Mutigrid Methods for Solving Differential Equations

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Mutigrid Methods for Solving Differential
Equations

• Ferien Akademie 05 Veselin Dikov

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Multigrid Methods

• Agenda
• Model problem
• Relaxation. Smoothing property
• Elements of Multigrid
• Multigrid schemes

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Model Problem

• 1D boundary problem of steady state temperature
  a long a uniform rod

• Discretization in n points, step h 1/n

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Multigrid Methods Model
Problem

• Av f, where

and

• Stencil notation

• A is Symmetric positive definite

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Multigrid Methods

• Agenda
• Model problem
• Relaxation. Smoothing property
• Elements of Multigrid
• Multigrid schemes

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Iterative
Methods

• Iterative vs Direct methods

More about iterative methods

• Jacobi and Gauss-Seidel methods

• Smoothing property

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Multigrid Methods Smoothing Property

• Error along the domain

After 35 sweeps with weighted Jacobi
Error was smoothed
Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing Property

• Smoothing property explained in four steps

• Fourier modes

• k wave number

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing Property

• Smoothing property explained in four steps

• Fourier modes

k 1
k 2
k 12
k 7
Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing Property

• Smoothing property explained in four steps

• Fourier modes

• smooth modes -

• oscillatory modes -

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing Property

• Smoothing property explained in four steps

• Fourier modes

• Modified model problem

• f 0, s 0 Au 0

• exact solution u 0

• error e u v -v
• we can trace the error!

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing Property

• Smoothing property explained in four steps

• Fourier modes

• Modified model problem

• Weighted Jacobi relaxation

• wJacobi step

• error

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing Property

• Smoothing property explained in four steps

• Fourier modes

• Modified model problem

• Weighted Jacobi relaxation

• Three experiments

• we relax with wJacobi with ? 2/3 on initial
  guesses respectively

iterations
Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing Property

• Smoothing property explained in four steps

• Fourier modes

• Modified model problem

• Weighted Jacobi relaxation

• Three experiments

• repeat the experiment with
• ? 2/3 and initial guess

iterations
Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing
Property

• Smoothing property explained in four steps

• Fourier modes

• Modified model problem

• Weighted Jacobi relaxation

• Three experiments

• Explanation

• R? has the same eigenvectors as A and they are
  the same as the wave vectors

• Recall that for the error e(m) Rme(0)

• Eigenvalues of R? ?

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing
Property

• Smoothing property explained in four steps

• Fourier modes

• Modified model problem

• Weighted Jacobi relaxation

• Three experiments

• Explanation

Eigenvalue
wavenumber k
Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Smoothing
Property

• Smoothing property explained in four steps

• Fourier modes

• Modified model problem

• Weighted Jacobi relaxation

• Three experiments

• Explanation

• Smoothing property

• Fast damping of oscillatory error modes

• Common for all iterative methods

• How to overcome the bad performance effect over
  smooth error modes?

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods

• Agenda
• Model problem
• Relaxation. Smoothing property
• Elements of Multigrid
• Multigrid schemes

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Multigrid Methods Elements of Multigrid

• Element I A smooth wave looks more oscillatory
  on a coarser grid

• Aliasing k looks like (n-k)

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Multigrid Methods Elements of Multigrid

• Element II Nested Iterations

• Problems?

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Multigrid Methods Elements of Multigrid

• Element III Correction scheme

• Residual equation Ae r

• The scheme

• Relax on Au f on to obtain an
  approximation .
• Compute .
• Relax on Ae r on to obtain an
  approximation to the error, .
• Correct the approximation .

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Elements of Multigrid

• Element IV Interpolation and restriction

• Interpolation

• Restriction

Injection
Full weighting

• Variational property

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Multigrid Methods

• Agenda
• Model problem
• Relaxation. Smoothing property
• Elements of Multigrid
• Multigrid schemes

Ferien Akademie 05 Veselin Dikov
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Multigrid Methods Two-Grid

• Two-Grid Corr.SchemeInterpolationRestriction

• Relax times on on
  with initial guess
• Compute and restrict
  .
• Solve on .
• Interpolate and correct
  .
• Relax times on on
  with initial guess

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Multigrid Methods Two-Grid -gt V-Cycle

• Two-Grid Scheme

• V-Cycle Recursive Two-Grid Scheme

V-Cycle
W-Cycle
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Multigrid Methods Full Multigrid(FMG)

• FMG V-Cycle nested iterations

FMG
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Multigrid Methods Costs

• V-Cycle costs

Storage
Computational cost

• FMG computational costs

• Speedup because working on smaller domains

Ferien Akademie 05 Veselin Dikov