Parallel Black Box H-LU
Preconditioning
Ronald Kriemann
MPI MIS Leipzig
Workshop “Complex Systems”
METU
2009-05-14/15
H
Lib
pro
Overview
1 H-Matrices
2 Algebraic Clustering
3 Algebraic Admissibility
4 Nested Dissection
5 Numerical Experiments
Parallel Black Box H-LU Preconditioning
2/26
H-Matrices
Parallel Black Box H-LU Preconditioning
3/26
H-Matrices
Model Problem
Uniformly elliptic 2nd order PDE
div α(x) ∇u(x) = f (x),
x∈Ω
with Dirichlet/Neumann boundary conditions.
Galerkin discretisation
Aij = h∇ϕi , α∇ϕj iL2 (Ω)
Ax = b,
with basis functions
ϕi : Ω →
R,
i ∈ I = {1, . . . , N }
Goal: Solve the system fast and robust using LU factorisation
of A as preconditioner.
Problem: LU factors are usually prohibitively dense.
Solution: Compute approximate LU factorisation using
H-matrices with (almost) linear complexity.
Parallel Black Box H-LU Preconditioning
4/26
H-Matrices
What are H-Matrices?
R
A matrix M ∈ n×m of rank ≤ k can
be represented as
M = UV T ,
U∈
Rn×k , V ∈ Rm×k
VT
U
I R(k)-matrix format
Parallel Black Box H-LU Preconditioning
5/26
H-Matrices
What are H-Matrices?
R
A matrix M ∈ n×m of rank ≤ k can
be represented as
M = UV T ,
U∈
Rn×k , V ∈ Rm×k
VT
U
I R(k)-matrix format
For a block-wise low-rank matrix M ∈
• each block is R(k)-matrix
Rn×m
• for small blocks: fullmatrix format
I H-matrix format with hierarchically block
organisation.
Needed: reordering (clustering) of index sets to
allow low-rank representation.
Parallel Black Box H-LU Preconditioning
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H-Matrices
Clustering
Domain
Construct cluster tree using geometrical data:
Matrix
Construct block cluster tree
Parallel Black Box H-LU Preconditioning
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H-Matrices
Clustering
Domain
Construct cluster tree using geometrical data:
cluster
Matrix
Construct block cluster tree
Parallel Black Box H-LU Preconditioning
6/26
H-Matrices
Clustering
Domain
Construct cluster tree using geometrical data:
cluster
Matrix
Construct block cluster tree
Parallel Black Box H-LU Preconditioning
6/26
H-Matrices
Clustering
Domain
Construct cluster tree using geometrical data:
Matrix
Construct block cluster tree
Parallel Black Box H-LU Preconditioning
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H-Matrices
Clustering
Domain
Construct cluster tree using geometrical data:
Ωt
Ωs
Matrix
Construct block cluster tree with admissibility condition
min(diam(t), diam(s)) ≤ η dist(t, s),
Parallel Black Box H-LU Preconditioning
η>0
6/26
H-Matrices
Clustering
Domain
Construct cluster tree using geometrical data:
Ωt
Ωs
Matrix
Construct block cluster tree with admissibility condition
min(diam(t), diam(s)) ≤ η dist(t, s),
Parallel Black Box H-LU Preconditioning
η>0
6/26
H-Matrices
Clustering
Domain
Construct cluster tree using geometrical data:
Matrix
Construct block cluster tree with admissibility condition
min(diam(t), diam(s)) ≤ η dist(t, s),
Parallel Black Box H-LU Preconditioning
η>0
6/26
H-Matrices
Matrix Structure
• O (n) blocks
Parallel Black Box H-LU Preconditioning
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H-Matrices
Matrix Structure
• O (n) blocks
• Small red blocks:
full matrices
Parallel Black Box H-LU Preconditioning
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H-Matrices
Matrix Structure
• O (n) blocks
• Small red blocks:
full matrices
• All other blocks:
R(k)-matrices
Parallel Black Box H-LU Preconditioning
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H-Matrices
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10 10 10 16
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10 10
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Parallel Black Box H-LU Preconditioning
8
11
exponential
decay of singular
values
7
13
16
9
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6
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7
10 16 9
8
10 9
6
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10 9
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16 10 10 9
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16 8
10 8
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16 10 10 10 10 8
10 16 10 10 10 8
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10 10 16 10
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10 10 10 16
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16 13
10
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10 10 9
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16 7
