AMG
(Ch. 8 in Briggs, Henson & McCormick)
1/29

Motivation, why do we need Algebraic Multigrid?
The problem.
Jacobi’s method revisited.
Algebraic smoothness.
Coarsening:
Strong dependencies between the variables?
Interpolation and restriction operators.
Colouring schemes.
AMG 2/29

Motivation
Geometric multigrid:
Relaxation schemes (Jacobi, Gauss-Seidel, etc.).
Sequence of grids.
Interpolation & restriction operators.
AMG 3/29

Motivation
Grids not available:
No grids associated with the problem.
Unstructured/irregular fine grid. Coarsening problematic.
Can we still apply multigrid techniques?
AMG 4/29

The problem
Starting point: A linear system is a matrix.
is a vector, unknown.
is a vector.
AMG 5/29

The problem
Must define:
Relaxation schemes (as in the geometric case!).
Algebraic smoothness.
Coarsening techniques. In terms of algebraic properties.
Interpolation and restriction operators.
Coarse scale approximations of .
AMG 6/29

Jacobi’s method revisited
Linear system
Splitting where is the diagonal of .
That is where .
Which motivates the scheme
AMG 7/29

Jacobi’s method revisited
Weighted Jacobi, where .
Which motivates the scheme
AMG 8/29

Jacobi’s method revisited
Note that
Hence
AMG
To summarise
9/29

Algebraic smoothness
We will assume that is a symmetric M-matrix:
Symmetric,
Positive definite,
.
.
for all .
for .
(Typically a discrete self-adjoint second order elliptic differential operator).
AMG 10/29

Algebraic smoothness
Geometric multigrid:
Errors associated the low-frequencies are not effectively reduced by relaxation.
The error appears to be smooth relatively to the grid.
Algebraic multigrid.
Stop relaxation as soon as the error is not effectively reduced.
The error not available.
In practice, use the residual
.
, or
AMG 11/29

Algebraic smoothness
Recall that
Since we find that
Hence
AMG 12/29

Algebraic smoothness
Error
Moreover,
Thus, the error propagation is
AMG 13/29

Algebraic smoothness
Error not reduced effectively by relaxation if
Thus or
Which gives the criterion
AMG 14/29

Algebraic smoothness
Note that
So the criterion can be written on the form or for all
AMG 15/29

Algebraic smoothness
“The error is algebraic smooth if, for all , can be approximated well by a weighted average of its neighbours”.
In practice or
AMG 16/29

Coarsening
How do we define a coarse appr. of
Fine scale: .
Fine scale indices
Algebraic coarsening:
Find an appropriate subset of
.
!
?
AMG 17/29

Coarsening
How do we find a coarse set of indices?
On which other variables are strongly dependent?
Colouring schemes.
Interpolation and restriction operators.
Coarse appr. of etc.
AMG 18/29

Dependence
Which variables have a strong influence on ?
is an M-matrix:
We associate the th equation with the th unknown
.
If is large, relative to the other coefficients, then strongly depends on .
AMG 19/29

Dependence
Definition: Given a threshold value strongly depends on the variable if
, the variable
AMG 20/29

Interpolation
Assume that the coarse scales indices have been determined.
Partitioning of the indices:
Disjoint union:
: Coarse scale indices.
.
: Fine scale indices that are only fine scale indices.
Our goal is to determine an interpolation operator
AMG 21/29

Interpolation
For a fine-grid point we define and , the neighbourhood of .
strongly influences , the coarse interpolatory set for .
strongly influences .
do not strongly influence , the weakly connected neighbours.
AMG 22/29

Interpolation
It seems to be reasonable to use the points in for interpolation, i.e. we are searching for an operator on the form if if where are weights which must now be determined.
AMG 23/29

Interpolation
Assume that .
Recall the smooth error condition
Hence
.
or
AMG
To obtain an interpolation formula we must “replace” the terms and
24/29

Interpolation
If is small.
then doesn’t depend strongly on . That is,
Thus, the approximation will introduce a relatively insignificant error.
It follows that
AMG 25/29

Interpolation
If
.
then there is a strong connection between and
It seems to be reasonable to approximate these
“pure” fine scale variables by a weighted sum of the coarse scale variables for
AMG 26/29

Interpolation
Combining this approximation and we find that
AMG 27/29

Interpolation
To summarise: The interpolation operator is given by if if where
AMG 28/29

Interpolation
The restriction operator is normally defined by i.e. the transposed of the interpolation operator.
AMG 29/29