inverse–based multilevel ilu preconditioners theory, applications...
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Inverse–Based Multilevel ILU PreconditionersTheory, Applications and Software
Matthias Bollh ofer Yousef SaadTU Berlin / TU Braunschweig Univ. of Minnesota
8-TH COPPERMOUNTAIN CONFERENCE ONITERATIVE METHODS
MARCH 28 – APRIL 02, 2004
April 02, 2004
OUTLINE
1. inverse–based ILUs, theoretical background
2. Multilevel algorithms
3. ILUPACK
4. Numerical experiments
5. Conclusions
supported by the DFG research center “Mathematics for key technologies” in Berlin 0
0
Inverse–Based ILUs
Ax = b
INCOMPLETE LU DECOMPOSITION
A =
(B F
E C
)= LDU + E ≈
(@
@@
@@
)(@
@@
@@
)(@
@@
@@
), where L, U are unit diagonal.
INVERSE–BASED DROPPING
At elimination step k, drop lik, ukj if
|lik| ∗ ‖e>k L−1‖ 6 τ |dkk|, |ukj| ∗ ‖U−1ek‖ 6 τ |dkk|,
where τ is a drop tolerance [B. ’01, ’03], [Li, Saad, Chow ’03].
=⇒ L−1 and U−1 serve as approximate inverses
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 1
1
Inverse–Based ILU — Approximate Inverse Relation
FACTORED APPROXIMATE INVERSE (AINV) , [Benzi, Tuma ’98]
W>AZ =
(@
@@
@@
)( )(@
@@
@@
)= D + E
W ←→ L−>, Z ←→ U−1
THEOREM [B., Saad ’01]. Suppose that the AINV process and the inverse–based ILU are computedusing the same approximate Schur complement.Let τ be the drop tolerance. Then
|(I −WL>)ij| 6 2(j− i)τ, |(I − ZU)ij| 6 2(j− i)τ ∀i < j.
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 2
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Impact on the Inverse Error
GROWTH OF THE INVERSE TRIANGULAR FACTORS
Problem. ‖L−1‖ and ‖U−1‖ may become very large.
⇒ pivoting recommended.
BUT WHAT KIND OF PIVOTING IS APPROPRIATE?
Consider the approximate factorization
A =
(B F
E C
)=
(LB 0
LE I
)︸ ︷︷ ︸
Lk
(DB 0
0 SC
)︸ ︷︷ ︸
Dk
(UB UF
0 I
)︸ ︷︷ ︸
Uk
+Ek,
where Lk and U>k are unit lower triangular matrices and the leading k × k part of Dk is diagonal.
INVERSE ERROR
Fk := L−1k EkU
−1k
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 3
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Inverse Error Bound
LEMMA . Denote by vk and wk the entries in column k of LkDk (resp. row k of DkUk) that are droppedat step k of the incomplete LU decomposition.
1. Suppose that the approximate Schur complement SC is defined via
SC = C − LEDBUF (S–version)
⇒ Fk =∑l6k
(L−1
k vle>l U−1
k + L−1k elw
>l U−1
k
)2. Suppose that the approximate Schur complement SC is defined via
SC =[−LEL−1
B I]A
[−U−1
B UF
I
](T–version)
⇒ Fk =∑l6k
(L−1
k vle>l + elw
>l U−1
k
)
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 4
4
Application: Inverse-Based Pivoting
CONSEQUENCES
• ‖L−1k ‖ and ‖U−1
k ‖ contribute to the inverse error
• For the S–version even their PRODUCT ‖L−1k ‖ · ‖U
−1k ‖!
INVERSE–BASED PIVOTING
Control ‖L−1‖, ‖U−1‖ 6 κ by a factor or skip strategy
At step k :
@@
@@
@@
@@
∗
↗
↘
@@
@@
@@
@@
∗
@@
@@
@@
@@
∗
factor
skip
Rows/columns that exceed the prescribed bound are pushed to the end
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 5
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Inverse–Based Multilevel ILUs
SKETCH OF THE TEMPLATES
1. Compute a static partial reordering
A→ P>AQ =
(B F
E C
)
2. Factor P>AQ and control ‖L−1k ‖, ‖U−1‖ 6 κ.
P>AQ −→ P>AQ =
(B F
E C
)=
(LB 0
LE I
)(DB 0
0 SC
)(UB UF
0 I
)
3. Apply strategy recursively to the approximate Schur complement SC (multilevel scheme)
A1.→ P>AQ
2.→ P>AQ3.→
(L ·D · U F
E S
)
S1.→ P>S SQS
2.→ P>S SQS3.→(
LS ·DS · US FS
ES T
)· · ·
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 6
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Inverse-Based Multilevel ILUs — Templates
TEMPLATE 1. STATIC REORDERINGS
Examples.