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10 10
10 10
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10 9
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10 10 7
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12 12 10
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10 8
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12 12
10 9
12 12 7
13 12
9
10 9
10 11
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10 10 9
8
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10
10
8
12
16
11 10
9
11
10 11
12
13
11 11 10
8
10 10
12
10 10
16 10 10 10
11
• block-wise:
13 12
10 10
8
16 10
10 10
12 13
15
8
10 10
12 13
16
13
12
10 10
9
10 11
16
13 12
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12 12
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12 12
10 10
10 12 12
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10 9
12 13
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10 8
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11 10
12 7
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16 10
12 13 11
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10 10
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10 8
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10 10 8
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10 16 9
12
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10 10 16
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11 11 6
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16 10 8
10 11
Matrix Structure
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10 13
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10 10
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11 11
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10 8
16 10 10 9
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12 12
11 10
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10 10 9
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10 9
16 10 10 9
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9
11
10 16 9
13
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11 13 12
10 9
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10 9
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10 11 9
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10 10 9
16 8
10 8
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10 16 8
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7/26
H-Matrices
Arithmetic
Due to hierarchical block structure, standard recursive block
algorithms can be used, e.g. for multiplication:
B11 B12
A11 · B11 + A12 · B21 A11 · B12 + A12 · B22
A11 A12
·
=
B21 B22
A21 · B11 + A22 · B21 A21 · B12 + A22 · B22
A21 A22
Parallel Black Box H-LU Preconditioning
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H-Matrices
Arithmetic
Due to hierarchical block structure, standard recursive block
algorithms can be used, e.g. for multiplication:
B11 B12
A11 · B11 + A12 · B21 A11 · B12 + A12 · B22
A11 A12
·
=
B21 B22
A21 · B11 + A22 · B21 A21 · B12 + A22 · B22
A21 A22
But, addition of low-rank matrices increases the rank and finally
produces full-rank matrices.
To limit complexity, a truncated addition is performed using SVD:
A1 B1T + A2 B2T =: CDT
→
U SV T
→
C 0 D0T
with (predefined) rank(C 0 D0T ) < rank(CDT ) .
Parallel Black Box H-LU Preconditioning
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H-Matrices
Arithmetic
Due to hierarchical block structure, standard recursive block
algorithms can be used, e.g. for multiplication:
B11 B12
A11 · B11 + A12 · B21 A11 · B12 + A12 · B22
A11 A12
·
=
B21 B22
A21 · B11 + A22 · B21 A21 · B12 + A22 · B22
A21 A22
But, addition of low-rank matrices increases the rank and finally
produces full-rank matrices.
To limit complexity, a truncated addition is performed using SVD:
A1 B1T + A2 B2T =: CDT
→
U SV T
→
C 0 D0T
with (predefined) rank(C 0 D0T ) < rank(CDT ) .
Complexity
truncation
storage
matrix × vector
addition
O (n)
O (n log n)
O (n log n)
O (n log n)
Parallel Black Box H-LU Preconditioning
multiplication
inversion
triangular solve
LU decomposition
O
O
O
O
n log2 n
n log2 n
n log2 n
n log2 n
8/26
H-Matrices
Summary
To solve Ax = b using H-LU factorisation:
1
construct cluster tree using geometrical data,
2
construct block cluster tree using admissibility condition
(based on geometrical data),
3
build H-matrix representation of A,
4
perform H-LU factorisation (with approximation due to
truncated addition),
5
solve Ax = b preconditioned with H-LU approximated A−1 .
