Element-Free AMGe: General
algorithms for computing interpolation
weights in AMG
Van Emden Henson
Panayot Vassilevski
Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
International Workshop on Algebraic Multigrid
Sankt Wolfgang, Austria
June 27, 2000
Overview




This work was performed under the auspices of the U. S. Department of
Energy by the University of California Lawrence Livermore National
Laboratory under contract number: W-7405-Eng-48. UCRL-VG-138290
AMG and AMGe
Element-Free AMGe: an interpolation rule based
on neighborhood extensions
Examples of extensions:
— A-extension,
— L 2-extension
— extensions from minimizing quadratic
functionals
Numerical experiments
CASC
VEH 2
AMG and AMGe



Assume a given sparse matrix
A
AMG, or Algebraic Multigrid is MG based only on
the matrix entries.
Essential components of AMG:
— a set, D, of fine-grid degrees of freedom
(dofs)
— a coarse grid, Dc ; typically a subset of D
— a prolongation operator P  Dc  D
— smoothing iterations; typically Gauß-Seidel or
Jacobi
— a coarse matrix given by Ac  P TA P
CASC
VEH 3
Coarse-grid selection

There are several ways to select the coarse grid

Dc is typically a maximal independent set

each fine-grid dof is typically interpolated from a
subset of its coarse nearest neighbors
CASC
VEH 4
Building the prolongation, P




Let i  D be a fine-grid dof
Let   i   D be a neighborhood of i
Let c i  be the coarse-grid dofs used to
interpolate a value at i
Examine rows of A corresponding to   i  \  c i  .
A 
CASC
Af f Af c 

 

 
}
}
}
  i  \  c i 
 c i 
D \   i
VEH 5
Prolongation in classical AMG


 i
is the minimal neighborhood of i
Replace A f f with modified version, A f f
— by adding to a i i all off-diagonal entries in i th
row that are weakly connected to i
— second, in all rows j for dofs strongly
connected to i :
– set
a 
a
jj

j jc
j c  c  i
– set off-diagonals to zero

ith row of P is ith row of
 1

  A f f Af c 


CASC
VEH 6
AMGe differs from standard AMG by
using finite element information



Traditional AMG uses the following heuristic (based
on M-matrices): smooth error varies slowest in the
direction of “large” coefficients
New heuristic based on multigrid theory:
interpolation must be able to reproduce a mode with
error proportional to the size of the associated
eigenvalue
Key idea of AMGe: We can use information carried
in the element stiffness matrices to determine
— the nature of the smooth error components
— accurate interpolation operators
— the selection of the coarse grids
CASC
VEH 7
AMGe uses finite element stiffness
matrices to localize the new heuristic

Local measure:
M i  max
e0
i Ti  I  Q e i Ti  I  Q e
A i e e
Ai 
where i are canonical basis vectors, A
and
i
are sums of local stiffness matrices


k i
Bk
If all local measures are small, the global measure
is bounded and small
good convergence!

Then solving a small local problem yields a row of
the optimal interpolation (for the given set of Cpoints).
CASC

VEH 8
AMGe uses a small local problem to
define prolongation

We can show that (for a given set of interpolation
points), the “optimal” prolongation is the set of
weights Q that satisfy the min/max problem:
min ma x
Q e0

i Ti  I  Q e i Ti  I  Q e
A i e e
Furthermore, solving the min/max problem is
exactly equivalent to the following small matrix
formulation
CASC
VEH 9
Prolongation in AMGe



The neighborhood   i  is the union of the
elements having i as vertex
We use the assembled neighborhood matrix A   i 
Partition the neighborhood matrix as before
Af f Af c
A  i 
 

}   i \  c  i
c
}
ith row of P is ith row of
1
  Af f Af c 
CASC
VEH 10
Prolongation in AMGe, cont.



