An
Multigrid Tutorial
Van Emden Henson
Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
vhenson@llnl.gov
http://www.casc.gov/CASC/people/henson
April 10, 1999
AMG:
What is Algebraic Multigrid??



Any multilevel method where geometry is not used
(and may not be available) to build coarse grids,
interpolation and restriction, or coarse-grid
operators.
“Classical” AMG was introduced by Brandt,
McCormick and Ruge in 1982. It was explored
early on by Stueben in 1983, and popularized by
Ruge and Stuben in 1987.
This tutorial will describe only the classical AMG
algorithm.
CASC
veh 2
AMG:
What is Algebraic Multigrid??

Many other algorithms qualify under the definition
given, that are important, or novel, or historical,
or downright weird:
•
•
•
•
•
•
•
•
Multigraph methods (Bank & Smith)
Aggregation methods (Braess; Chan & Zikatanov & Xu )
Smoothed Aggregation methods (Mandel & Brezina & Vanek)
Black Box Multigrid (Dendy, Dendy & Bandy)
Algebraic Multilevel Recursive Solver (Saad)
Element based algebraic multigrid (Chartier; Cleary et al)
MultiCoarse correction with Suboptimal Operators (Sokol)
Multilevel block ILU methods (Jang & Saad; Bank & Smith &
Wagner; Reusken)
• AMG based on Element Agglomeration (Jones & Vassilevski)
• Sparse Approximate Inverse Smoothers (Tang & Wan)
•Algebraic Schur-Complement approaches (Axelsson &
Vassilevski & Neytcheva)
CASC
veh 3
Highlights of Multigrid:
The 1-d Model Problem

Poisson’s equation:   u  f in [0,1], with
boundary conditions u 0  u 1  0 .
Discretized as:
 ui  1  2 ui  ui  1
u0  uN  0
 fi
h2

Leads to the Matrix equation

CASC
 2
 1

1
A 
2
h 



1
2 1
1 2 1






1 2 1

1 2 
Au  f
u1
u2
u3








u 



u
 N 2
u

 N 1
, where
 f1 
 f 
 2 
 f 
f  3 


 f N 2


f
N

1


veh 4
Highlights of Multigrid:
Weighted Jacobi Relaxation

Consider the iteration:

2h
Letting A = D+L+U, the matrix form is:


n
e
w


o
l
d

ui
  1   ui

 ui ol1d  ui ol1d  f i 
2
u  n ew   1   I   D  1 L  U u  o ld   D  1f
 G u  old   D  1f

It is easy to see that if
then
.
e  u  ex a c t   u
 a p p ro x 
,
e  n ew  G e  old
CASC
veh 5
Highlights of Multigrid:
Relaxation Typically Stalls

The eigenvectors of G are the same as those of A ,
and are Fourier Modes: vi  sin  ik  N  k  1 2  N  1

The eigenvalues of G are 1  2  s i n 2  k   2N , so
the effect of relaxation on the modes is:
1
N = 400
0 .5
  1 3
  1 2
  2 3
0
-0 .5
-1
CASC
0
2 0 0
4 0 0
Note: No value
of  will damp
out the low
frequency
waves
  1
veh 6
Highlights of Multigrid:
Relaxation Smooths the Error
Initial error,

2
1
0
- 1
- 2
0
0 . 5
1
Error after 35 iteration sweeps:

2
1
0
- 1
Many relaxation
schemes
have the smoothing
property, where
oscillatory
modes of the error
are
eliminated
effectively, but
smooth modes are
damped
very slowly.
- 2
0
CASC
0 . 5
1
veh 7
Highlights of Multigrid: Smooth error
can be represented on a coarse grid
A smooth function:

1
0
. 5
0
- 0
. 5
- 1
0
0
. 5
1
Can be represented by linear
interpolation from a coarser grid:

1
0
. 5
0
- 0
. 5
- 1
0
CASC
0
. 5
1
On the coarse grid, the
smooth error appears to
be relatively higher in
frequency: in the example
it is the 4-mode, out of
a possible 16, on the fine
grid, 1/4 the way up the
spectrum. On the coarse
grid, it is the 4-mode out
of a possible 8, hence it
is 1/2 the way up the
spectrum.
Relaxation will be more
effective on this mode if
done on the coarser grid!!
veh 8
Highlights of Multigrid:
Coarse-grid Correction



h
h
Perform relaxation on A uh  f on fine grid until
error is smooth.
h  f h Ahuh
r
Compute residual,
and transfer to the
2h
coarse grid r 2h  I h r h.
Solve the coarse-grid residual equation to obtain the
error:
2h 2h
2h 1 2h
2h
2h
A e

r 
 e
 A 
r
Interpolate the error to the fine grid and correct the
fine-grid solution:
u h  u h  I 2hh e 2h
CASC
veh 9
Highlights of Multigrid:
Coarse-grid Correction
h h
h
A
u

f
Relax on
h h hh
Compute r  f A u
uh 
Correct
u h  eh
Interpolate
Restrict
r 2h  I h2h r h
eh  I2hh e2h
Solve
A2h e2h  r2h
e 2h   A 2h  1 r 2h
CASC
veh 10
Highlights of Multigrid:
Tools Needed

Interpolation and restriction operators:
 0 5

 1 0



 0 5 0 5



h
I 2h  
1 0


0 5 0 5 


1 0 


0 5 


0 1 0

0 1 0


0 1 0

I h2h  
0 25 1 0 0 25

0 25 1 0 0 25


0 25 1 0 0 25 

I h2h  
Linear
Injection
Full-weighting
Interpolation
2h
A
Coarse-grid Operator
. Two methods:
(1) Discretize equation at larger spacing
(2) Use Galerkin Formula:
A 2h  I h2h A h I 2hh
CASC
veh 11
Highlights of Multigrid:
Recursion: the (  ,0) V-cycle

Major question: How do we “solve” the coarse-grid
residual equation? Answer: recursion!
u h  G   A h f h 
uh  uh  eh
e h  I 2hh u 2h
f 2h  I h2h  f h  A h u h 
u 2h  G   A 2h  f 2h 
u 2h  u 2h  e 2h
f 4h  I 24hh  f 2h  A 2h u 2h 
e 2h  I 42hh u 4h
u 4h  u 4h  e 4h
u 4h  G   A 4h  f 4h 
f 8h  I 48hh  f 4h  A 4h u 4h 
u 8h  G   A 8h  f 8h 
e 4h  I 84hh u 8h
u 8h  u 8h  e 8h
e H   A H  1 f H
CASC
veh 12
Algebraic multigrid:
for unstructured-grids

Automatically defines coarse “grid”

AMG has two distinct phases:

AMG approach is opposite of geometric
MG
— setup phase: define MG components
— solution phase: perform MG cycles
— fix relaxation (point Gauss-Seidel)
— choose coarse “grids” and prolongation, P,
so that error not reduced by relaxation is
in range(P)
— define other MG components so that
coarse-grid correction eliminates error in
range(P) (i.e., use Galerkin principle)
(in contrast, geometric MG fixes coarse grids,
then defines suitable operators and smoothers)
CASC
veh 13
AMG has two phases:

Setup Phase
m1
 m  1 2   
— Select Coarse “grids,” 
— Define interpolation,
m
I m  1  m  1 2   
— Define restriction and coarse-grid operators
T
m1
m
Im
  Im

1

m1 m m
A m  1  Im
A Im  1
Solve Phase
— Standard multigrid operations, e.g., V-cycle, Wcycle, FMG, etc
CASC
veh 14
In AMG, we choose relaxation first:

Typically, pointwise Gauss-Seidel is used
A   D  L  U

The iteration is developed:
Ax  b
 D  L x  b  Ux
1
1
n
e
w
x
  D  L b   D  L Ux old

Add and subtract  D  L  1 D  L x old to get:
 1 old
n
e
w
o
l
d
x
 x
  D  L
r
CASC
veh 15
Gauss-Seidel relaxation
error propagation:

The iteration is:
 1 old
n
e
w
o
l
d
x
 x
  D  L
r

Subtracting both sides from the exact solution:
x ex a ct  x n ew  x ex a ct   x old   D  L   1 r old 
enew  e old   D  L  1 r old