Parallel Black Box H-LU Preconditioning
9/26
H-Matrices
Summary
To solve Ax = b using H-LU factorisation:
1
construct cluster tree using geometrical data,
2
construct block cluster tree using admissibility condition
(based on geometrical data),
3
build H-matrix representation of A,
4
perform H-LU factorisation (with approximation due to
truncated addition),
5
solve Ax = b preconditioned with H-LU approximated A−1 .
But what to do if no geometry information is available?
Parallel Black Box H-LU Preconditioning
9/26
Algebraic Clustering
Parallel Black Box H-LU Preconditioning
10/26
Algebraic Clustering
Motivation
Consider
−∆u = 0
in Ω = [0, 1]2
Using a uniform grid with step width h and standard piecewise
linear finite elements with nodal points xi , i ∈ I, one obtains the
stiffness matrix A ∈ I×I as
R
Parallel Black Box H-LU Preconditioning
11/26
Algebraic Clustering
Motivation
Consider
−∆u = 0
in Ω = [0, 1]2
Using a uniform grid with step width h and standard piecewise
linear finite elements with nodal points xi , i ∈ I, one obtains the
stiffness matrix A ∈ I×I as
R
Define the matrix graph G(A) = (VA , EA ) of A as
VA := I,
EA := {(i, j) : i 6= j ∧ aij 6= 0},
i.e. graph corresponds to sparsity pattern of stiffness matrix.
Parallel Black Box H-LU Preconditioning
11/26
Algebraic Clustering
Motivation
Consider
−∆u = 0
in Ω = [0, 1]2
Using a uniform grid with step width h and standard piecewise
linear finite elements with nodal points xi , i ∈ I, one obtains the
stiffness matrix A ∈ I×I as
R
Define the matrix graph G(A) = (VA , EA ) of A as
VA := I,
EA := {(i, j) : i 6= j ∧ aij 6= 0},
i.e. graph corresponds to sparsity pattern of stiffness matrix.
Parallel Black Box H-LU Preconditioning
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Algebraic Clustering
Motivation
Define distance distG (i, j) between nodes i, j ∈ I as length of
shortest path in G(A). Then, for i, j ∈ I we have:
kxi − xj k2 ≤ distG (i, j)h,
i.e. distance in
k
R2 is mapped to distance in G(A):
j
kxi − xj k2 =
kxi − xk k2 =
√
√
13h, distG (i, j) = 5
5h,
distG (i, k) = 3
i
Parallel Black Box H-LU Preconditioning
12/26
Algebraic Clustering
Motivation
Define distance distG (i, j) between nodes i, j ∈ I as length of
shortest path in G(A). Then, for i, j ∈ I we have:
kxi − xj k2 ≤ distG (i, j)h,
i.e. distance in
k
R2 is mapped to distance in G(A):
j
kxi − xj k2 =
kxi − xk k2 =
√
√
13h, distG (i, j) = 5
5h,
distG (i, k) = 3
i
In model problem: since nodes in G(A) with small distance are also
geometrically neighboured, one can use graph distance to cluster
indices.
