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

2. 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

3. 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

4. TRAJ06B • optimal control (Boeing) • condest = 1.4e6 • inertia = (794, 0, 871)

5. TRAJ06B permuted • optimal control (Boeing) • condest = 1.4e6 • inertia = (794, 0, 871) • 794 2-by-2 pivots, 77 1-by-1 pivots

6. 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

7. 3 1 2 3 4 5 1 4 5 2 3 4 2 5 1 A G(A) Symmetric matrix and graph • Hollow vertex = zero diagonal element

8. 3 1 2 3 4 5 1 4 5 2 3 4 2 5 1 A G(A) Symmetric matrix, nonsymmetric matching • Perfect matching in A = disjoint directed cycles that cover every vertex of G

9. Symmetric matrix, symmetric matching 3 1 2 3 4 5 1 4 5 2 3 4 2 5 1 A G(A) • Theorem (easy): Any even cycle can be converted to a set of 2-cycles without decreasing the weight of the matching

10. 1 1 2 3 4 5 1 5 2 2 3 4 3 5 4 A G(A) What about odd cycles? • Breaking up odd cycles may decrease weight

11. What about odd cycles? 1 1 2 3 4 5 1 5 2 2 3 4 3 5 4 A G(A) • Breaking up odd cycles may decrease weight • Can break up odd cycles of length more than k and keep weight at least (k+1)/(k+2) times max

12. What about odd cycles? 1 1 2 3 4 5 1 5 2 2 3 4 3 5 4 A G(A) • Breaking up odd cycles may decrease weight • Can break up odd cycles of length more than k and keep weight at least (k+1)/(k+2) times max • But, may need to defer singular pivots

13. 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 = L*D*L’ without row/col interchanges (possibly fixing up bad pivots) • Solve, and improve solution by iterative refinement

14. 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

15. 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)

16. 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)

17. 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)

18. 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

19. 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

20. 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 “We’ll hope to get some boundedness from hyperbolic sines”[Burns & Demmel 2002]

21. 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 “We’ll hope to get some boundedness from hyperbolic sines”[Burns & Demmel 2002]

22. 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

23. 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)

24. 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 =

25. 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

26. 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

27. 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