Using
r  Ae
e n ew 
CASC
this can be written as:
I  D L  1 A
e o ld
veh 16
An observation: error that is slow
to converge  “small” residuals

Consider the iterative method error recurrence
1
k

1
e
  I Q A  ek

Error that is slow to converge satisfies
 I Q 1A  e  e

Q


1
r  0
Perhaps a better viewpoint is
1
 I Q 1A  e  e

CASC
Ae 0
Q
A e A e
«
e  Ae
veh 17
Some implications
of slow convergence


For most iterations (e.g., Jacobi or Gauss-Seidel)
this last holds if D  1 A e  A e « e  A e . (1)
Hence
N ri2
 a ii
i1
«
N
 ri ei implying
i 1
that, on average,
 ri  « aii  ei 

An implication is that, if e is an error slow to
converge, then locally at least, ei can be wellapproximated by an average of its neighbors:
CASC
veh 18
In Multigrid, error that is slow
to converge is geometrically smooth


Combining the algebraic property that slow
convergence implies “small residuals” with the
observation above, in AMG we DEFINE smooth
error:
Smooth error is that error which is slow to
converge under relaxation, that is,
 I Q 1A  e  e
  I
CASC
or, more precisely,
1
 Q A e A   e A
veh 19
AMG uses dependence (influence) to
determine MG components



We need to choose a subset of the gridpoints
(coarse grid) that can be used 1) to represent
smooth errors, and 2) to interpolate these errors
to the fine grid.
Intuitively, a point u j is a good candidate for a Cpoint if its value is important in determining the
value of another point, ui in the ith equation.
If the a i j coefficient is “large” compared to the
other off-diagonal coefficients in the ith equation
then u j influences ui (or ui depends on u j ).
CASC
veh 20
Dependence and smooth error

For M-matrices, we define “i depends on j ” by
 aij   max
k i
  aik  
0  1
alternatively, “ j influences i. ”

It is easy to show from (1) that smooth error
satisfies
(2)
A e  e « De  e
CASC
veh 21
Dependence and smooth error

For M-matrices, we have from (2)
1
2
2
  a i j   ei  e j 
  2 a ii   ei  « 1
i j
— If ei does not depend on ej then the inequality
may be satisfied because a i j is “small”.
— If ei does depend on ej, then a i j need not be
small, and the inequality must be satisfied by

This implies that smooth error varies slowly in the
direction of dependence.
CASC
veh 22
Some useful definitions

The set of dependencies of a variable ui , that is,
the variables upon whose values the value of ui
depends, is defined as


S i   j   a i j  ma x   a i k  
ki



The set of points that ui influences is denoted:
S iT   j  i  S j 
CASC
veh 23
More useful definitions

The set of coarse-grid variables is denoted C .

The set of fine-grid variables is denoted F.

The set of coarse-grid variables used to
interpolate the value of the fine-grid variable
called the coarse interpolatory set for i, is
denoted C i .
CASC
ui ,
veh 24
Two Criteria for Choosing
the Coarse Grid Points

First Criterion: F - F dependence
— (C1) For each i  F , each point j  S i should
either be in C itself or should depend on at
least one point in C i .
j  Si
k  Ci
i F
n  Ci
CASC
Since the value of ui depends on the
value of uj , the value of uj must be
represented on the coarse-grid for
good interpolation. If j isn’t a C-point,
it should depend on a point in C i so its
value is “represented” in the
interpolation.
veh 25
Two Criteria for Choosing
the Coarse Grid Points

Second Criterion: Maximal Subset
— (C2) C should be a maximal subset with the
property that no C -point depends on another.
— (C1) tends to increase the number of C -points.
H
In general, the more C-points on  , the
better the h-level convergence.
— But more C-points means more work for
relaxation and interpolation.
— (C2) is designed to limit the size (and work) of
the coarse grid.
CASC
veh 26
Two Criteria for Choosing
the Coarse Grid Points



It is sometimes not possible to satisfy both
criteria simultaneously (an example will be seen
shortly).
In those cases, we choose to satisfy (C1), the
requirement that F-F dependencies be represented
in the coarse-interpolatory set, while using (C2) as
a guide.
This choice leads to somewhat larger coarse grids,
but tends to preserve good convergence properties.
CASC
veh 27
Choosing the Coarse Grid Points