Parallel Black Box H-LU Preconditioning
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Algebraic Clustering
Clustering via Breadth First Search
Algorithm:
1 determine two nodes i, j ∈ VA with (almost) maximal
distance,
Parallel Black Box H-LU Preconditioning
13/26
Algebraic Clustering
Clustering via Breadth First Search
Algorithm:
1 determine two nodes i, j ∈ VA with (almost) maximal
distance,
2 perform simultaneous BFS from i and j to construct sub
clusters:
• per step, add unvisited neighbours of nodes in sub clusters
Parallel Black Box H-LU Preconditioning
13/26
Algebraic Clustering
Clustering via Breadth First Search
Algorithm:
1 determine two nodes i, j ∈ VA with (almost) maximal
distance,
2 perform simultaneous BFS from i and j to construct sub
clusters:
• per step, add unvisited neighbours of nodes in sub clusters
Parallel Black Box H-LU Preconditioning
13/26
Algebraic Clustering
Clustering via Breadth First Search
Algorithm:
1 determine two nodes i, j ∈ VA with (almost) maximal
distance,
2 perform simultaneous BFS from i and j to construct sub
clusters:
• per step, add unvisited neighbours of nodes in sub clusters
Parallel Black Box H-LU Preconditioning
13/26
Algebraic Clustering
Clustering via Breadth First Search
Algorithm:
1 determine two nodes i, j ∈ VA with (almost) maximal
distance,
2 perform simultaneous BFS from i and j to construct sub
clusters:
• per step, add unvisited neighbours of nodes in sub clusters
Parallel Black Box H-LU Preconditioning
13/26
Algebraic Clustering
Clustering via Breadth First Search
Algorithm:
1 determine two nodes i, j ∈ VA with (almost) maximal
distance,
2 perform simultaneous BFS from i and j to construct sub
clusters:
• per step, add unvisited neighbours of nodes in sub clusters
Parallel Black Box H-LU Preconditioning
13/26
Algebraic Clustering
Clustering via Breadth First Search
Algorithm:
1 determine two nodes i, j ∈ VA with (almost) maximal
distance,
2 perform simultaneous BFS from i and j to construct sub
clusters:
• per step, add unvisited neighbours of nodes in sub clusters
3
recurse in sub graphs
Parallel Black Box H-LU Preconditioning
13/26
Algebraic Clustering
Clustering via General Graph Partitioning
In graph theory, the graph partitioning problem is defined as:
Given a graph G = (V, E) a partitioning P = {V1 , V2 },
with V1 ∩ V2 = ∅ and V = V1 ∪ V2 , of V is sought, such
that
#V1 ∼ #V2
and
#{(i, j) ∈ E : i ∈ V1 ∧ j ∈ V2 } = min .
(edge-cut)
A small edge-cut corresponds to a low-rank coupling of matrix
blocks.
Parallel Black Box H-LU Preconditioning
14/26
Algebraic Clustering
Clustering via General Graph Partitioning
In graph theory, the graph partitioning problem is defined as:
Given a graph G = (V, E) a partitioning P = {V1 , V2 },
with V1 ∩ V2 = ∅ and V = V1 ∪ V2 , of V is sought, such
that
#V1 ∼ #V2
and
#{(i, j) ∈ E : i ∈ V1 ∧ j ∈ V2 } = min .
(edge-cut)
A small edge-cut corresponds to a low-rank coupling of matrix
blocks.
Although the graph partitioning problem is NP-hard good
approximation algorithms exist, e.g. multilevel or spectral methods.
Furthermore, they are available in open source packages, e.g.
METIS, Chaco or Scotch.
Parallel Black Box H-LU Preconditioning
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Algebraic Admissibility
Parallel Black Box H-LU Preconditioning
15/26
Algebraic Admissibility
Definition
To apply the standard admissibility condition
min(diam(t), diam(s)) ≤ η dist(t, s)
for a block cluster (t, s) ∈ V × V , one needs to define distance and
diameter of clusters in a graph.
Parallel Black Box H-LU Preconditioning
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Algebraic Admissibility
Definition
To apply the standard admissibility condition
min(diam(t), diam(s)) ≤ η dist(t, s)
for a block cluster (t, s) ∈ V × V , one needs to define distance and
diameter of clusters in a graph.
• For V1 , V2 ⊂ V , the distance between V1 and V2 is defined as
distG (V1 , V2 ) :=
min
i∈V1 ,j∈V2
distG (i, j).
• The diameter of a sub graph induced by V 0 ⊆ V is defined as
diamG (V 0 ) := max0 distG (i, j).
i,j∈V
Parallel Black Box H-LU Preconditioning
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Algebraic Admissibility
Definition
To apply the standard admissibility condition
min(diam(t), diam(s)) ≤ η dist(t, s)
for a block cluster (t, s) ∈ V × V , one needs to define distance and
diameter of clusters in a graph.