Note that there is no need to modify
Af f
Knowledge of the element matrices (used to create
the assembled neighborhood matrix) carries with it
implicitly the correct assignment and treatment of
“weak” and “strong” connection. This is the main
contribution of AMGe methods
AMGe produces superior prolongation. The goal of
this work is to accomplish the superior prolongation
without the knowledge of the element matrices
CASC
VEH 11
AMGe - Richardson vs Gauß-Seidel
Two-level
Multilevel
height
amg
amge amg
amge
1
0.97
0.49
0.98
0.65
1/4
0.97
0.48
0.98
0.68
1/8
0.98
0.47
0.99
0.64
1/16
0.97
0.49
0.99
0.58
1/32
0.97
0.45
0.98
0.51
1/64
0.98
0.39
0.98
0.39
d
h
1
1/4
1/8
1/16
1/32
1/8
1/16
1/64
CASC
AMG
interpolation
0.60
0.95
0.90
0.92
AMGe
interpolation
0.20
0.25
0.26
0.26
Richardson (1,0) cycle
AMGe CG choice
and interpolation
0.10
0.09
0.08
0.08
Gauß-Seidel (1,0) cycle
VEH 12
Prolongation in Element-free AMGe:
based on extensions

Let i be the f-point
to which we wish to
interpolate
CASC
VEH 13
Prolongation in Element-free AMGe:
based on extensions


Let i be the f-point
to which we wish to
interpolate
  i  is the set of
points in the
neighborhood of
CASC
i
VEH 14
Prolongation in Element-free AMGe:
based on extensions



Let i be the f-point
to which we wish to
interpolate
  i  is the set of
points in the
neighborhood of
i
c  i is the set of
coarse nearest
neighbors of i
CASC
VEH 15
Prolongation in Element-free AMGe:
based on extensions

Define  X  i  , the
set of “exterior”
points for the
neighborhood of i :
the set of points j
such that j is
connected to a fine
point in the
neighborhood of i
 X  i   j    i  a j k  0  j   i \  C  i 
CASC
VEH 16
Prolongation in Element-free AMGe:
based on extensions

Define  X  i  , the
set of “exterior”
points for the
neighborhood of i :
the set of points j
such that j is
connected to a fine
point in the
neighborhood of i
 X  i   j    i  a j k  0  j   i \  C  i 
CASC
VEH 17
Prolongation in Element-free AMGe:
based on extensions

We use the following window of the matrix A
Af f
Af c Af X 0




AX f AX c AX X 




 i \ c  i
c  i
X  i
everything else on grid
where we will only be interested in the blocks
shown.
CASC
VEH 18
Prolongation in Element-free AMGe:
based on extensions

Assume that an extension mapping is available:
I
0
I
E 0
EX f EX c
i.e., we interpolate the exterior dofs (“X”) from
the interior dofs f and c, by the rule
v X  EX f v f  EX c v c
CASC
VEH 19
Prolongation in Element-free AMGe:
based on extensions

We construct the prolongation operator on the
basis of the modified matrix
Af f  Af c

Af f  Af c  Af X
I
0
0
I
EX f EX c
that is,
A f f  A f f  A f X EX f
and
A f c  A f c  A f X EX c
CASC
VEH 20
Prolongation in Element-free AMGe:
based on extensions

Then the ith row of the prolongation matrix P is
taken as the ith row of the matrix
 1

  Aff Af c 



To make use of this method we must determine
useful ways in which to build the extension
operator
I
0
I
E 0
EX f EX c
CASC
VEH 21
Examples of extension: A-extension



Given v defined on   i , we wish to extend it
to v X defined on  X  i  .
Let i X be an exterior dof and
define S   j  a i  j  0  to be
X
entries of A to which i X is
connected.
 iX
.The A-extension is:
vi
CASC
X

S
1
S  a i X  j
 ai  j  v j

S
X

VEH 22
Examples of extension: L 2-extension



Given v defined on   i , we wish to extend it
to v X defined on  X  i  .
Let i X be an exterior dof and
define S   j  a i  j  0  to be
X
entries of A to which i X is
connected.
The L 2-extension is (a simple average):
vi
CASC
X

1
S
S
 iX
v
j

S
1
VEH 23
Examples of extensions: based on
minimizing a quadratic functional


This is the method Achi Brandt proposed in 1999.
Given v defined on
finding
  i ,
find the extension v X by
mi n Q vX 
where

