DMMG solver convergence with single precision?

Barry Smith bsmith at mcs.anl.gov
Tue Dec 2 22:43:46 CST 2008 

   I have reproduced this. I expect that you would not be able to  
decrease the residual by more than a factor
of 10^6 but the results below are worse than I expected.

   After first running it I thought I would have to say "I have no  
idea". But then I ran

barry-smiths-macbook-pro-17:ksp/examples/tutorials] bsmith% ./ex34 - 
ksp_monitor_true_residual -dmmg_nlevels 6 -ksp_type fgmres - 
ksp_gmres_restart 4
     0 KSP preconditioned resid norm 4.451733827591e-02 true resid  
norm 4.451733827591e-02 ||Ae||/||Ax|| 1.000000000000e+00
     1 KSP preconditioned resid norm 2.803983807098e-04 true resid  
norm 2.803510287777e-04 ||Ae||/||Ax|| 6.297569256276e-03
     2 KSP preconditioned resid norm 7.431745416397e-06 true resid  
norm 7.503178949264e-06 ||Ae||/||Ax|| 1.685450988589e-04
     3 KSP preconditioned resid norm 4.979479399481e-06 true resid  
norm 6.953502179385e-06 ||Ae||/||Ax|| 1.561976241646e-04
     4 KSP preconditioned resid norm 6.971503353270e-06 true resid  
norm 6.971503353270e-06 ||Ae||/||Ax|| 1.566019782331e-04
     5 KSP preconditioned resid norm 4.069146442021e-08 true resid  
norm 8.315442414641e-07 ||Ae||/||Ax|| 1.867910941655e-05
Residual norm 8.31544e-07
Error norm 0.000372035
Error norm 9.54604e-05
Error norm 1.38928e-07

   Part of the problem is coming from the gmres not generating a good  
orthonormal basis of directions, so I ran
[barry-smiths-macbook-pro-17:ksp/examples/tutorials] bsmith% ./ex34 - 
ksp_monitor_true_residual -dmmg_nlevels 6 -ksp_type fgmres - 
ksp_gmres_modifiedgramschmidt
     0 KSP preconditioned resid norm 4.451733827591e-02 true resid  
norm 4.451733827591e-02 ||Ae||/||Ax|| 1.000000000000e+00
     1 KSP preconditioned resid norm 2.803631359711e-04 true resid  
norm 2.803224197123e-04 ||Ae||/||Ax|| 6.296926643699e-03
     2 KSP preconditioned resid norm 6.318659416138e-06 true resid  
norm 6.395913715096e-06 ||Ae||/||Ax|| 1.436724269297e-04
     3 KSP preconditioned resid norm 5.441804660222e-06 true resid  
norm 5.559954843193e-06 ||Ae||/||Ax|| 1.248941407539e-04
     4 KSP preconditioned resid norm 5.441734174383e-06 true resid  
norm 5.580118795478e-06 ||Ae||/||Ax|| 1.253470836673e-04
     5 KSP preconditioned resid norm 4.763217020809e-06 true resid  
norm 4.965124844603e-06 ||Ae||/||Ax|| 1.115323830163e-04
     6 KSP preconditioned resid norm 1.149661784439e-06 true resid  
norm 1.742098334034e-06 ||Ae||/||Ax|| 3.913302862202e-05
     7 KSP preconditioned resid norm 1.129046836468e-06 true resid  
norm 1.607488229638e-06 ||Ae||/||Ax|| 3.610926069086e-05
     8 KSP preconditioned resid norm 1.129005681832e-06 true resid  
norm 1.682757897470e-06 ||Ae||/||Ax|| 3.780005499721e-05
     9 KSP preconditioned resid norm 9.863339300864e-07 true resid  
norm 1.657332745708e-06 ||Ae||/||Ax|| 3.722892870428e-05
    10 KSP preconditioned resid norm 3.080494650476e-07 true resid  
norm 1.298029587815e-06 ||Ae||/||Ax|| 2.915784352808e-05
Residual norm 1.29803e-06
Error norm 0.00100342
Error norm 0.000631685
Error norm 6.85996e-07

It looks like using the modified gram-schmidt orthogonalization helps  
(that is it takes less iterations than with the classical (default) BUT
looking more closely we see that it actually terminated too early and  
the answer is crappy.