• For V1 , V2 ⊂ V , the distance between V1 and V2 is defined as
distG (V1 , V2 ) :=
min
i∈V1 ,j∈V2
distG (i, j).
• The diameter of a sub graph induced by V 0 ⊆ V is defined as
diamG (V 0 ) := max0 distG (i, j).
i,j∈V
Problem: diameter and distance in G costs O n2 .
Parallel Black Box H-LU Preconditioning
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Algebraic Admissibility
Testing
Solution: approximate cluster diameter and construct cluster
surrounding ensuring admissibility.
For testing admissibility of block cluster (t, s) ∈ V × V
• choose i ∈ t and compute j ∈ t with
distG (i, j) = max,
s0
s
]
• diamG (t) ≤ 2 distG (i, j) =: diam,
• build surrounding t̃ around t with
layers,
1]
η diam
t
• if t̃ ∩ s = ∅ then (t, s) is admissible.
Parallel Black Box H-LU Preconditioning
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Algebraic Admissibility
Testing
Solution: approximate cluster diameter and construct cluster
surrounding ensuring admissibility.
For testing admissibility of block cluster (t, s) ∈ V × V
• choose i ∈ t and compute j ∈ t with
distG (i, j) = max,
s0
s
]
• diamG (t) ≤ 2 distG (i, j) =: diam,
• build surrounding t̃ around t with
layers,
1]
η diam
t
• if t̃ ∩ s = ∅ then (t, s) is admissible.
Parallel Black Box H-LU Preconditioning
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Algebraic Admissibility
Testing
Solution: approximate cluster diameter and construct cluster
surrounding ensuring admissibility.
For testing admissibility of block cluster (t, s) ∈ V × V
• choose i ∈ t and compute j ∈ t with
distG (i, j) = max,
s0
s
]
• diamG (t) ≤ 2 distG (i, j) =: diam,
• build surrounding t̃ around t with
layers,
1]
η diam
t
• if t̃ ∩ s = ∅ then (t, s) is admissible.
Parallel Black Box H-LU Preconditioning
17/26
Algebraic Admissibility
Testing
Solution: approximate cluster diameter and construct cluster
surrounding ensuring admissibility.
For testing admissibility of block cluster (t, s) ∈ V × V
• choose i ∈ t and compute j ∈ t with
distG (i, j) = max,
s0
s
]
• diamG (t) ≤ 2 distG (i, j) =: diam,
• build surrounding t̃ around t with
layers,
1]
η diam
t
• if t̃ ∩ s = ∅ then (t, s) is admissible.
Parallel Black Box H-LU Preconditioning
17/26
Algebraic Admissibility
Testing
Solution: approximate cluster diameter and construct cluster
surrounding ensuring admissibility.
For testing admissibility of block cluster (t, s) ∈ V × V
• choose i ∈ t and compute j ∈ t with
distG (i, j) = max,
s0
s
]
• diamG (t) ≤ 2 distG (i, j) =: diam,
• build surrounding t̃ around t with
layers,
1]
η diam
t
• if t̃ ∩ s = ∅ then (t, s) is admissible.
With usual FEM sparsity patterns, this procedure has complexity
O (#t) .
Parallel Black Box H-LU Preconditioning
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Nested Dissection
Parallel Black Box H-LU Preconditioning
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Nested Dissection
Vertex Separator
In nested dissection the two constructed sub graphs of a partition
have to be separated by a (minimal) vertex separator.
Matrix graph:
Matrix:
Parallel Black Box H-LU Preconditioning
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Nested Dissection
Vertex Separator
In nested dissection the two constructed sub graphs of a partition
have to be separated by a (minimal) vertex separator.
Matrix graph:
Matrix:
Parallel Black Box H-LU Preconditioning
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Nested Dissection
Vertex Separator
In nested dissection the two constructed sub graphs of a partition
have to be separated by a (minimal) vertex separator.
Matrix graph:
Matrix:
Advantages of Nested Dissection
• zero blocks do not fill up during H-LU factorisation,
• blocks can be computed in parallel.