Q v X 


 ai  j   vi  v j  2
X
X
ix  X  i
j    i
This is a “simultaneous” extension, and is more
expensive than A-extension or L 2-extension
CASC
VEH 24
Examples of extensions: minimizing a
“cut-off” quadratic functional

Given v defined on
that satisfies
  i ,
mi n B v
find the extension v X
T
vX
v
 f 
v   vc 
 vX 
A
  i
Bv
I
B 
I
X
where
and
is
a diagonal matrix. A good choice is the vector
1
X   A X X A X f  A X c
1f
1c
note that this is also a “simultaneous” extension
(but less expensive than the “full” quadratic)
CASC
VEH 25
Classical AMG viewed as extension

The classical (Ruge-Stüben) AMG corresponds to
selecting
  i    i    c i 
and defining an A-extension by setting vi  vi
X
if i X is weakly connected to i , and setting
vi 
X
1
 aiX  j
 aiX  j v j
if i X is strongly connected to
CASC
i
VEH 26
Properties of the extensions


All the extensions described ensure that if v is
constant in   i  then the extension v X is the
same constant in  X  i , an important property
for second-order elliptic PDEs.
For systems of PDEs we use the same extension
mappings, based on the blocks of A associated
with a given physical variable. That is, the
extension to a dof i X corresponding to the
physical variable k is based on dofs from   i 
that describe the same physical variable k.
CASC
VEH 27
A simple example: the stretched
quadrilateral


Consider the 2-D Laplacian operator, finiteelement formulation on regular quadrialteral
elements that are greatly stretched: hx » hy
hx
  , the operator stencil tends to:
As
hy
1 4 1
2
8
2
1 4 1
CASC
VEH 28
A simple example: the stretched
quadrilateral: geometric approach


For this problem the standard geometric multigrid
approach is to semicoarsen:
And the interpolation stencil is:
0 0 5 0
PA 

0 0 5 0
CASC
VEH 29
A simple example: the stretched
quadrilateral: the AMG stencil

A simple calculation shows that classical AMG
yields the interpolation stencil:
PA M G 
CASC
1 1 1
12 3 12

1 1 1
12 3 12

 083  333  083

 083  333  083
VEH 30
A simple example: the stretched
quadrilateral: the neighborhood

Let  i  i  c i  i  N S SW  NW  SE  NE
NW
N
E
W
SW

NE
S
SE
Define  i    W  E 
then
Af f   8 
Af c    4  4  1  1  1  1 
Af X   2 2 
CASC
VEH 31
A simple example: the stretched
quadrilateral: the A-extension

For the A-extension the extension operators are:
1 2
1 1 1 4 4
EX f 
EX c 
12 2
12 1 1
4 4
from which
104
Aff 
12
, and
1
Afc 
3
 11
 11
1
1
1
1
yielding the interpolation stencil:
PA 
CASC
1 11 1
26 26 26

1 11 1
26 26 26

 038  423  038

 038  423  038
VEH 32
A simple example: the stretched
quadrilateral: the L 2-extension

A similar calculation for the weights using the
extension yields the interpolation stencil:
PL 
2
CASC
3 16 3
44 44 44

3 16 3
44 44 44

L2
 068  364  068

 068  364  068
VEH 33
A simple example: the stretched
quadrilateral: the cut-off quadratic min

For the cut-off extension the extension operators
are:
1 1 1 4 4
1 1
EX c 
EX f  
8 1 1
4 4
4 1
from which
Aff 
7
, and
1
Afc 
7 7 0 0 0 0
2
yielding the interpolation stencil:
PA 
CASC
0 5 0

0 5 0
VEH 34
The stencil produced by various
extension methods:

Classical AMG
PA 


 038  423  038

 038  423  038
L2-extension

CASC
PA M G 
PA 
0 5 0

0 5 0
 083  333  083

 083  333  083
A-extension
PL 
2
 068  364  068

 068  364  068
cut-off quadratic
VEH 35
Numerical experiments

Second order elliptic operator
— Unstructured triangular mesh (400, 1600,
6400, and 25600 fine-grid elements)
— coarsened using agglomeration method of Jones
& Vassilevski
CASC
VEH 36
AMGe requires coarse-grid elements &
stiffness matrices.

