Multi - Grid Methods Press et al., Numerical Recipes in C , Section - - PowerPoint PPT Presentation

Multi-Grid Methods

Press et al., “Numerical Recipes in C”, Section 19.6

http://www.nrbook.com/a/bookcpdf.php

A Multigrid Tutorial,

http://www.llnl.gov/casc/people/henson/mgtut/welcome.html

1 Tuesday, June 26, 2007

Multi-Grid Methods

• Introduced by Achi Brandt, “Multi-Level

Adaptive Solutions to Boundary V alue Problems”, Math. Comp. 1977

• Solve elliptic PDEs (such as the Poisson

equation) on N grid points in O(N)

• General framework, whose components must

be adjusted to solve specific problems

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Motivation

• Large sparse linear systems might converge

slowly using standard iterative relaxation methods (such as Jacobi, Gauss-Seidel).

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Example

Au = 0, ui−1 −2ui +ui+1 = 0

0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

• 1
• 0 . 8
• 0 . 6
• 0 . 4
• 0 . 2

0 . 2 0 . 4 0 . 6 0 . 8 1

k = 3

k = 1

k = 6

20 40 60 80 100 120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

k = 1

k = 3

k = 6

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Multi-Grid

• Many relaxation schemes have the smoothing

property, where oscillatory modes of the error are eliminated effectively, but smooth modes are damped very slowly!

• Idea: use a sequence of progressively coarser

grids in order to quickly get rid of smooth (low- frequency) errors!

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Two-Grid Case

• Linear elliptic problem:
• Discretized on a uniform grid with spacing h

between grid points:

• Let be an approximate solution.
• Error/correction:
• Residual/defect:

L u = f Lhuh = fh ˜ uh vh = uh − ˜ uh dh = Lh ˜ uh − fh

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Use the result to approximate the correction, and correct the fine grid solution!

Two-Grid Case (continued)

Error may be computed by solving: Lhvh = −dh Idea: use a coarser grid to solve:

LHvH = −dH ˜ unew

h

= ˜ uh + ˜ vh dh = Lh ˜ uh − fh = Lh (uh −vh)− fh = Lhuh −Lhvh − fh = −Lhvh

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Two-Grid Case (continued)

How do we switch between coarse and fine grids? Need to define two linear operators:

• Restriction/injection/fine-to-coarse operator R
• Prolongation/interpolation/coarse-to-fine
• perator P

dH = Rdh ˜ vh = P˜ vH

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Two-Grid Case (summary)

Iterate until convergence:

• Relaxation/smoothing on the fine grid
• Compute the defect on fine grid
• Restrict defect to coarse grid
• Solve the coarse problem
• Interpolate the correction to fine grid
• Apply correction to get a new

approximation

dh = Lh ˜ uh − fh dH = Rdh ˜ vh = P˜ vH LHvH = −dH ˜ unew

h

= ˜ uh + ˜ vh

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Relaxation / Smoothing

Relaxation (smoothing) operator: typically a few iterations of Gauss-Seidel:

ui = − 1 Lii

• ∑

j=i

Lij u j − fi

• i = 1,...,N

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Relaxation / Smoothing

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Prolongation

Prolongation operator in 1D: typically linear interpolation. Specified by the following stencil (symbol):

• 1/2

1 1/2

• 12

Tuesday, June 26, 2007

Example

The prolongation operator, written in matrix form (in 1D, for interpolating 3 variables into 7):

          1/2 1 1/2 1/2 1 1/2 1/2 1 1/2          

7×3

  vH,1 vH,2 vH,3   =           vh,1 vh,2 vh,3 vh,4 vh,5 vh,6 vh,7          

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Prolongation in 2D

Prolongation operator in 2D: typically bi-linear interpolation.

  0.25 0.5 0.25 0.5 1 0.5 0.25 0.5 0.25  

Specified by the following stencil (symbol), on a 2D grid:

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Restriction

Simplest operator: straight injection

[1]

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Restriction

Another operator: full-weighting [1/4

1/2 1/4]

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Example

The restriction operator, written in matrix form (in 1D, for restricting 7 variables into 3):

  1/4 1/2 1/4 1/4 1/2 1/4 1/4 1/2 1/4  

3×7

          vh,1 vh,2 vh,3 vh,4 vh,5 vh,6 vh,7           =   vH,1 vH,2 vH,3  

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Restriction in 2D

Simplest operator: straight injection

[1]

Other choices: Full weighting Half weighting

  1/16 1/8 1/16 1/8 1/4 1/8 1/16 1/8 1/16     1/8 1/8 1/2 1/8 1/8  

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Prolongation & Restriction

1 2           1 2 1 1 2 1 1 2 1          

P cPT

1 4   1 2 1 1 2 1 1 2 1  

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Operator Coarsification

• Discretize the continuous linear operator on

progressively coarser grids

• Use the prolongation/restriction operators:

LH = RLh P

• r

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Cycle Strategies: V and W

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Cycle Strategies: FMG

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