Parallel Black Box H-LU Preconditioning
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Nested Dissection
Cluster Tree for the Vertex Separator
A vertex separator can be obtained by computing a vertex cover of
the edge-cut between both node sets in a partition.
But for H-matrices the vertex separator has to be further
partitioned to form a cluster tree.
Parallel Black Box H-LU Preconditioning
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Nested Dissection
Cluster Tree for the Vertex Separator
A vertex separator can be obtained by computing a vertex cover of
the edge-cut between both node sets in a partition.
But for H-matrices the vertex separator has to be further
partitioned to form a cluster tree.
Problem: restricting G to nodes in vertex separator V might
remove important edges, e.g.
G|V
Parallel Black Box H-LU Preconditioning
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Nested Dissection
Cluster Tree for the Vertex Separator
A vertex separator can be obtained by computing a vertex cover of
the edge-cut between both node sets in a partition.
But for H-matrices the vertex separator has to be further
partitioned to form a cluster tree.
Problem: restricting G to nodes in vertex separator V might
remove important edges, e.g.
Solution: modify previous BFS based algorithm to perform
partitioning in a surrounding of the vertex separator.
Parallel Black Box H-LU Preconditioning
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Numerical Experiments
Parallel Black Box H-LU Preconditioning
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Numerical Experiments
Comparison with Geom. Clustering
Solving model problem:
N
2532
3582
5112
7292
10232
403
513
643
813
1023
Geometric
Time (s) Mem (MB)
0.9
51
1.9
86
4.5
212
9.6
371
20.2
878
12.6
99
46.9
300
117.4
592
269.8
1410
752.3
3020
Algebraic
Time (s) Mem (MB)
1.3
47
2.9
94
6.5
198
15.0
402
31.6
819
32.7
135
97.6
323
289.1
719
804.3
1570
1907.3
3370
Accuracy of H-arithmetic chosen such that
kI − (LH UH )−1 Ak2 ≤ 10−4
Parallel Black Box H-LU Preconditioning
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Numerical Experiments
Comparison with Direct Solvers
Solving
in Ω = [0, 1]2
−∆u + λu = f
700
Time for setup in sec.
600
500
H-Matrix
Pardiso
MUMPS
UMFPACK
SuperLU
Spooles
400
300
200
100
0
5.0 . 105
1.0 . 106
1.5 . 106
2.0 . 106
No. of Unknowns
Parallel Black Box H-LU Preconditioning
23/26
Numerical Experiments
Comparison with Direct Solvers
Solving
in Ω = [0, 1]3
−∆u + λu = f
H-Matrix
Pardiso
MUMPS
UMFPACK
SuperLU
Spooles
18000
Time for setup in sec.
16000
14000
12000
10000
8000
6000
4000
2000
0
5.0 . 105
1.0 . 106
1.5 . 106
2.0 . 106
No. of Unknowns
Parallel Black Box H-LU Preconditioning
24/26
Numerical Experiments
Parallel Performance
Parallel speedup for algebraic H-LU factorisation in
30
R2 and R3.
2
n = 2895
n = 1283
Speedup
25
20
15
10
5
5
10
15
20
25
30
No. of Processors
Parallel Black Box H-LU Preconditioning
25/26
Literature
L. Grasedyck, R. Kriemann and S. Le Borne,
Domain Decomposition Based H-LU Preconditioning,
to appear in “Numerische Mathematik”.
L. Grasedyck, R. Kriemann and S. Le Borne,
Parallel Black Box H-LU Preconditioning for Elliptic Boundary Value
Problems,
“Computing and Visualization in Science”, 11(4-6), pp. 273–291, 2008.
L. Grasedyck, W. Hackbusch and R. Kriemann,
Performance of H-LU Preconditioning for Sparse Matrices,
to appear in “Computational Methods in Applied Mathematics”.
pro
H-Lib
http://www.hlibpro.org
H
Lib
pro
Parallel Black Box H-LU Preconditioning
26/26