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

RDF 2University of Colorado

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

/


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


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


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.


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


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

∑


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


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


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


Agglomerations for triangular elements, both structured & unstructured

Element-Free AMGe


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


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


Algebraically determine relaxation blocks

Use strong connections to determine blocks.

Relax on block perhaps using AMG (nested).

Compatible Relaxation (CR)


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


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


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.


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

/

algebraic multigrid amg

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