• RCM (Reverse Cuthill–McKee), MMD (Multiple Minimum Degree), AMF (Approximate MinimumFill), INDSET (independent set, ARMS), MC64 (from HSL).
• ddPQ [Saad ’03]. Permutations P, Q as a compromise between diagonal dominance and fill
1. Define numerical weights
ρi =max
j|aij|∑
j |aij|, γj =
maxi|aij|∑
i |aij|2. define fill weights, e.g.
fi = |{j : aij 6= 0}, gj = |{i : aij 6= 0}, hij = (fi − 1)(gj − 1) + 1, . . .
3. order ratios in decreasing order, e.g. order
{ ρi
gki
∀i}, {γj
flj
∀j}, {ρiγki
hi,ki
∀i,ρljγj
hlj ,j∀j}, . . .
4. use associated indices (i, j) and skip duplicate entries
Further selection criteria, e.g. enforce submatrix to be close to lower triangular
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 7
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Inverse–Based Multilevel ILUs — Templates
TEMPLATE 2. PILUC — INVERSE-BASED ILU THAT CONTROLS‖L−1k ‖,‖U
−1k ‖
• ILU (Crout–version)??
−→
-- ↓
• ‖L−1k ‖, ‖U
−1k ‖ 6 κ controlled by diagonal pivoting
@@
@@
@@
@@
∗
↗
↘
@@
@@
@@
@@
∗
@@
@@
@@
@@
∗
factor
skip
• ‖L−1k ‖, ‖U
−1k ‖ estimated [Cline, Moler, Stewart, Wilkinson ’77], [B. ’03]
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 8
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Inverse–Based Multilevel ILUs — Templates
TEMPLATE 3. MULTILEVEL SCHEME
(P>AQ)−1 =
(B F
E C
)−1
≈{(
LB 0
LE I
)(DB 0
0 S
)(UB UF
0 I
)}−1
≈(
B−1 0
0 0
)+
(−B−1F
I
)S−1
C
(−EB−1 I
), where B = LBDBUB
Recursive application to SC −→ (additive) algebraic multigrid scheme (or block ILU)
• use concept of ARMS [Saad, Suchomel ’99], i.e. skip LE, UF .
• After AMG is set up, SC can be discarded.
• Without LE, UF , forward/backward substitution twice as expensive, but more accurate and lessmemory needed.
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 9
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Inverse–Based Multilevel ILUs — Templates
APPROXIMATE SCHUR COMPLEMENTS
1. SC = C − LEDBUF
use analogous approximation scheme for B ≈ LBDBUB (S–version)
2. SC =[−LEL−1
B I]A
[−U−1
B UF
I
]use analogous approximation scheme for B ≈ LBDBUB (T–version)
3. Use S–version for the computation of B ≈ LBDBUB (M–version)Use T–version for the computation of SC
APPROXIMATION ORDER
Denote by τ the drop tolerance and let B = LBDBUB + EB
• If we use 1., then ‖EB‖ = O(τ ), ‖L−1B EBU−1
B ‖ = O(τκ2)
• if we use 2., then ‖EB‖ = O(τ 2), ‖L−1B EBU−1
B ‖ = O(τ 2κ)
• Similar arguments apply to SC
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 10
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Preconditioning Package: ILUPACK
Preconditioning software package (C/FORTRAN 77) based on inverse–based multilevel ILUs
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
S P A R S K I T RA M S BLASMC64SPARSPAK
Appr. Min. Fill
Interfaces
ILUPACK
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 11
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ILUPACK — General
NAMES AND FORMATS
1. ILUPACK currently supports real single/double precision and complex single/double precision.
2. Apart from the arithmetic ILUPACK distinguishes between two classes of matrices, namelygeneral matrices and symmetric (Hermitian) positive definite matrices.
3. matrices are stored in SPARSKIT format (compressed sparse row format).
SPARSKIT, ARMS, . . .
• Several routines from SPARSKIT incorporated (GMRES, FGMRES, ILUT, ILUTP,. . . )
• Reorderings (INDSET, ddPQ) and several tools from ARMS adapted
• Most common reorderings supported or at least interfaces are provided
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ILUPACK — Templates
TEMPLATE 1
• Preconditioner allows three arbitrarily chosen permutation/scaling functions
1. Initial preprocessing (applied only once)2. Regular reordering (applied to any subsequent system)3. Final pivoting (if the regular reordering fails)
• Custom reorderings possible
TEMPLATE 2
Partial ILUC (Crout version) with static permutation and diagonal pivoting (F77 core routine)
TEMPLATE 3
• Three different versions of approximate Schur complements are supported (S–, M– and T–version). M–version is default.