Assign to each gridpoint k a “value” equal to the
number of points that depend on k.
Choose the first point with global maximum value
as a C-point.
The new C-point can be used to interpolate values
of points it influences. Assign them all as Fpoints.
Other points influencing these new F-points can be
used in their interpolation. Increment their value.
Repeat until all points are C- or F-points.
CASC
veh 28
Ruge AMG:
start
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
 select
C-pt with
maximal measure
 select
5
8
8
8
8
8
5
5
8
8
8
8
8
5
5
8
8
8
8
8
5
3
5
5
5
5
5
3
CASC
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 29
Ruge AMG:
select C-pt 1
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
5
8
8
8
8
8
5
5
8
8
8
8
8
5
5
8
8
8
8
8
5
3
5
5
5
5
5
3
CASC
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 30
Ruge AMG:
select F-pt 1
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
 select
C-pt with
maximal measure
 select
5
CASC
8
8
8
8
8
5
8
8
8
5
8
8
8
5
5
5
5
3
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 31
Ruge AMG:
update F-pt neighbors 1
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
 select
C-pt with
maximal measure
 select
7
CASC
11
10
9
8
8
5
10
8
8
5
11
8
8
5
7
5
5
3
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 32
Ruge AMG:
select C-pt 2
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
7
11
10
9
8
8
5
10
8
8
5
8
8
5
5
5
3
7
CASC
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 33
Ruge AMG:
select F-pt 2
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
7
11
10
9
8
8
5
8
5
8
5
5
3
CASC
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 34
Ruge AMG:
update F-pt neighbors 2
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
7
11
11
11
10
9
5
10
5
11
5
6
3
CASC
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 35
Ruge AMG: select C-pt, F-pts,
update neighbors 3
3
5
5
5
5
5
3
5
8
8
8
8
8
5
5
8
8
8
8
8
5
7
11
11
11
11
11
7
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
CASC
veh 36
Ruge AMG: select C-pt, F-pts,
update neighbors 4
3
7
5
11
5
10
5
5
5
3
9
8
8
5
10
8
8
5
13
11
11
7
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
CASC
veh 37
Ruge AMG: select C-pt, F-pts,
update neighbors 5
3
7
5
11
5
11
5
11
5
10
5
3
9
5
10
5
13
7
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
CASC
veh 38
Ruge AMG: select C-pt, F-pts,
update neighbors 6,7,8,9
 select
next C-pt
with maximal
measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
CASC
veh 39
Examples: Laplacian Operator
5-pt FD, 9-pt FE (quads), and 9-pt FE (stretched quads)
CASC
5-pt FD
9-pt FE (quads)
1


 1 4 1


1
1 1 1


 1 8 1 
 1 1 1 
veh 40
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
 select
C-pt with
maximal measure
 select
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
CASC
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 41
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
CASC
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 42
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
CASC
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 43
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
10
8
8
10
11
10
9
8
8
9
10
8
8
8
8
8
8
8
8
8
8
8
8
8
8
11
10
9
8
8
9
10
10
8
8
10
11
8
8
11
CASC
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of F-pt neighbors
veh 44
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
10
10
11
11
11
10
9
9
10
8
8
8
8
8
8
8
8
8
8
8
8
8
8
11
11
11
10
9
9
10
10
10
11
11
CASC
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of new F-pt
neighbors
veh 45
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
11
11
11
11
11
11
11
8
8
8
8
8
8
8
8
8
8
8
8
8
8
11
11
11
11
11
11
11
CASC
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of new F-pt
neighbors
veh 46
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
11
11
11
11
11
11
11
11
10
9
8
8
9
10
10
8
8
10
13
11
11
13
CASC
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of new F-pt
neighbors
veh 47
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
11
11
11
11
11
11
11
11
11
11
10
9
9
10
10
10
13
13
CASC
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of new F-pt
neighbors
veh 48
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
11
11
11
11
11
11
11
11
11
11
11
11
11
11
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of new F-pt
neighbors
CASC
veh 49
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
13
11
11
13
13
11
11
13
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of new F-pt
neighbors
CASC
veh 50
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
13
13
13
13
 select
C-pt with
maximal measure
 select
neighbors
as F-pts
 update
measures
of new F-pt
neighbors
CASC
veh 51
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
 Modulo
periodicity,
it’s the same
coarsening as in
the Dirichlet case.
 However,
it has
many F-F
connections that do
not share a common
C-point
CASC
veh 52
Coarse-grid selection: 9pt Laplacian,
periodic boundary conditions
A
second pass is made
in which some F-points
are made into C-points
to enforce (C1).
 Goals
of the second
pass include minimizing
C-C connections, and
minimizing the number
of C-points converted
to F-points.
CASC
veh 53
Prolongation
F
F
C
i
C
C
F
 iC
ei

