[Trilinos-Users] Behavior of ML preconditioner

[Eric Marttila]

Hello,

I am seeing some interesting behavior with the ML preconditioner.

I have two matrices which appear to be nearly identical, but when I 
create the ML preconditioner with matrix1, I see a warning message from ML:
          Gen_Prolongator warning : max eigen <= 0.0

I do not see the warning message when I create the preconditioner with 
matrix2.

When I use matrix1 (along with the ML preconditioner generated with 
matrix1) to perform a linear solve with AztecOO, the solution will converge.

However, when I use matrix2 (along with the ML preconditioner generated 
with matrix2) to perform a linear solve with AztecOO, the solution will 
*not* converge.

So my questions are:

1) Why does ML generate two (apparently) very different preconditioners 
when the two input matrices appear to be nearly identical?

2) In the case with matrix1 (where I get the Gen_Prolongator warning), 
is ML entering some sort of 'fail-safe' mode in order to generate a 
usable preconditioner?

3) Are there any ML settings that I should use to help ML generate a 
usable preconditioner for these types of matrices?

I've attached the two matrices (they are fairly small), as well as some 
source code I've used to generate the preconditioner. Also attached is 
the output of running the code with the two different matrices.

Any thoughts are greatly appreaciated!

Thanks,

Eric Marttila

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1074 1074 1.0000000000000000e+00
1075 1075 1.0000000000000000e+00
1076 1076 1.0000000000000000e+00
1077 1077 1.0000000000000000e+00
1078 1078 1.0000000000000000e+00
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Reading A matrix from mat1.mtx ...
Creating the preconditioner...
------------------------------------------------------------------------------
Using `cg' for eigen-computations
*** 
*** ML_Epetra::MultiLevelPreconditioner
***
Matrix has 1078 rows and 2170 nonzeros, distributed over 1 process(es)
The linear system matrix is an Epetra_CrsMatrix
** Leaving column map of Main linear system matrix unchanged
Default values for `SA'
Maximum number of levels = 10
Using increasing levels. Finest level  = 0, coarsest level = 9
Number of applications of the ML cycle = 1
Number of PDE equations = 1
Aggregation threshold = 0
Max coarse size = 128
R and P smoothing : P = (I-\omega A) P_t, R = P^T
R and P smoothing : \omega = 1.333/lambda_max
Null space type      = default (constants)
Null space dimension = 1
-----------------------------------------------------------------------
ML_Gen_MultiLevelHierarchy (level 0) : Gen Restriction and Prolongator
-----------------------------------------------------------------------
ML_Aggregate_Coarsen (level 0) begins
ML_Aggregate_CoarsenUncoupled : current level = 0
ML_Aggregate_CoarsenUncoupled : current eps = 0.000000e+00
Aggregation(UVB) : Total nonzeros = 2170 (Nrows=1078)
Aggregation(UC) : Phase 0 - no. of bdry pts  = 343 
Aggregation(UC) : Phase 1 - nodes aggregated = 714 (1078)
Aggregation(UC) : Phase 1 - total aggregates = 357 
Aggregation(UC_Phase2_3) : Phase 1 - nodes aggregated = 714 
Aggregation(UC_Phase2_3) : Phase 1 - total aggregates = 357
Aggregation(UC_Phase2_3) : Phase 2a- additional aggregates = 0
Aggregation(UC_Phase2_3) : Phase 2 - total aggregates = 378
Aggregation(UC_Phase2_3) : Phase 2 - boundary nodes   = 343
Aggregation(UC_Phase2_3) : Phase 3 - leftovers = 21 and singletons = 0
Aggregation(UC) : QR factorization - too small aggregates = 0
Calculating eigenvalue estimate using CG method
Gen_Prolongator warning : max eigen <= 0.0 
Gen_Prolongator (level 0) : Max eigenvalue = 1.0000e+00

Prolongator/Restriction smoother (level 0) : damping factor #1 = 1.3330e+00
Prolongator/Restriction smoother (level 0) : ( = 1.3330e+00 / 1.0000e+00)