Element Agglomeration: Use graph theory to create coarse
elements first, then select coarse-grid by abstracting geometric
concepts of face, edge, vertex
CASC
VEH 37
Agglomeration coarsening

CASC
Finest grid
— 1600 elements
— 861 dofs
— 5781 nonzero
entries
VEH 38
Agglomeration coarsening

CASC
1st coarse grid
— 382 elements
— 330 dofs
— 2602 nonzero
entries
VEH 39
Agglomeration coarsening

CASC
2nd coarse grid
— 93 elements
— 143 dofs
— 1397 nonzero
entries
VEH 40
Agglomeration coarsening

CASC
3rd coarse grid
— 33 elements
— 64 dofs
— 634 nonzero
entries
VEH 41
Agglomeration coarsening

CASC
4th coarse grid
— 15 elements
— 32 dofs
— 304 nonzero
entries
VEH 42
Agglomeration coarsening

CASC
5th coarse grid
— 7 elements
— 16 dofs
— 126 nonzero
entries
VEH 43
Agglomeration coarsening

CASC
6th coarse grid
— 3 elements
— 8 dofs
— 46 nonzero
entries
VEH 44
Agglomeration coarsening

CASC
coarsest grid
— 1 elements
— 4 dofs
— 16 nonzero
entries
VEH 45
Coarsening history, unstructured 2nd
order PDE


unstructured
triangular grid
2nd order PDE
CASC
VEH 46
Convergence results, unstructured 2nd
order PDE

2nd order PDE problem
— unstructured triangular grid
— V(1,1) cycles
— Gauß-Seidel smoothing
CASC
VEH 47
Numerical experiments

2d elasticity (including thin-body)
— structured quadrilateral grid, h x  h y
— d   0 1  is the beam thickness
— coarsened using agglomeration method
— the same coarse-grid is used for AMGe
(vertices of agglomerated elements)
CASC
VEH 48
Coarsening history, elasticity problem

Structured
rectangular
grid

2-d elasticity

d = 1
CASC
VEH 49
Coarsening history, elasticity problem



Structured
rectangular
grid
2-d elasticity
thin body
— d = 0.05
CASC
VEH 50
Convergence results, elasticity

2d elasticity problem
—d = 1
— V(1,1) cycles
— Gauß-Seidel smoothing
CASC
VEH 51
Convergence results, elasticity

2d thin-body elasticity problem
— d = 0.05
— V(1,1) cycles
— Gauß-Seidel smoothing
CASC
VEH 52
Conclusions




AMGe produces superior prolongation, but requires
extra information, i.e., the element matrices
For purely algebraic problems, element-free AMGe
provides a technique for building pseudo-element
matrices based on several extension schemes
Preliminary experimental results suggest that
element-free AMGe also produces superior
prolongation, competitive to AMGe results
Systems of PDEs can be handled in a very natural
way
CASC
VEH 53
Some papers of interest

Pre/Re-prints Available at
http://www.llnl.gov/CASC/linear_solvers

V. Henson and P. Vassilevski, “Element-free AMGe: General Algorithms for computing
Interpolation Weights in AMG,” submitted to SIAM Journal on Scientific Computing.

M. Brezina, A. Cleary, R. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick, and J.
Ruge, “Algebraic Multigrid Based on Element Interpolation (AMGe),” to appear in SIAM
Journal on Scientific Computing.

J. Jones and P. Vassilevski, “AMGe Based on Element Agglomeration,” submitted to
SIAM Journal on Scientific Computing.

A. Cleary, R. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick, G. Miranda, and J.
Ruge, “Robustness and Scalability of Algebraic Multigrid,” SIAM Journal on Scientific
Computing, vol. 21 no. 5, pp. 1886-1908, 2000

A. Cleary, R. Falgout, V. Henson, J. Jones, “Coarse-Grid Selection for Parallel Algebraic
Multigrid,” in Proc. of the Fifth International Symposium on: Solving Irregularly Structured
Problems in Parallel, Lecture Notes in Computer Science, Springer-Verlag, New York, 1998.
CASC
VEH 54