 P e  
i   ik ek  iF
kCi
The interpolated value at
point i is just ei if i is a Cpoint. If i is an F-point,
the value is a weighted sum
of the values of the points in
the coarse interpolatory set
C .
i
CASC
veh 54
To define prolongation at i, we must
examine the types of connections of ui.
Sets of connection types:
F
F
C
i
C
C
F
CASC
Ci
i is dependent on these
coarse interpolatory Cpoints.
Dis
i is dependent on these Fpoints.
Diw
i does not depend on these
“weakly connected” points,
which may be C- or Fpoints.
veh 55
Prolongation is based on smooth error
and dependencies (from M-matrices)
Recall that smooth error is characterized by
“small” residuals:
ri  a i i ei 

j N i
a i j ej  0
which we can rewrite as:
a i i ei  
 a i j ej
j i
We base prolongation on this formula by
“solving” for ei and making some approximating
substitutions.
CASC
veh 56
Prolongation is based on smooth error
and dependencies (from M-matrices)
We begin by writing the smooth-error relation:
a i i ei  
a i j ej

j i
Identifying its component sums:
a i i ei  
 aij ej  s aij ej  waij ej
j Ci
Coarse
interpolatory
set
j Di
j Di
F-point
dependencies
Weak
connections
We must approximate e in each of the last
j
two sums in terms of ei or of e for j  Ci .
j
CASC
veh 57
For the weak connections:
let ej  ei .
a i i ei  
F
F
 aij ej
j Ci
Coarse
interpolatory
set

s aij ej
j Di
F-point
dependencies

w aij ej
j Di
Weak
connections
C
i
C
C
F
Effectively, this throws the weak
connections onto the diagonal:

 aii   ai j

j  Diw

 ei    a i j ej   a i j ej

j  Ci
j  Dis
This approximation can’t hurt too much:
• Since the connection is weak.
• If i depended on points in Diw, smooth error varies
slowly in the direction of dependence
CASC
veh 58
For the F-point dependencies:
use a weighted avg. of errors in Ci  Cj.
F
F


a

a
 ii
 w i j  ei    a i j ej   s a i j ej


j  Ci
j  Di
j  Di
C
Weak
connections
Coarse
interpolatory set
i
C
C
F
Approximate e j by a
weighted average of
the ek in the coarse
interpolatory set
Ci  C j .
ej 
F-point
dependencies


  a j k ek 
 k Ci



  ajk 
 k Ci

It is for this reason that the intersection of the
coarse interpolatory sets of two F-points with a
dependence relationship must be nonempty (C1).
CASC
veh 59
Finally, the prolongation
weights are defined
Making the previous substitution, and with a bit of
messy algebra, the smooth error relation can be
“solved” for ei to yield the interpolation formula:
ei 

j  Ci
i j ej
where the prolongation weights are given:
aij 
i j  
aikak j
s 
a
j Di mC k m
i
aii 
 w ain
n Di
CASC
veh 60
Highlights of Multigrid:
Storage: f h, u h must be stored each level

In 1-d, each coarse grid has about half
the number of points as the finer grid.
 In 2-d, each coarse grid has about one-
fourth the number of points as the finer
grid.
 In d-dimensions, each coarse grid has
about 2  d
finer grid.
the number of points as the
d
 Storage cost: 2N  1  2
d
2
 2d
2
 3d
 2
 Md
 
