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Sparse factorizationusing low rank submatrices
Cleve AshcraftLSTC
cleve@lstc.com
2010 MUMPS User Group MeetingApril 15-16, 2010
Toulouse, FRANCE
ftp.lstc.com:outgoing/cleve/MUMPS10 Ashcraft.pdf
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LSTCLivermore Software Technology Corporation
Livermore, California 94550
• Founded by John Hallquist of LLNL, 1980’s
• Public domain versions of DYNA and NIKE codes
• LS-DYNA : implicit/explicit,nonlinear finite element analysis code
• Multiphysics capabilities
– Fluid/structure interaction
– Thermal analysis
– Acoustics
– Electromagnetics
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Multifrontal Algorithm
• very large, sparse LDLT and LDU factorizations
• tree structure organizes factor storage,solve and factor operations
• medium-to-large sparse linear systemslocated at each leaf node of the tree
• medium-sized dense linear systemslocated at each interior node of the tree
• dense matrix-matrix operationsat each interior node
• sparse matrix-matrix adds between nodes
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Multifrontal tree
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Multifrontal tree – polar representation
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Multifrontal Algorithm
• 40M equations, present frontier at LSTC
• serial, SMP, MPP, hybrid, GPU
• large problems, > 1M - 10M dof,require out-of-core storage of factor entries,even on distributed memory systems
• IO cost largely hidden during the factorization
• IO cost dominant during the solves
• eigensolver =⇒ several right hand sides
• many applications, e.g., Newton’s method,have a single right hand side
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Low Rank Approximations
• hierarchical matrices,Hackbusch, Bebendorf, Leborne, others
• semi-separable matrices, Gu, others
• submatrices are numerically rank deficient
• method of choice for Boundary Elements (BEM)
• now applied to Finite Elements (FEM)
• A ≈ XY TAm
n
= m X
r
Y Tn
r
• storage = r(m + n) vs mn
• reduction in ops =r
min(m,n)7
Multifrontal Algorithm + Low Rank Approximations
• At each leaf node in the multifrontal tree —use standard multifrontal
• At each interior node in the multifrontal tree —low rank matrix-matrix multiplies and sums
• Between nodes —low rank matrix sums
• Dramatic reduction in storage
• Dramatic reduction in operations
• Excellent approximation propertiesfor finite element operators
• Our experience is with potential equations,elasticity with solids and shells
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Outline
• graph, tree, matrix perspectives
• experiments – 2-D potential equation
• low rank computations
• blocking strategies
• summary
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One subgraph
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
One subtree
ΩI ΩJ
@@
@@
@@@I
S
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∂S
One submatrix
AΩI ,ΩIAΩI ,S
AΩI ,∂S
AΩJ ,ΩJAΩJ ,S AΩJ ,∂S
AS,ΩIAS,ΩJ
AS,S AS,∂S
A∂S,ΩIA∂S,ΩJ
A∂S,S A∂S,∂S
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Portion of original matrix
0 200 400 600 800 1000 1200
0
200
400
600
800
1000
1200
nz = 9052
A ordered by domains, separator and external boundary
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Portion of factor matrix
0 200 400 600 800 1000 1200
0
200
400
600
800
1000
1200
nz = 48899
L ordered by domains, separator and external boundary
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Schur complement matrix
[AS,S AS,∂S
A∂S,S A∂S,∂S
]=
0 20 40 60 80 100 120 140 160
0
20
40
60
80
100
120
140
160
nz = 20659
Schur complement, separator and external boundary
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Outline
• graph, tree, matrix perspectives
• experiments – 2-D potential equation
• low rank computations
• blocking strategies
• summary
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Computational experiments
• Compute LS,S, L∂S,S and A∂S,∂S
• Find domain decompositions of S and ∂S
• Form block matrices, e.g., LS,S =∑
K≥J
LK,J
• Find singular value decompositions of each LK,J
• Collect all singular values
σ =∑
K≥J
min(|K|,|J |)∑
i=1
σ(K,J)i
• Split matrix LS,S = MS,S + NS,S using singularvalues σ.
