Algebraic Multigrid AMG
description
Transcript of Algebraic Multigrid AMG
Steve McCormick, Tom Manteuffel, John Ruge+++
Applied Math DepartmentUniversity of Colorado @ Boulder
January, 2002
Algebraic MultigridAMG
+++Center for Applied Scientific ComputingLawrence Livermore National Laboratory
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OutlineAlphabet Soup
AMG Classical AMG
AMGe Element Interpolation AMG
AMGe Element-Free AMGe
SA Smoothed Aggregation
aAMG Algebraic Relaxation AMG
cAMG Compatible Relaxation AMG
AMGe Spectral AMGe
AMGe Adaptive AMG
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Multigrid for discretized PDEs L
hu h = b
h
smoothing
Finest Grid
First Coarse Grid
restriction
prolongation(interpolation)
The MultigridV-cycle
Note:smaller grid
MG scalability comes from using a family of grids. Each grid efficiently computes features at its own scale.
Key: “good” transfer of residual equationL
he h = r
h f h - L
hu h
to “good” coarse grid
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Algebraic multigrid for unstructured grids
Ax = b Standard AMG only uses matrix info
AMG automatically coarsens “grids”
DYNA3D
Accurate characterization of smooth error is crucial
AMG FrameworkR n
algebraically smooth
error
Fixe
d!error dampedquickly
by pointwise relaxation
Choose coarse grids, transfer
operators, etc. to eliminate
Classical AMG
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Capturing Smooth Error by Interpolation
M-Matrices: Poisson on unstructured grid.
Choose interpolation to capture ‘smooth’
error:
e = Pe2h.
But what characterizes smoothness?
‘Small’ residual: Ae 0.
M-matrices: Smooth error varies slowly along
‘strong coupling’: |aij | ≥ aii.
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Coarsening (Coarse-Grid Correction)
Ax = b
Ae = b - Ax x exact = x + e
APe2h = b - Ax smooth e e = Pe2h
(PTAP)e2h = PT(b - Ax) applying PT to both
sides
A2he2h = b2h redefining terms
x x + Pe2h correcting fine grid
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Prolongation based on smooth error & variable interdependence (weighted
graph)
i
CC
CF
F
F
Sets:Strongly connected -pts.
Strongly connected -
pts. Weakly connected points.
Ci C
DsiDwi
F
Strong C Strong F Weak pts.
Ae ≈ 0
€
aiiei ≈ − aikk∈Ci
∑ ek − aijj∈F
∑ ej − aiωeωj∈W
∑
AMGe
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Good local characterization of smooth error is key to robust AMG
AMG uses heuristics based on M-matrices: smooth error varies slowest in the direction of “large” coefficients.
AMGe heuristics based on multigrid theory: interpolation must reproduce a mode with error proportional to the associated eigenvalue.
A
−−−
−−−=
141282141
Stretched quad example ( ):Direction of strength not
apparent.Worse for systems.
∞→Δx
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AMGe coarsening uses elements to localize & approximate modes with error
Use local measure to construct AMGe components:
IPQeeA
eQIeQIM
i
Ti
Ti
ei =;,
)−(ε,)−(ε= 0xam
≠0
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Computing interpolation in practice
Partition local matrix by F & C-pts:
Interpolation to point i is defined by
Perfect interpolation of the local problem.
AA
AAA
ccfc
cfffi =
AAq
ifffci ε−=
0−1
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Agglomerations for triangular elements, both structured & unstructured
Element-Free AMGe
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The Assumptions: smooth error from low energy modes of local Ai; no elements!
We construct the prolongation operator on the basis of the modified local matrix
Then the ith row of the prolongation matrix P is taken as the ith row of the matrix.
EEI
IAAAAA
cXfX
Xfcfffcfff ,,=, 00
⎞⎟⎠
⎛⎜⎝
− AA cfff− 1
Smoothed Aggregation
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Uses simple interpolation & smoothing
Choose simple interpolation (e.g., piecewise constant) based on element agglomerates.
Smooth: relax a couple of times on the simple basis elements.
Algebraic Relaxation AMG
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Algebraically determine relaxation blocks
Use strong connections to determine blocks.
Relax on block perhaps using AMG (nested).
Compatible Relaxation (CR)
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How good are the C points?
Global martix partition:
Relax on Aff xf = 0.
If CR is slow to converge, either increase
the coarse-grid size or do more relaxation.
Can be generalized (Brandt).
AA
AAA =
ccfc
cfff
Spectral AMGe
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AMGe: Take eigenvectors as the basis!
As with AMGe, use elements to localize the problem of determining & matching smooth error.
Coarse dofs are no longer subsets of fine dofs: coefficients of local eigenvectors become the dofs.
Currently expensive, but potentially very robust.
Adaptive AMGe
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Adaptive or Bootstrap or Calibration or Prerelaxation or Feedback or … AMG
Test your AMG on a problem whose solution you know: Ax = 0.
If it works after a few cycles, stop. Else, x is a good bad guy: it’s an algebraically
smooth error in the sense that AMG cannot quickly reduce it.
Now adjust the coarse grid (primarily interpolation) so that it matches x well. The trick is to do this locally & to continue it on coarser levels.
Our early 80’s scheme recovered fast convergence for mis-scaled scalar problems. Now working on systems.
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Alphabet Soup
AMG Classical AMG
AMGe Element Interpolation AMG
AMGe Element-Free AMGe
SA Smoothed Aggregation
aAMG Algebraic Relaxation AMG
cAMG Compatible Relaxation AMG
AMGe Spectral AMGe
AMGe Adaptive AMG
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