2N d
1  2d
less than 2, 4/3, 8/7 the cost of storage on the fine grid
for 1, 2, and 3-d problems, respectively.
CASC
veh 61
AMG storage:
grid complexity


For AMG there is no simple predictor for total
m
m 1 m 1
storage costs. u m, f m , and A m  I m
A
Im
1
must be stored on all levels.
Define , the grid complexity, as the total
number of unknowns (gridpoints) on all levels,
divided by the number of unknowns on the finest
level. Total storage of the vectors u and f
occupy 2 storage locations.
CASC
veh 62
AMG storage:
operator complexity

Define  A, the operator complexity, as the total m
number of nonzero coefficients of all operators A
divided by the number of nonzero coefficients in
0
the fine-level operator A . Total storage of the
operators occupies  A storage locations.
CASC
veh 63
AMG storage:
interpolation


We could define  I , an interpolation complexity,
as the total number of nonzero coefficients of all
m1
operators I m divided by the number of nonzero
1
coefficients in the operator I 0 . This measure is
not generally cited, however (like most
multigridders, the AMG crowd tends to ignore the
cost of intergrid transfers).
Two measures that occasionally appear are  A ,
the average “stencil size,” and  I, the average
number of interpolation points per F-point.
CASC
veh 64
AMG Setup Costs:
flops


Flops in the setup phase are only a small portion of
the work, which includes sorting, maintaining
linked-lists, keeping counters, storage
manipulation, and garbage collection.
Estimates of the total flop count to define
interpolation weights (I ) and the coarse-grid
operators ( A) are:
A  N  I  2  I   A   I   3  I   A 
and
 I  N  I  3  A   I   2 
CASC
veh 65
AMG setup costs:
a bad rap

Many geometric MG methods need to compute
prolongation and coarse-grid operators

The only additional expense in the AMG setup phase
is the coarse grid selection algorithm

AMG setup phase is only 10-25% more
expensive than in geometric MG and may
be considerably less than that!
CASC
veh 66
Highlights of Multigrid:
Computation Costs




Let 1 Work Unit (WU) be the cost of one
relaxation sweep on the fine-grid.
Ignore the cost of restriction and interpolation
(typically about 20% of the total cost). (See?)
Consider a V-cycle with 1 pre-Coarse-Grid
correction relaxation sweep and 1 post-CoarseGrid correction relaxation sweep.
Cost of V-cycle (in WU):
2 1 2

d
2
 2d
2
 3d
   2  Md  
2
1 2d
Cost is about 4, 8/3, 16/7 WU per V-cycle in 1,
2, and 3 dimensions.
CASC
veh 67
AMG Solve Costs:
flops per cycle


The approximate number of flops in on level m for
one relaxation sweep, residual transfer, and
interpolation are (respectively)
F
A
A
F
NmC  2  I Nm
2N m
2 Nm
 2  I Nm
A
N
where m is the number of coefficients in A m
m
C
F

and N m  N m are the numbers of C-, F-points on
.
The total flop count for a  1  2  V-cycle, noting
F
that  N m  N and letting   1  2 is
m
approximately
N  2    1  A    4  I     1 
CASC
veh 68
AMG Solve Costs:
flops per cycle, again



All that is very well, but in practice we find the
solve phase is generally dominated by the cost of
relaxation and computing the residual.
Both of those operations are proportional to the
number of nonzero entries in the operator matrix
on any given level.
Thus the best measure of the ratio of work done
on all levels to the work done on the finest level is
operator complexity:
A
CASC
veh 69
Highlights of Multigrid: difficultiesanisotropic operators and grids

Consider the operator :

2u
x 2

2u
y 2
 If  «  then the GS-smoothing
factors in the x- and y-directions are
shown at right.
Note that GS relaxation does not
damp oscillatory components in the
x-direction.
 f  x  y
1.2
1
0.8
0.6
0.4
0.2
0
-100
-50
0
50
100
The same phenomenon
occurs for grids with much
larger spacing in one direction
than the other:

CASC
veh 70
Highlights of Multigrid: difficultiesdiscontinuous or anisotropic coefficients

Consider the operator :     D x  y   u  where
 d11 x  y  d12 x  y  
D x  y   

d

x

y

d

x

y

22
 21

 Again, GS-smoothing factors in the x- and y-directions
can be highly variable, and very often, GS relaxation does
not damp oscillatory components in the one or both
directions.

