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

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RDF 3University of Colorado

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

RDF 4University of Colorado

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

RDF 6University of Colorado

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.

RDF 7University of Colorado

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

RDF 8University of Colorado

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

RDF 10University of Colorado

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

RDF 11University of Colorado

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

RDF 12University of Colorado

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

RDF 13University of Colorado

Agglomerations for triangular elements, both structured & unstructured

Element-Free AMGe

RDF 15University of Colorado

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

RDF 17University of Colorado

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

RDF 19University of Colorado

Algebraically determine relaxation blocks

Use strong connections to determine blocks.

Relax on block perhaps using AMG (nested).

Compatible Relaxation (CR)

RDF 21University of Colorado

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

RDF 23University of Colorado

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

RDF 25University of Colorado

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.

RDF 26University of Colorado

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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