• After the computation of the approximate Schur complement the parts LE and UF are discarded.
• if the fill exceeds a certain amount, the algorithm switches to full-matrix processing.
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 13
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ILUPACK — Experiments
PRECONDITIONERS
1. inverse–based multilevel ILU (referred as “ILUPACK”)
2. ILUTP
3. ILUC (single–level, inverse–based)
ORDERINGS
PQ, RCM, . . . + row/column scaling
ITERATIVE SOLVERS
• GMRES, PCG. Iteration stopped if ‖b− Ax(k)‖ 6√
eps ‖b− Ax(0)‖
• Iterative solver treated as failure, if a break down occurred or more than 500 steps were needed.
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ILUPACK — Application to Unstructured Systems
EXAMPLE . Matrix ATT/TWOTONE
10−6
10−4
10−2
0
50
100
150
200
250
300
350
400 ILUC, MMD+MC64ILUTP, MMD+MC64ILUPACK−M, PQILUPACK−S, PQ
10−6
10−4
10−2
0
2
4
6
8
10
12ILUC, MMD+MC64ILUTP, MMD+MC64ILUPACK−M, PQILUPACK−S, PQ
Computation time Memory requirement
drop tol. drop tol.
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ILUPACK — Application to Unstructured Systems
EXAMPLE . Matrix VAVASIS/AV41092
10−6
10−4
10−2
0
100
200
300
400
500
600ILUTP, RCMILUPACK−M, PQILUPACK−S, PQ
10−6
10−4
10−2
0
2
4
6
8
10
12ILUTP, RCMILUPACK−M, PQILUPACK−S, PQ
Computation time Memory requirement
drop tol. drop tol.
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 16
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ILUPACK — dependence on inverse bound κ
Almost no difference between κ = 10 , κ = 25 , κ = 50 , κ = 100
64 LARGE TEST MATRICES, NUMBER OF SUCCESSFUL COMPUTATIONS
1 2 3 4 50
10
20
30
40
50
60
70
1 2 3 4 50
10
20
30
40
50
60
70
ILUPACK M–version ILUPACK S–version
drop tol. [10−∗] drop tol. [10−∗]
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 17
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ILUPACK — dependence on inverse bound κ
Almost no difference between κ = 10 , κ = 25 , κ = 50 , κ = 100
64 LARGE TEST MATRICES, NUMBER OF SUCCESSFUL COMPUTATIONS
1 2 3 4 50
10
20
30
40
50
60
70
1 2 3 4 50
10
20
30
40
50
60
70
ILUPACK M–version ILUTP + RCM
drop tol. [10−∗] drop tol. [10−∗]
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 18
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ILUPACK — dependence on inverse bound κ
Almost no difference between κ = 10 , κ = 25 , κ = 50 , κ = 100
64 LARGE TEST MATRICES, NUMBER OF SUCCESSFUL COMPUTATIONS
1 2 3 4 50
10
20
30
40
50
60
70
1 2 3 4 50
10
20
30
40
50
60
70
ILUPACK M–version ILUTP + RCM + MC64
drop tol. [10−∗] drop tol. [10−∗]
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 19
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ILUPACK — Abused as AMG
EXAMPLE . cylindric shell problem
Tight bounds on ‖U−1‖ 6 κ implicitly yield FINE GRID/COARSE GRID selection
10−2
10−1
4
6
8
10
12
14
16κ= 2κ= 5κ= 10κ= 25κ= 50κ=100
10−2
10−1
0.5
1
1.5
2
2.5
3κ= 2κ= 5κ= 10κ= 25κ= 50κ=100
Computation time Memory requirement
drop tol. drop tol.
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ILUPACK — Abused as AMG
EXAMPLE . cylindric shell problem
Number of levels. κ = 2, κ = 5, κ = 10, κ = 25, κ = 50, κ = 100.
1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4Number of levels depending on κ
drop tol. 0.1 0.05 0.025 0.01 0.005
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Conclusions
1. Multilevel preconditioners using inverse–based ILUs
2. numerical experiments indicate higher robustness with respect to parameter tuning
(time dependent problems, nonlinear equations)
3. Main objective: inverse triangular factors are kept bounded
4. AMG-like preconditioners realized in the software package ILUPACK that offers inverse–basedmultilevel ILUs for a broad class of matrices
5. Still to do: reordering strategies for PDE-type problems
ILUPACK BEING RELEASED BYAPRIL/MAY, 2004
http://www.math.tu-berlin.de/ilupack/
M. Bollh ofer /Y. Saad Inverse–Based Multilevel ILUs — Theory, Applications and Software 22
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