Algebraic MultiGrid

Algebraic MultiGrid – AMG (Brandt 1982)

General structure Choose a subset of variables: the C-points

such that every variable is “strongly connected” to this subset

Define the interpolation (aggregation) weights of each fine variables to the C-points

Construct the coarse level equations Repeat until a small enough problem Interpolate (disaggregate) to the finer level Classical versus weighted aggregation

Data structureFor each node i in the graph keep1. A list of all the graph’s neighbors: for each neighbor

keep a pair of index and weight2. …3. …4. Its current placement5. The unique square in the grid the node belongs to

For each square in the grid keep1. A list of all the nodes which are mostly within Defines the current physical neighborhood 2. The total amount of material within the square

Data structureFor each node i in the graph keep1. A list of all the graph’s neighbors: for each neighbor

keep a pair of index and weight2. A list of finer level vertices belonging to i 3. A list of coarse level aggregates i contributes to4. Its current placement5. The unique square in the grid the node belongs to

For each square in the grid keep1. A list of all the nodes which are mostly within Defines the current physical neighborhood 2. The total amount of material within the square

Influence of (pointwise) Gauss-Seidelrelaxation on the error

Poisson equation, uniform grid

Error of initial guess Error after 5 relaxation

Error after 10 relaxations Error after 15 relaxations

The basic observations of ML Just a few relaxation sweeps are needed to

converge the highly oscillatory components of the error

=> the error is smooth Can be well expressed by less variables Use a coarser level (by choosing every other

line) for the residual equation Smooth component on a finer level becomes

more oscillatory on a coarser level=> solve recursively The solution is interpolated and added

TWO GRID CYCLE

Approximate solution:hu~

hhh u~UV hhh RVL

hhhh u~LFR

Fine grid equation: hhh FUL

2. Coarse grid equation: hhh RVL 22

hh2

h2v~~~ hold

hnew uu h

h2

Residual equation:Smooth error:

1. Relaxation

residual:

h2v~Approximate solution:

3. Coarse grid correction:

4. Relaxation

TWO GRID CYCLE

Approximate solution:hu~

hhh u~UV hhh RVL

hhhh u~LFR

Fine grid equation: hhh FUL

2. Coarse grid equation: hhh RVL 22

hh2

hold

hnew uu h2v~~~ h

h2

Residual equation:Smooth error:

1. Relaxation

residual:

h2v~Approximate solution:

3. Coarse grid correction:

4. Relaxation

1

2

34

5

6

by recursion

MULTI-GRID CYCLE

Correction Scheme

interpolation (order m)of corrections relaxation sweeps

residual transfer

ν ν enough sweepsor direct solver*

.. .

*

1ν

1ν

1ν

2ν

2ν

2ν

Vcyclemultigrid

h0

h0/2

h0/4

2h

h

V-cycle: V

Hierarchy of

graphs

Apply grids in all scales: 2x2, 4x4, … , n1/2xn1/2

Coarsening Interpolate and relax

Solve the large systems of equations by multigrid!

G1

G2

G3

Gl

G1

G2

G3

Gl

Graph drawing example