j + wi −1,j + = fi,j , ∆x 2 ∆y 2 How up-to-date the values are define different schemes We can over-relax to improve convergence q +1 q ∗ wi,j = (1 − ω )wi,j + ωwi,j

and also solve a line (or block) using a direct method. Next up - convergence properties...

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Dr. Johnson

MATH65241