Symmetric Weighted Matching for Indefinite Systems

1
Symmetric Weighted Matching for Indefinite Systems

• Iain Duff, RAL and CERFACS
• John Gilbert, MIT and UC Santa Barbara
• June 21, 2002

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Symmetric indefinite systems Block LDLT

• Optimization, interior eigenvalues, . . .
• Factor A L D L
• L is unit lower triangular
• D is block diagonal with small symmetric blocks

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Symmetric indefinite systems Block LDLT

• Optimization, interior eigenvalues, . . .
• Factor A L D L
• L is unit lower triangular
• D is block diagonal with small symmetric blocks

Static pivoting after preprocessing by weighted
matching
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TRAJ06B

• optimal control (Boeing)
• condest 1.4e6
• inertia (794, 0, 871)

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TRAJ06B permuted

• optimal control (Boeing)
• condest 1.4e6
• inertia (794, 0, 871)
• 794 2-by-2 pivots, 77 1-by-1 pivots

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TRAJ06B factor, solve, iterative refinement

• optimal control (Boeing)
• condest 1.4e6
• inertia (794, 0, 871)
• 794 2-by-2 pivots, 77 1-by-1 pivots
• factor nz 29,842
• max L 550
• iterations 2
• rel residual 3.7e-14
• AGL nz 40,589
• GESP nz 53,512
• GEPP nz 432,202

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Symmetric matrix and graph

• Hollow vertex zero diagonal element

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Symmetric matrix, nonsymmetric matching

• Perfect matching in A disjoint directed
  cycles that cover every vertex of G

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Symmetric matrix, symmetric matching
3
1
5
2
3
4
1
4
5
2
3
4
2
1
5
G(A)
A

• Theorem (easy) Any even cycle can be
  converted to a set of 2-cycles without
  decreasing the weight of the matching

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What about odd cycles?

• Breaking up odd cycles may decrease weight

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What about odd cycles?
1
1
5
2
3
4
1
5
2
2
3
4
3
4
5
G(A)
A

• Breaking up odd cycles may decrease weight
• Can break up odd cycles of length more than k
  and keep weight at least (k1)/(k2) times max

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What about odd cycles?
1
1
5
2
3
4
1
5
2
2
3
4
3
4
5
G(A)
A

• Breaking up odd cycles may decrease weight
• Can break up odd cycles of length more than k
  and keep weight at least (k1)/(k2) times
  max
• But, may need to defer singular pivots

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BlockPerm Static block pivot permutation

• Find nonsymmetric max weight matching (MC64)
• Break up long cycles
• Permute symmetrically to make pivot blocks
  contiguous
• Permute blocks symmetrically for low fill
  (approx minimum degree)
• Move any singular pivots to end (caused by odd
  cycles or structural rank deficiency)
• Factor A LDL without row/col interchanges
  (possibly fixing up bad pivots)
• Solve, and improve solution by iterative
  refinement

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SAWPATH

• optimization (RAL)
• condest 2e17
• inertia (776, 0, 583)
• 580 2-by-2 pivots,199 1-by-1 pivots
• factor nz 5,988
• max L 9e14
• iterations 1
• rel residual 4.4e-9
• MA57 nz 8,355 (best u)
• GESP nz 34,060
• GEPP nz 331,344

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LASER

• optimization
• inertia (1000, 2, 2000)
• 1000 2-by-2 pivots,1002 1-by-1 pivots (2
  singular)
• factor nz 7,563
• max L 1.6
• iterations 0
• rel residual 6.8e-17
• GESP nz 11,001
• GEPP nz 10,565 (rank wrong)

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DUFF320

• optimization (RAL)
• inertia (100, 120, 100)
• 100 2-by-2 pivots (3 singular)
• 120 1-by-1 pivots (all singular)
• factor nz 685
• max L 3.4e8
• iterations 2
• rel residual 3.0
• MA57 nz 2,848
• GESP nz 1,583
• GEPP nz 1,500 (no solve)

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DUFF320, take 2 Magic permutation

• optimization (RAL)
• inertia (100, 120, 100)
• 100 2-by-2 pivots (none singular)
• 120 1-by-1 pivots (all singular)

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DUFF320, take 2 Magic permutation

• optimization (RAL)
• inertia (100, 120, 100)
• 100 2-by-2 pivots (none singular)
• 120 1-by-1 pivots (all singular)
• factor nz 722
• max L 1.0
• iterations 0
• rel residual 0.0

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DUFF320, take 3 Identity block

• optimization (RAL)
• inertia (100, 0, 220)
• 100 2-by-2 pivots
• 120 1-by-1 pivots
• factor nz 1907
• max L 1
• iterations 1
• rel residual 1.7e-15
• GESP nz 2,929
• GEPP nz 3,554

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Questions

• Judicious dynamic pivoting (e.g. MA57)
• Preconditioning iterative methods
• Combinatorial methods to seek good block diagonal
  directly (high weight, well-conditioned, sparse)
• Symmetric equilibration / scaling
• Better fixups for poor static pivots
• Well hope to get some boundedness from
  hyperbolic sines Burns Demmel 2002

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Questions

• Judicious dynamic pivoting (e.g. MA57)
• Preconditioning iterative methods
• Combinatorial methods to seek good block diagonal
  directly (high weight, well-conditioned, sparse)
• Symmetric equilibration / scaling
• Better fixups for poor static pivots
• Well hope to get some boundedness from
  hyperbolic sines Burns Demmel 2002

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BCSSTK24

• structures (Boeing)
• condest 6e11
• inertia (0, 0, 3562)
• no 2-by-2 pivots,3562 1-by-1 pivots
• factor nz 289,191
• max L 410
• iterations 2
• rel residual 2.2e-16
• AGL nz 299,000
• Cholesky nz 289,191
• GEPP nz 1,217,475

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DUFF320, take 2 Magic permutation

• optimization (RAL)
• inertia (100, 120, 100)
• 100 2-by-2 pivots (none singular)
• 120 1-by-1 pivots (all singular)

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DUFF33

• optimization (RAL)
• inertia (9, 15, 9)
• 9 2-by-2 pivots (1 singular)
• 15 1-by-1 pivots (all singular)
• factor nz 62
• max L 7.0e7
• iterations 1
• rel residual 3.0
• MA57 nz
• GESP nz
• GEPP nz

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DUFF33, take 2 Magic permutation

• optimization (RAL)
• inertia (9, 15, 9)
• 9 2-by-2 pivots (none singular)
• 15 1-by-1 pivots (all singular)
• factor nz 51
• max L 1.0
• iterations 0
• rel residual 0.0

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DUFF33, take 2 L with magic permutation

• optimization (RAL)
• inertia (9, 15, 9)
• 9 2-by-2 pivots (none singular)
• 15 1-by-1 pivots (all singular)
• factor nz 51
• max L 1.0
• iterations 0
• rel residual 0.0

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DUFF33, take 2 D with magic permutation

• optimization (RAL)
• inertia (9, 15, 9)
• 9 2-by-2 pivots (none singular)
• 15 1-by-1 pivots (all singular)
• factor nz 51
• max L 1.0
• iterations 0
• rel residual 0.0