Solutions: line-relaxation (where whole gridlines of
values are found simultaneously), and/or semicoarsening (coarsening only in the strongly coupled
direction).
CASC
veh 71
AMG does semi-coarsening
automatically!

Consider the operator :


2u
x 2

2u
y 2
 f  x  y
In the limit, as 
stencil becomes:
  , the

AMG
automatically
produces a semicoarsened grid!!
1 4 1


8
2 
 2
 1 4 1 
CASC
veh 72
How’s it perform (vol I)?
Regular grids, plain, old, vanilla problems

The Laplace Operator:
Convergence
Stencil
per cycle
5-pt
0.054
0.067
0.078
0.109
5-pt skew
9-pt (-1,8)
9-pt (-1,-4,20)

Anisotropic Laplacian:
Time
Setup
Complexity per Cycle Times
2.21
2.12
1.30
1.30
0.29
0.27
0.26
0.26
1.63
1.52
1.83
1.83
  Ux x  Uy y
Epsilon 0.001
0.01
0.1
0.5
1
2
10
100
1000
Convergence/cycle 0.084
0.093
0.058
0.069
0.056
0.079
0.087
0.093
0.083
CASC
veh 73
How’s it perform (vol II)?
Structured Meshes, Rectangular Domains

5-point Laplacian on regular rectangular grids
Convergence factor (y-axis) plotted against number of nodes (x-axis)
0 .1 6
0 .1 4
0 .1 3 5
0 .1 4 2
0 .1 4 6
0 .1 3 3
0 .1 2
0 .1
0 .0 8
0 .0 8 2
0 .0 6
0 .0 5 5
0 .0 4
0 .0 2
0
0
CASC
500000
1000000
veh 74
How’s it perform (vol III)?
Unstructured Meshes, Rectangular Domains

Laplacian on random unstructured grids (regular
triangulations, 15-20% nodes randomly collapsed into neighboring nodes)
Convergence factor (y-axis) plotted against number of nodes (x-axis)
0 .3
0 .2 5 3
0 .2 5
0 .2 2 3
0 .2
0 .1 7 5
0 .1 5
0 .1 1 1
0 .0 9 7
0 .1
0 .0 5
0
0
CASC
20000
40000
60000
veh 75
How’s it perform (vol IV)?
Isotropic diffusion, Structured/Unstructured Grids
   d  x  y   u  on structured, unstructured grids
0 . 4
0 . 3 5
6
6
7
7
8
8
9
9
0 . 3
0 . 2 5
0 . 2
0 . 1 5
0 . 1
0 . 0 5
a
b
a
b
a
b
a
b
0
N=16642
Structured
N=66049
Structured
N=13755
Unstruct.
N=54518
Unstruct.
Problems used: “a” means parameter c=10, “b” means c=1000
6
d  x  y   1 0  c x  y
1 0 x  0 5
7 d  x  y   
 c x  0 5
CASC
1 0 0 125  max  x  0 5  y  0 5   0 25
8 d x y  
otherwise
c

9 d x y  1 0 0 125 
c
 x  0 5 2   y  0 5 2  0 25
otherwis e
veh 76
How’s it perform (vol IVa)?
Isotropic diffusion, Structured/Unstructured Grids
   d  x  y   u  on structured, unstructured grids
0 .4 5
0 .4
0 .3 5
1
1
1
1
1
1
0 .3
0 .2 5
0 .2
0 .1 5
0 .1
0 .0 5
0
0
1
1
2
2
a
b
a
b
a
b
,
,
,
,
,
,
n
n
n
n
n
n
=
=
=
=
=
=
2
2
1
1
5
5
0
0
0
0
0
N=16642
Structured
N=66049 N=13755
Structured Unstruct.
N=54518
Unstruct.
Problem used: “a” means parameter c=10, “b” means c=1000
“Checkerboard” of coefficients 1.0 and c, squares sized 1/n:
j 1
i
i 1 j

