Preconditioning Symmetric Indefinite Systems with Maximum Weight Matching

1
Preconditioning Symmetric Indefinite Systems with
Maximum Weight Matching

• Han Chen Lin Tan
• Dept. of Computer Science
• University of California, Santa Barbara
• March 10, 2004

2
Outline

• The Problem
• Motivations
• Maximum Weight Matching
• Results
• Conclusions and Future Work

3
The Problem

• Consider a large sparse linear system
• Ax b (1)
• where A is symmetric and indefinite
• Our goal is to develop general purpose
  preconditioners for Krylov subspace methods for
  solving (1) using maximum weight matching

4
The Problem (cont.)

• The most widely used Krylov methods
• MINRES
• SYMMLQ
• Simplified QMR(SQMR)
• But what about preconditioning?

5
The Problem (cont.)

• Consider for instance left preconditioning
• MINRES and SYMMLQ require M to be SPD
• SQMR is the simplification of QMR that takes
  advantage of the symmetric. SQMR is the only safe
  solver for a general symmetric preconditioner M.
  (A Matlab function sqmr is written based on
  Matlabs qmr.)
• Use a nonsymmetric solver like GMRES or BiCGSTAB
  ? less efficient

6
Motivations

• Symmetric indefinite systems naturally arise in
  many important applications
• Structural analysis
• Acoustics
• Electromagnetics
• Fluid flow Stokes
• Mixed FEM for elliptic equations
• Optimization (KKT)
• Control
• Etc.
• In contrast to the positive definite case, few
  widely applicable techniques of preconditioning
  exist.

7
SymPerm Symmetric permutation
Maximum Weight Matching

• Find nonsymmetric max weight matching (MC64)
• Find disjoint cycles in the nonsymmetric matching
• Break up long cycles ( length gt 3 )
• Permute A symmetrically s.t. A LDLT
• L is the lower triangular of A
• D is a block diagonal with small symmetric blocks
• Solve (1) by using D as a preconditioner in SQMR

8
Symmetric matrix and graph
Maximum Weight Matching
3
1
5
2
3
4
1
4
5
2
3
4
2
1
5
G(A)
A

• Hollow vertex zero diagonal element

9
Symmetric matrix, nonsymmetric matching
Maximum Weight Matching

• Perfect matching in A disjoint directed
  cycles
• that cover every vertex of G
• P 2,5,3,1,4 ( nonsymmetric )
• A(,P) Max weight Matching, but not symmetric

10
Symmetric matrix, symmetric matching
Maximum Weight 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
• Q 1,2,5,4,3 ( symmetric )
• A(Q,Q) Max weight matching, and symmetric

11
What about odd cycles?
Maximum Weight Matching
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
• P 2,3,4,5,1 ( nonsymmetric )
• A(,P) Max weight matching, but not symmetric

12
What about odd cycles?
Maximum Weight Matching
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
• Q 1,2,3,4,5 ( symmetric )
• A(Q,Q) symmetric, may lose some weight

13
SymPerm Symmetric permutation
Maximum Weight Matching

• Find nonsymmetric max weight matching (MC64)
• Find disjoint cycles in the nonsymmetric matching
• Break up long cycles
• Even cycle 2 ways to break (equivalent)
• Odd cycle n ways to break ( which is better? Max
  Weight)
• Factor A LDLT
• L is lower triangular of A
• D is block diagonal with small symmetric blocks
• Solve (1) by using D as preconditioner in SQMR

14
Break long odd cycles n 5
Maximum Weight Matching
Max Diagonal Element
Can break up odd cycles of length more than k and
keep weight at least (k1)/(k2) times max.
15
Results

• Optimal control (Boeing)
• 871 positive eigenvalues
• 794 negative eigenvalues
• Condest1.3e6

SQMR converges very slowly (1e4 iterations).
16
Results (cont.)

• 77 1x1 pivots
• 794 2x2 pivots

• Use C as a preconditioner for Axb?

SQMR still converges very slowly.

• The 1x1/2x2 diagonal D of C is singular. Cant
  use D as a preconditioner. Cant use the block
  SSOR preconditioner.

• ILU of C doesnt help.

17
Results (cont.)

• SQMR converges still slowly.

• Use M as a preconditioner for ?

Failed. M is singular!
18
Results (cont.)

• Optimization (RAL)
• 583 positive eigenvalues
• 776 negative eigenvalues
• Condest2e17

SQMR converges after 291 iterations. (QMR
stagnated.)
19
Results (cont.)

• 199 1x1 pivots
• 580 2x2 pivots

• SQMR didnt converge after 3000 iterations, with
  C and D as preconditioners.

• Solving Cxb also stagnated, with and without
  preconditioner D.

• Since D is nonsingular, the block SSOR
  preconditioner can be used.

20
Results (cont.)
Block SSOR preconditioning
where L is strictly lower triangular and D is
block diagonal with 1x1 and 2x2 blocks. The
preconditioner is then
For SAWPATH, SQMR stagnated with the block SSOR
preconditioner.
21
Results (cont.)

• Symmetrized WATT
• 128 positive eigenvalues
• 1728 negative eigenvalues
• Condest 5.4e9

• SQMR converges after 295 iterations.
• Identical after symmetric permutation.
  Symmetrized WATT already has the max weight
  matching of 1x1 pivots on its diagonal!
• Preconditioning with D yields a faster
  convergence (117 iterations).

22
Results (cont.)

• 10 other symmetric indefinite sparse
  matrices from Harwell-Boeing ASH292,BCSPWR01,BCSP
  WR05,BCSSTK08,CAN_1054,DWT_869,LSHP1009, etc.
• Most of the Ds are singular and cannot be used
  as preconditioners, and the block SSOR
  preconditioners cannot be used either.
• Solving Cyb is not any faster.
• Preconditioning with C makes SQMR converge slower
  or stagnate.
• D is a good preconditioner only when A already
  has the MWM on its diagonal and the
  symmetric/non-symmetric permutation doesnt make
  any change to it.

23
Conclusions

• Use of MWM for symmetric permutation makes pivot
  blocks contiguous.
• Preconditioning with the permuted matrix C
  doesnt help.
• Most of the block diagonal matrices D of the
  permutated matrix are singular.
• D can be a good preconditioner only when the
  permuted matrix has only 1x1 pivots.

24
Future work

• LDL factorization using MWM and symmetric
  permutation.
• Use of larger block sizes, e.g. 1x1, 2x2 and 3x3
  blocks.