• We want ‖NS,S‖F ≤ 10−14 ‖LS,S‖F
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255 × 255 diagonal block factor LS,S
43% dense, relative accuracy 10−14
LS,S =
0 50 100 150 200 250
0
50
100
150
200
250
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754 × 255 lower block factor L∂S,S
16% dense, relative accuracy 10−14
L∂S,S =
0 100 200
0
100
200
300
400
500
600
700
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754 × 754 update matrix A∂S,∂S
21% dense, relative accuracy 10−14
A∂S,∂S =
0 100 200 300 400 500 600 700
0
100
200
300
400
500
600
700
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255 × 255 factor matrix LS,S
storage vs accuracy
0 0.1 0.2 0.3 0.4 0.5 0.6−16
−14
−12
−10
−8
−6
−4
−2
0
fraction of dense storage
log 10
(1 −
||M
|| F /
||A|| F
)
approximating 255 x 255 separator factor matrix L3,3
2 x 2 blocks3 x 3 blocks4 x 4 blocks5 x 5 blocks6 x 6 blocks7 x 7 blocks8 x 8 blocks
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754 × 255 factor matrix L∂S,S
storage vs accuracy
0 0.05 0.1 0.15 0.2−16
−14
−12
−10
−8
−6
−4
−2
0
fraction of dense storage
log 10
(1 −
||M
|| F /
||A|| F
)
approximating 754 x 255 interface−separator factor matrix L4,3
3 x 1 blocks6 x 2 blocks9 x 3 blocks12 x 4 blocks
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754 × 754 update matrix A∂S,∂S
storage vs accuracy
0 0.1 0.2 0.3 0.4 0.5−16
−14
−12
−10
−8
−6
−4
−2
0
fraction of dense storage
log 10
(1 −
||M
|| F /
||A|| F
)
approximating 754 x 754 schur complement matrix Ahat44
2 x 2 blocks3 x 3 blocks4 x 4 blocks5 x 5 blocks6 x 6 blocks7 x 7 blocks8 x 8 blocks
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Outline
• graph, tree, matrix perspectives
• experiments – 2-D potential equation
• low rank computations
• blocking strategies
• summary
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How to compute low rank submatrices ?
• SVD – singular value decomposition
A = UΣUT
where U and V are orthogonal and Σ is diagonal
• Gold standard, expensive, O(n3) ops
• QR factorization
AP = QR
where Q is orthogonal and R is upper triangular,and P is a permutation matrix
• Silver standard, moderate cost, O(rn2) ops
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row norms of R vs singular values of A
754 × 754 update matrix A∂S,∂S
94 × 94 submatrix sizenear, mid and far interactions
0 2 4 6 8 10 12 14 16 18 20−20
−18
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−14
−12
−10
−8
−6
−4
−2
0row norms of R versus singular values of 754 x 754 A using 8 segments
near ||R
i,:||
2
near σi
mid ||Ri,:
||2
mid σi
far ||Ri,:
||2
far σi
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SVD vs QR with column pivoting — Conclusions :
• Column pivoting QR factorization does well.
• Row norms of R track singular values σ
• The numerical rank of R is greater than needed,but not that much greater
• For more accuracy/less storagetwo sided orthogonal factorizations
– AP = ULV , U and V orthogonal, L triangular
– PAQ = UBV ,U and V orthogonal, B bidiagonal
track the singular values very closely.
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Type of approximation of submatrices
• submatrix LI,J of L∂S,S, ‖LI,J‖F = 2.92 × 10−2
factorization numerical rank total entriesnone — 4900QR 20 2681
ULV 18 2680SVD 18 2538
• submatrix LI,J of L∂S,S, ‖LI,J‖F = 2.04 × 10−3
factorization numerical rank total entriesnone — 4900QR 14 1896
ULV 10 1447SVD 10 1410
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Operations with low rank submatrices
A = UAV TA , B = UBV T
B , C = UCV TC
See Bebendorf, “Hierarchical Matrices”, Chapter 1
• Multiplication A = B C, rank(A) ≤ min(rank(B), rank(C))
A = UAV TA =
(UBV T
B
) (UCV T
C
)= BC
= UB
(V T
B UC
)V T
C
= UB
((V T
B UC
)V T
C
)
=(UB
(V T
B UC
))V T
C
• Addition A = B + C, rank(A) ≤ rank(B) + rank(C)
A = UAV TA =
(UBV T
B
)+
(UCV T
C
)= B + C
=[UB UC
] [VB VC
]T
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Multiplication A = B C
A = UAV TA , B = UBV T
B , C = UCV TC
A = B C =
=(
(m × h) (h × l))(
(l × k) (k × n))
= =
= (m × min(h, k)) (min(h, k) × n)
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Near-near matrix product
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near−near interaction, L2,1
L2,1T
σ(L21)
σ(L21*L21T)
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mid-near matrix product
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mid−near interaction, L2,1
L1,1T
σ(L21)σ(L11)
σ(L21*L11T)
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far-near matrix product
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far−near interaction, L5,1
L6,1T
σ(L51)σ(L61)
σ(L51*L61T)
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mid-mid matrix product
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mid−mid interaction, L1,1
L1,1T
σ(L11)
σ(L11*L11T)
