GMRES left-preconditioned with ILU1 versus ILU0

Fri May 16 11:34:43 CDT 2008

On Fri, May 16, 2008 at 11:22 AM, Stephane Aubert
<stephane.aubert at fluorem.com> wrote:
> Hi,
> I tried to improve the convergence of a rather badly conditioned linear
> system by increasing the ILU(k) level from k=0 to k=1.
> And, whereas I got a convergence for k=0, I ended up with an explosive
> divergence for k=1!!!
> The PETCS version is 2.3.2-p8.
> The matrix type is MPIBAIJ (block size=7,mesh points=23010,non-empty
> blocks=297454)
> The common command line options are:
>
>   * KSP="-ksp_type gmres -ksp_max_it 800 -ksp_gmres_restart 800
>     -ksp_rtol 1.0e-12 -ksp_left_pc -ksp_gmres_modifiedgramschmidt
>     -ksp_gmres_cgs_refinement_type REFINE_NEVER -ksp_singmonitor
>     -ksp_compute_eigenvalues": I'm forcing 800 krylovs without restart
>     to get the condition number of the pre-conditioned system (if I
>     understand correctly the man page of -ksp_singmonitor)
>   * PC="-pc_type asm -pc_asm_overlap 2": I'm planning to run with more
>     than 1 partition, but for the time being, only one partition is used.
>   * BLK_KSP="-sub_ksp_type preonly": Because of GMRES+ILU
>
> For ILU(0), I'm using:
>
>   * BLK_PC="-sub_pc_type ilu -sub_pc_factor_levels 0
>     -sub_pc_factor_fill 1.00 -sub_pc_factor_shift_nonzero
>     -sub_pc_factor_mat_ordering_type rcm -sub_pc_factor_pivot_in_blocks"
>
> and I got for convergence:
>  0 KSP Residual norm 1.687258996558e+00 % max 1 min 1 max/min 1
>  1 KSP Residual norm 1.687132576728e+00 % max 67.8829 min 67.8829 max/min 1
>  2 KSP Residual norm 1.685760733293e+00 % max 3496.78 min 19.5582 max/min
> 178.789
>  3 KSP Residual norm 1.668552043995e+00 % max 3604.26 min 12.0073 max/min
> 300.174
>  4 KSP Residual norm 1.578511835381e+00 % max 3639.92 min 7.35118 max/min
> 495.148
> ....
> 795 KSP Residual norm 1.465607932165e-09 % max 18209.9 min 0.00612973
> max/min 2.97075e+06
> 796 KSP Residual norm 1.390602265424e-09 % max 18227.3 min 0.00612913
> max/min 2.97388e+06
> 797 KSP Residual norm 1.320529491862e-09 % max 18231.9 min 0.0061286 max/min
> 2.97489e+06
> 798 KSP Residual norm 1.253371917713e-09 % max 18234.8 min 0.00612856
> max/min 2.97538e+06
> 799 KSP Residual norm 1.188955299647e-09 % max 18278.5 min 0.00612594
> max/min 2.98378e+06
> 800 KSP Residual norm 1.118294486519e-09 % max 18278.5 min 0.00612475
> max/min 2.98437e+06
>
> and the iterative solution compares very well with the one computed using
> complete LU factorization.
>
> For ILU(1), I'm using:
>
>   * BLK_PC="-sub_pc_type ilu -sub_pc_factor_levels 1
>     -sub_pc_factor_fill 3.81 -sub_pc_factor_shift_nonzero
>     -sub_pc_factor_mat_ordering_type rcm
>     -sub_pc_factor_pivot_in_blocks": RCM gives the smallest fill value.
>
> and I got for "convergence":
>  0 KSP Residual norm 7.095990612421e+126 % max 1 min 1 max/min 1
>  1 KSP Residual norm 3.313547979190e+123 % max 1.68012e+135 min 1.68012e+135
> max/min 1
>  2 KSP Residual norm 1.257750994639e+119 % max 6.34518e+135 min 3.55953e+131
> max/min 17825.9
>  3 KSP Residual norm 5.233083258710e+118 % max 1.25538e+136 min 1.42732e+127
> max/min 8.79538e+08
>  4 KSP Residual norm 1.938981257595e+118 % max 1.44472e+136 min 4.82369e+125
> max/min 2.99506e+10
>  5 KSP Residual norm 3.371270926617e+116 % max 1.45841e+136 min 1.79839e+125
> max/min 8.1095e+10
>  6 KSP Residual norm 2.179293254483e+115 % max 1.45842e+136 min 5.16422e+122
> max/min 2.82408e+13
>  7 KSP Residual norm 2.120598006522e+115 % max 1.46024e+136 min 3.67337e+122
> max/min 3.97521e+13
>  8 KSP Residual norm 1.486820601733e+115 % max 1.461e+136 min 1.02249e+122
> max/min 1.42886e+14
>  9 KSP Residual norm 7.653834441859e+114 % max 1.46138e+136 min 4.93314e+121
> max/min 2.96237e+14
> 10 KSP Residual norm 5.001204920218e+114 % max 1.47243e+136 min 4.89675e+121
> max/min 3.00694e+14
>
> My guess is that the ILU(1) is singular (zero as diagonal elements?), but I
> thought that the options "-sub_pc_factor_shift_nonzero
> -sub_pc_factor_pivot_in_blocks" were taking care of that... I got lost in
> the source files trying to find out who at the end is computing and applying
> the ILU for MPIBAIJ format (replaced by SEQBAIJ with only one partition, I'm
> guessing).
>
> The question is: What am I doing wrong? I never heard that ILU(1) was worst
> than ILU(0)!

It is not uncommon. There are no theoretical guarantees for ILU(k), which is why
I dislike it so much. ILU(1) can indeed be worse than ILU(0), depending on the
type of matrix you have and the iterative solver used.

  Matt

> Stef.
>
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>

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which
their experiments lead.
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