-----------------------------------------------------------------------
ML_Gen_MultiLevelHierarchy (level 1) : Gen Restriction and Prolongator
-----------------------------------------------------------------------
ML_Aggregate_Coarsen (level 1) begins
ML_Aggregate_CoarsenUncoupled : current level = 1
ML_Aggregate_CoarsenUncoupled : current eps = 0.000000e+00
Aggregation(UVB) : Total nonzeros = 1680 (Nrows=378)
Aggregation(UC) : Phase 0 - no. of bdry pts  = 21 
Aggregation(UC) : Phase 1 - nodes aggregated = 357 (378)
Aggregation(UC) : Phase 1 - total aggregates = 84 
Aggregation(UC_Phase2_3) : Phase 1 - nodes aggregated = 357 
Aggregation(UC_Phase2_3) : Phase 1 - total aggregates = 84
Aggregation(UC_Phase2_3) : Phase 2a- additional aggregates = 0
Aggregation(UC_Phase2_3) : Phase 2 - total aggregates = 84
Aggregation(UC_Phase2_3) : Phase 2 - boundary nodes   = 21
Aggregation(UC_Phase2_3) : Phase 3 - leftovers = 0 and singletons = 0
Aggregation(UC) : QR factorization - too small aggregates = 0
Calculating eigenvalue estimate using CG method
Gen_Prolongator (level 1) : Max eigenvalue = 2.7293e+00

Prolongator/Restriction smoother (level 1) : damping factor #1 = 4.8841e-01
Prolongator/Restriction smoother (level 1) : ( = 1.3330e+00 / 2.7293e+00)

Smoothed Aggregation : operator complexity = 1.909677e+00.
Time to build the hierarchy = 0.00148201 (s)
Number of actual levels : 3

Smoother (level 0) : # global rows = 1078, # estim. global nnz = 2170, # nnz per row = 2.01
Smoother (level 0) : symmetric Gauss-Seidel (sweeps=2,omega=1,both)
Smoother (level 0) : Epetra_CrsMatrix detected, using Ifpack implementation
Smoother (level 0) : Setup time : 0.000121117 (s)

Smoother (level 1) : # global rows = 378, # estim. global nnz = 1680, # nnz per row = 4.44
Smoother (level 1) : ML symmetric Gauss-Seidel (sweeps=2,omega=1,both)
Smoother (level 1) : Setup time : 3.8147e-05 (s)

Amesos (level 2) : NumGlobalRows = 84
Amesos (level 2) : NumGlobalNonzeros = 294
Amesos (level 2) : Fill-in = 4.16667 %
Amesos (level 2) : Building KLU
Amesos (level 2) : Time for factorization  = 0.000102997 (s)


Cumulative timing for construction so far: 
- for initial setup   = 0.00304794 (s)
- for hierarchy setup = 0.00149989 (s)
- for smoothers setup = 0.000159264 (s)
- for coarse setup    = 0.000235081 (s)
- for final setup     = 2.40803e-05 (s)
Total for this setup  = 0.00510097 (s)
------------------------------------------------------------------------------
Done.
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Reading A matrix from mat2.mtx ...
Creating the preconditioner...
------------------------------------------------------------------------------
Using `cg' for eigen-computations
*** 
*** ML_Epetra::MultiLevelPreconditioner
***
Matrix has 1078 rows and 2170 nonzeros, distributed over 1 process(es)
The linear system matrix is an Epetra_CrsMatrix
** Leaving column map of Main linear system matrix unchanged
Default values for `SA'
Maximum number of levels = 10
Using increasing levels. Finest level  = 0, coarsest level = 9
Number of applications of the ML cycle = 1
Number of PDE equations = 1
Aggregation threshold = 0
Max coarse size = 128
R and P smoothing : P = (I-\omega A) P_t, R = P^T
R and P smoothing : \omega = 1.333/lambda_max
Null space type      = default (constants)
Null space dimension = 1
-----------------------------------------------------------------------
ML_Gen_MultiLevelHierarchy (level 0) : Gen Restriction and Prolongator
-----------------------------------------------------------------------
ML_Aggregate_Coarsen (level 0) begins
ML_Aggregate_CoarsenUncoupled : current level = 0
ML_Aggregate_CoarsenUncoupled : current eps = 0.000000e+00
Aggregation(UVB) : Total nonzeros = 2170 (Nrows=1078)
Aggregation(UC) : Phase 0 - no. of bdry pts  = 343 
Aggregation(UC) : Phase 1 - nodes aggregated = 714 (1078)
Aggregation(UC) : Phase 1 - total aggregates = 357 
Aggregation(UC_Phase2_3) : Phase 1 - nodes aggregated = 714 
Aggregation(UC_Phase2_3) : Phase 1 - total aggregates = 357
Aggregation(UC_Phase2_3) : Phase 2a- additional aggregates = 0
Aggregation(UC_Phase2_3) : Phase 2 - total aggregates = 378
Aggregation(UC_Phase2_3) : Phase 2 - boundary nodes   = 343
Aggregation(UC_Phase2_3) : Phase 3 - leftovers = 21 and singletons = 0
Aggregation(UC) : QR factorization - too small aggregates = 0
Calculating eigenvalue estimate using CG method
Gen_Prolongator (level 0) : Max eigenvalue = 8.6788e+20