1 0 n  x  n  n  y  n  i  j ev en
d x  y  
j 1
i
i 1 j
c
n  x  n  n  y  n  i  j odd

CASC
veh 77
How’s it perform (vol V)?
Laplacian operator, unstructured Grids
Convergence factor
0.3
0.2754
0.25
0.2198
0.2237
0.2
0.1888
0.1715
0.15
0.1002
0.1
0.05
0
0
50000 100000 150000 200000 250000 300000 350000
Gridpoints
CASC
veh 78
So, what could go wrong?
Strong F-F connections: weights are dependent on each other



For point the value ej is interpolated from k 1, k 2 ,
and is needed to make the interpolation weights for
approximating ei .
For point j the value ei is interpolated from k 1 , k 2 ,
and is needed to make the interpolation weights for
approximating ej .
It’s an implicit system!
i
k 1  Ci
k 1  Cj
k 3  Ci
j
i
k4  Cj
k 2  Ci
CASC
k 2  Cj
veh 79
Is there a fix?


A Gauss-Seidel like iterative approach to weight
definition is implemented. Usually two passes
suffice. But does it work?
Frequently, it does:
Convergence factors for
Laplacian, stretched quadrilaterals
theta Standard Iterative
 x  10  y
0.25
0.5
0.47
0.24
0.14
0.14
 x  100  y
0.25
0.5
0.83
0.53
0.82
0.23
CASC
veh 80
AMG for systems

How can we do AMG on systems?
 A 11 A 12   u   f 
A A  v  g
 21 22     

Naïve approach: “Block” approach (block Gauss-Seidel,
using scalar AMG to “solve” at each cycle)
u   A 11 
1
 f  A 12 v 
v   A 22  1  g  A 21 u 

Great Idea! Except that it doesn’t work! (relaxation
does not evenly smooth errors in both unknowns)
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AMG for systems: a solution



To solve the system problem, allow interaction
between the unknowns at all levels:
k
k
k

I

0 
k

1
A
A



u
11 12
k
I

Ak  

and
k 1 
k
k
k 
0
 Ik  1 

 A 21 A 22 
v
This is called the “unknown” approach.
Results: 2-D elasticity, uniform quadrilateral
mesh:
mesh spacing 0.125 0.0625 0.03135 0.015625
Convergence factor
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0.22
0.35
0.42
0.44
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So, what else can go wrong?
Ouch! Thin body elasticity!

Elasticity, 3-d, thin bodies!
1
1
 uyy  uz z  
 v x y  wx z   f 1
2
2
1
1
v yy 
 vxx  vz z  
 ux y  wy z   f 2
2
2
1
1
wz z 
 wx x  wy y  
 ux z  v yz   f 3
2
2
ux x 

Slide surfaces, Lagrange multipliers,
force balance constraints:
 S T   x 1   F1 

 x    F 
 U V  2  2

Wanted:
Good solution
method
for this
problem.
REWARD
S is “generally” positive definite, V
can be zero, U T  T
.
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veh 83
Needed: more robust methods for
characterizing smooth error

Consider quadrilateral finite elements on a
stretched 2D Cartesian grid (dx -> infinity):
A 
1 4 1
2 8 2
1 4 1

Direction of dependence is not apparent here
Iterative weight interpolation will sometimes
compensate for mis-identified dependence

Elasticity problems are still problematic

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Scalability is central for large-scale
parallel computing


A code is scalable if it can effectively use
additional computational resources to solve larger
problems
Many specific factors contribute to scalability:
— architecture of the parallel computer
— parallel implementation of the algorithms
— convergence rates of iterative linear solvers
Linear solver convergence can be discussed independent of parallel
computing, and is often overlooked as a key scalability issue.
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In Conclusion,
AMG Rules!




Interest in AMG methods is high, and probably
still rising, because of the increasing importance
of terra-scale simulations on unstructured grids.
AMG has been shown to be a robust, efficient
solver on a wide variety of problems of real-world
interest.
Much research is underway to find effective ways
of parallelizing AMG, which is essential to largescale computing.
This work was performed under the auspices of the U. S.
Department of Energy under contract number W-7405-Eng-48.
Release number UCRL MI 133749.
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