a multigrid tutorial - lawrence livermore national laboratory (llnl)
TRANSCRIPT
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Properties of the Grid TransferOperators: Interpolation
• Interpolation: or
: ℜ N/2-1 ℜ N-1
• For N=8,
• has full rank and null space .
I2hh
I 22
hhhh Ω→Ω:
I2hh =
1211
211
21
21
I2hh φ
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Spectral properties of .
• How does act on the eigenvectors of ?
• Consider , 1 ≤ k ≤ N/2-1, 0 ≤ j ≤ N/2
• Good Exercise: show that the modes of areNOT preserved by , but that the space ispreserved:
I2hh
I2hh A 2h
/
π
=)(Nkj
wj
hk2
2nis
A 2h
I2hh W k
wN
kw
Nk
wI 2222
hk
hk
hk
hh
π
−
π
=2
nis2
soc ′
−= wswc hkk
hkk ′
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Spectral properties of (cont.).
• Interpolation of smooth modes excitesoscillatory modes on .
• Note that if ,
• is second-order interpolation.
I2hh
wswcI2hkk
hkk
hh −= ′
Ω2h
Ωh
Nk
2«
wN
kOw
N
kOwI
2
2
2
22
2hk
hk
hk
hh
+
−
= 1 ′
≈ whk
I2hh
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The Range of .
• The range of is the span of the columns of
• Let be the ith column of .
• All the Fourier modes of are needed torepresent Range( )
I2hh
I2hh I2
hh
ξi I2hh
:ξi
I2hh
≠,=ξ khkk
N
k
hi
−
=
1
1bwb 0
A h
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Use all the facts to analyze thecoarse-grid correction scheme1) Relax times on :
2) Compute and restrict residual
3) Solve residual equation
4) Correct fine-grid solution .
• The entire process appears as
• The exact solution satisfies
Ωh vRv hh ← α1
vAfIf 22 hhhhh
h )−(←
fAv 2122 hhh )(=−
vIvv hhh
hh +← 22
vRAfIAIvRv hhhhh
hhh
hh )−()(+← α−α 2122
uRAfIAIuRu hhhhh
hhh
hh )−()(+← α−α 2122
α
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Error propagation of the coarse-grid correction scheme
• Subtracting the previous two expressions, we get
• How does CG act on the modes of ? Assumeconsists of the modes and forand .
• We know how act on and .
eRAIAIIe hhhh
hhh
h )(−← 2122
α−
eGCe hh←
A h
whk′wh
k 21 /≤≤ NkkNk −=′
IAIAR−α ,)(,,, h
hhh
hh
2122
whk wh
k′
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Error propagation of CG• For now, assume no relaxation . Then
is invariant under CG.
where
=α 0wwW h
khkk ,= naps ′
wswswGC hkk
hkk
hk += ′
wcwcwGC hkk
hkk
hk ′′ +=
Nk
ck
π
=2
soc 2N
ksk
π
=2
nis 2
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CG error propagation for k << N
• Consider the case k << N (extremely smooth andoscillatory modes):
• Hence, CG eliminates the smooth modes but doesnot damp the oscillatory modes of the error!
wN
kOw
N
kOw kkk
+
→2
2
2
2
′
wN
kOw
N
kOw kkk
−
+
−
→ 112
2
2
2
′
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Now consider CG with relaxation
• Next, include relaxation sweeps. Assume thatthe relaxation preserves the modes of(although this is often unnecessary). Letdenote the eigenvalue of associated with .For k < < N/2,
αR A h
λkR wk
wswsw kkkkkkk λ+λ→ ′αα
wcwcw kkkkkkk ′α′
α′′ λ+λ→
Small!
Small!
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The crucial observation:
• Between relaxation and the coarse-grid correction, both smooth andoscillatory components of the errorare effectively damped.
• This is essentially the “spectral”picture of how multigrid works. Weexamine now another viewpoint, the“algebraic” picture of multigrid.
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Recall the variational properties
• All the analysis that follows assumes that thevariational properties hold:
IAIA 222 h
hhh
hh =
IcI 22
Thh
hh )(=
(Galerkin Condition )
for c in ℜ
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Algebraic interpretation ofcoarse-grid correction
• Consider the subspaces that make up andΩh Ω2h
IR )( 2hh IN )( h
h2Ωh
Ω2h IR )( hh2
I2hh I h
h2
For the rest of this talk, ‘R( )’refers to the Range of a linearoperator.
From the fundamental theoremof linear algebra:
orIRIN ))((⊥)( Thh
hh
22
IRIN )(⊥)( hh
hh 22
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Subspace decomposition of .• Since has full rank, we can say, equivalently,
( where means that ).