Prolongator/Restriction smoother (level 0) : damping factor #1 = 1.5359e-21
Prolongator/Restriction smoother (level 0) : ( = 1.3330e+00 / 8.6788e+20)

-----------------------------------------------------------------------
ML_Gen_MultiLevelHierarchy (level 1) : Gen Restriction and Prolongator
-----------------------------------------------------------------------
ML_Aggregate_Coarsen (level 1) begins
ML_Aggregate_CoarsenUncoupled : current level = 1
ML_Aggregate_CoarsenUncoupled : current eps = 0.000000e+00
Aggregation(UVB) : Total nonzeros = 1680 (Nrows=378)
Aggregation(UC) : Phase 0 - no. of bdry pts  = 21 
Aggregation(UC) : Phase 1 - nodes aggregated = 357 (378)
Aggregation(UC) : Phase 1 - total aggregates = 84 
Aggregation(UC_Phase2_3) : Phase 1 - nodes aggregated = 357 
Aggregation(UC_Phase2_3) : Phase 1 - total aggregates = 84
Aggregation(UC_Phase2_3) : Phase 2a- additional aggregates = 0
Aggregation(UC_Phase2_3) : Phase 2 - total aggregates = 84
Aggregation(UC_Phase2_3) : Phase 2 - boundary nodes   = 21
Aggregation(UC_Phase2_3) : Phase 3 - leftovers = 0 and singletons = 0
Aggregation(UC) : QR factorization - too small aggregates = 0
Calculating eigenvalue estimate using CG method
Gen_Prolongator (level 1) : Max eigenvalue = 2.2156e+02

Prolongator/Restriction smoother (level 1) : damping factor #1 = 6.0165e-03
Prolongator/Restriction smoother (level 1) : ( = 1.3330e+00 / 2.2156e+02)

Smoothed Aggregation : operator complexity = 1.909677e+00.
Time to build the hierarchy = 0.00148201 (s)
Number of actual levels : 3

Smoother (level 0) : # global rows = 1078, # estim. global nnz = 2170, # nnz per row = 2.01
Smoother (level 0) : symmetric Gauss-Seidel (sweeps=2,omega=1,both)
Smoother (level 0) : Epetra_CrsMatrix detected, using Ifpack implementation
Smoother (level 0) : Setup time : 0.000120163 (s)

Smoother (level 1) : # global rows = 378, # estim. global nnz = 1680, # nnz per row = 4.44
Smoother (level 1) : ML symmetric Gauss-Seidel (sweeps=2,omega=1,both)
Smoother (level 1) : Setup time : 4.29153e-05 (s)

Amesos (level 2) : NumGlobalRows = 84
Amesos (level 2) : NumGlobalNonzeros = 294
Amesos (level 2) : Fill-in = 4.16667 %
Amesos (level 2) : Building KLU
Amesos (level 2) : Time for factorization  = 0.000102997 (s)


Cumulative timing for construction so far: 
- for initial setup   = 0.00311685 (s)
- for hierarchy setup = 0.00149894 (s)
- for smoothers setup = 0.000163078 (s)
- for coarse setup    = 0.000232935 (s)
- for final setup     = 2.31266e-05 (s)
Total for this setup  = 0.00516582 (s)
------------------------------------------------------------------------------
Done.