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far-mid matrix product
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mid−far interaction, L6,1
L1,1T
σ(L61)
σ(L61*L61T)
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far-far matrix product
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−14
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far−far interaction, L6,1
L6,1T
σ(L61)
σ(L61*L61T)
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Addition A = B + C
rank(A) ≤ rank(B) + rank(C)
A = UAV TA =
(UBV T
B
)+
(UCV T
C
)= B + C
=[UB UC
] [VB VC
]T
= (Q1R1) (Q2R2)T
= Q1
(R1R
T2
)QT
2
= Q1 (Q3R3) QT2
= (Q1Q3)(R3Q
T2
)
= (Q1Q3)(Q2R
T3
)T
• R1RT2 usually has low numerical rank
• examples follow35
update matrix, diagonal blockA3,3 = L3,1L
T3,1 + L3,2L
T3,2 + L3,3L
T3,3
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log 10
(σ)
update matrix −− self interaction
σ(near)σ(sum)σ(mid)σ(far)
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update matrix, mid-distance off-diagonal blockA5,3 = L5,1L
T3,1 + L5,2L
T3,2 + L5,3L
T3,3
0 2 4 6 8 10 12−20
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−16
−14
−12
−10
−8
−6
−4
−2
log 10
(σ)
update matrix −− mid−range interaction
σ(sum)σ(near)σ(mid)σ(far)
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update matrix, far-distance off-diagonal blockA7,3 = L7,1L
T3,1 + L7,2L
T3,2 + L7,3L
T3,3
0 2 4 6 8 10 12−20
−18
−16
−14
−12
−10
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−6
−4
−2
log 10
(σ)
update matrix −− far−range interaction
σ(sum)σ(near)σ(mid)σ(far)
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Outline
• graph, tree, matrix perspectives
• experiments – 2-D potential equation
• low rank computations
• blocking strategies
• summary
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Blocking Strategies
• Active data structures[LJ,J
L∂J,J A∂J,∂J
]
• partition J and ∂J independently
• for 2-d problems, J and ∂J are 1-d manifolds
• for 3-d problems, J and ∂J are 2-d manifolds
• we need mesh partitioning of the separatorand the boundary of a region J
• each index set of a partition is a segment
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Segment partition
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
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1 segment wirebasket domain decomposition
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•
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0
50
100
150
200
250
LJ,J =∑
σ,τ
Lσ,τ σ × τ ⊆ J × J
•
0 100 200
0
100
200
300
400
500
600
700
L∂J,J =∑
σ,τ
Lσ,τ σ × τ ⊆ ∂J × J
•
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0
100
200
300
400
500
600
700
A∂J,∂J =∑
σ,τ
Aσ,τ σ×τ ⊆ ∂J×∂J
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Strategy I
• The partition of ∂J (local to node J)conforms to the partitions of ancestors K,K ∩ ∂J 6= ∅
• Advantages :
– Update assembly is simplified
A(p(J))σ2,τ2 = A
(p(J))σ2,τ2 + A
(J)σ1,τ1
where σ1 ⊆ σ2, τ1 ⊆ τ2
– One destination for each A(J)σ1,τ1
• Disadvantages :
– Partition of ∂J can be fragmented,more segments, smaller size,less efficient storage
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Strategy II
• The partition of ∂J (local to node J)need not conform to the partitions of ancestors
• Advantages :
– Partition of ∂J can be optimized bettersince ∂J is small and localized
• Disadvantages :
– Update assembly is more complex
A(p(J))σ1∩σ2,τ1∩τ2
= A(p(J))σ1∩σ2,τ1∩τ2
+ A(J)σ1∩σ2,τ1∩τ2
where σ1 ∩ σ2 6= ∅, τ1 ∩ τ2 6= ∅
– Several destinations for a submatrix A(J)σ1,τ1
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Outline
• graph, tree, matrix perspectives
• experiments – 2-D potential equation
• low rank computations
• blocking strategies
• summary
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Multifrontal tree Multifrontal Factorization
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• each leaf node is ad-dimensional sparseFEM matrix
• each interior node isa (d − 1)-dimensionaldense BEM matrix
• use low rank storageand computation ateach interior node
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Call to action
• progress to date in low rank factorizationsdriven by iterative methods
• work needed from direct methods community
• start from industrial strength multifrontal code
– serial, SMP, MPP, hybrid, GPU
– pivoting for stability, out-of-core,singular systems, null spaces
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Call to action
• Many challenges
– partition of space, partition of interface
– added programming complexityof low rank matrices
– challenges to pivot for stability
– challenges/opportunitiesto implement in parallel
• Payoff will be huge
– reduction in storage footprint
– reduction in computational work
– take direct methods to next levelof problem size
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