• Therefore, any can be written as where and .
• Hence,
Ωh
A h
AINIR )(⊥)( 22
hhhA
hh h
yx ⊥A h yxA h =, 0
eh tse hhh +=IRs h
hh )(∈ 2 AINt hh
hh )(∈ 2
)(⊕)(=Ω hhh
hh
h AINIR 22
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Characteristics of the subspaces
• Since for some , weassociate with the smooth components of .But, generally has all Fourier modes in it (recallthe basis vectors for ).
• Similarly, we associate with oscillatorycomponents of , although generally has allFourier modes in it as well. Recall that isspanned by therefore isspanned by the unit vectorsfor odd i, which “look” oscillatory.
qIs hhh
h = 22 q 22 hh Ω∈
sh eh
shI2
hh
t h
eh t h
IN )( hh2
=η ih
i eA AIN )( hhh2
e Thi ),...,,,,...,,(= 001000
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qIAIAIIsGC hhh
hhh
hhh
h )(−= 22
2122
−
Algebraic analysis of coarse-gridcorrection
• Recall that (without relaxation)
• First note that if then .This follows since for someand therefore
• It follows that , that is, the nullspace of coarse-grid correction is the range ofinterpolation.
AIAIIGC )(−= 2122
hhh
hhh
−
IRs hh
h )(∈ 2 sGC h = 0qIs hh
hh = 2
2 q 22 hh Ω∈
= 0
IRGCN )(=)( 2hh
A 2h by Galerkin property
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More algebraic analysis ofcoarse-grid correction
• Next, note that if then
• Therefore .
• CG is the identity on
AINt hhh
h )(∈ 2
tAIAIItGC hhhh
hhh
h )(−= 2122
−
ttGC hh =0
AIN )( hhh2
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How does the algebraic picturefit with the spectral view?
• We may view in two ways:
=
that is,
or as
Ωh
Low frequency modes1 ≤ k ≤ N/2
High frequency modesN/2 < k < N Ωh ⊕
⊕=Ωh HL
)(⊕)(=Ω hhh
hh
h AINIR 22
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Actually, each view is just partof the picture
• The operations we have examined work ondifferent spaces!
• While is mostly oscillatory, it isn’t .and while is mostly smooth, it isn’t .
• Relaxation eliminates error from .
• Coarse-grid correction eliminates error from
H
H
IR )( 2hh L
AIN )( hhh2
IR )( 2hh
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Relaxation eliminates H, butincreases the error in .
IR )( 2hh
AIN )( hhh2
eh
sh
t h
H
L
IR )( 2hh
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Difficulties:anisotropic operators and grids
• Consider the operator :
• The same phenomenonoccurs for grids with muchlarger spacing in one directionthan the other:
),(=∂∂β−
∂∂
α−2
2
2
2yxf
yu
xu
-100 -50 0 50 1000
0.2
0.4
0.6
0.8
1
1.2 If then the GS-smoothingfactors in the x- and y-directions areshown at right.Note that GS relaxation does notdamp oscillatory components in the x-direction.
βα «
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Difficulties: discontinuous oranisotropic coefficients
• Consider the operator :where
• Solutions: line-relaxation (where whole gridlinesof values are found simultaneously), and/or semi-coarsening (coarsening only in the strongly coupleddirection).
Again, GS-smoothing factors in the x- and y-directionscan be highly variable, and very often, GS relaxation doesnot damp oscillatory components in the one or bothdirections.
)∇),((•∇− uyxD
yxdyxd
yxdyxdyxD
),(),(
),(),(
=),(2212
2111
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For nonlinear problems, use FAS (Full Approximation Scheme)
• Suppose we wish to solve: wherethe operator is non-linear. Then the linearresidual equation does not apply.
• Instead, we write the non-linear residual equation:
• This is transferred to the coarse grid as:
• We solve for and transfer theerror (only!) to the fine grid:
reA =
fuA =)(
ruAeuA =)(−)+(
uAfIeuA 2222 hhhhh
hhh ))(−(=)+(
euw 222 hhh +≡
uIwIuu hhh
hhh
hh )−(+← 222
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Multigrid: increasingly, the righttool!
• Multigrid has been proven on a wide variety ofproblems, especially elliptic PDEs, but has also foundapplication among parabolic & hyperbolic PDEs,integral equations, evolution problems, geodesicproblems, etc.
• With the right setup, multigrid is frequently anoptimal (i.e., O(N)) solver.
• Multigrid is of great interest because it is one of thevery few scalable algorithms, and can be parallelizedreadily and efficiently!