So lets turn off ALL the GMRES stuff
[barry-smiths-macbook-pro-17:ksp/examples/tutorials] bsmith% ./ex34 - 
ksp_monitor_true_residual -dmmg_nlevels 6 -ksp_type richardson - 
mg_levels_ksp_type richardson
     0 KSP preconditioned resid norm 3.698072814941e+02 true resid  
norm 4.451733827591e-02 ||Ae||/||Ax|| 1.000000000000e+00
     1 KSP preconditioned resid norm 4.145919799805e+01 true resid  
norm 9.632258675992e-03 ||Ae||/||Ax|| 2.163709551096e-01
     2 KSP preconditioned resid norm 4.848244190216e+00 true resid  
norm 1.894293935038e-03 ||Ae||/||Ax|| 4.255182296038e-02
     3 KSP preconditioned resid norm 5.902504324913e-01 true resid  
norm 3.106203221250e-04 ||Ae||/||Ax|| 6.977513432503e-03
     4 KSP preconditioned resid norm 7.403665781021e-02 true resid  
norm 4.669937698054e-05 ||Ae||/||Ax|| 1.049015438184e-03
     5 KSP preconditioned resid norm 9.483236819506e-03 true resid  
norm 6.756537914043e-06 ||Ae||/||Ax|| 1.517731725471e-04
     6 KSP preconditioned resid norm 1.249413820915e-03 true resid  
norm 1.256137807104e-06 ||Ae||/||Ax|| 2.821682210197e-05
Residual norm 1.25614e-06
Error norm 0.000369181
Error norm 9.46438e-05
Error norm 1.37756e-07

The final residual doesn't look that great but if you look at the  
error norms you see that the solution has fully converged.

This final run is actually classical multigrid. So it looks like that  
is the way to go; BUT you have to be very careful with how
you decide when it has converged. Obviously you want to stop as soon  
as the algebraic error is smaller than the truncation error.
But, sadly we don't know what the algebraic error is (we only know the  
residual) and we do not know what the truncation error is.

My hack convergence criteria would be to stop as soon as the residual  
norm DOES NOT decrease by at least (say) 20%

To do a little better we can use SOR as the smoother instead of ILU
[barry-smiths-macbook-pro-17:ksp/examples/tutorials] bsmith% ./ex34 - 
ksp_monitor_true_residual -dmmg_nlevels 6 -ksp_type richardson  - 
mg_levels_ksp_type richardson  -mg_levels_pc_type sor
     0 KSP preconditioned resid norm 3.202437744141e+02 true resid  
norm 4.451733827591e-02 ||Ae||/||Ax|| 1.000000000000e+00
     1 KSP preconditioned resid norm 1.194643592834e+01 true resid  
norm 2.088646404445e-03 ||Ae||/||Ax|| 4.691759496927e-02
     2 KSP preconditioned resid norm 4.755175709724e-01 true resid  
norm 1.131777826231e-04 ||Ae||/||Ax|| 2.542330417782e-03
     3 KSP preconditioned resid norm 1.988304778934e-02 true resid  
norm 6.019943612046e-06 ||Ae||/||Ax|| 1.352269464405e-04
     4 KSP preconditioned resid norm 8.689521928318e-04 true resid  
norm 8.886481737136e-07 ||Ae||/||Ax|| 1.996184437303e-05
Residual norm 8.88648e-07
Error norm 0.000368883
Error norm 9.4803e-05
Error norm 1.37996e-07

Hope this helps,

   Barry





On Dec 2, 2008, at 7:52 PM, Chris Cooper wrote:

> Hi,
>
> Just wondering if there are any techniques for improving DMMG  
> convergence on a linear problem when operating in single precision?
>
> eg using ksp example 34 in Petsc 2.3.3 with multigrid levels changed  
> to 6...
>
> double precision:
> 0 KSP Residual norm 4.451733404502e-02
> 1 KSP Residual norm 2.801676364873e-04
> 2 KSP Residual norm 3.215426166316e-06
> 3 KSP Residual norm 5.446277176355e-08
> Residual norm 5.44628e-08
> Error norm 0.000356798
> Error norm 9.21724e-05
> Error norm 1.34208e-07
>
> single precision:
> 0 KSP Residual norm 4.450287297368e-02
> 1 KSP Residual norm 2.799208450597e-04
> 2 KSP Residual norm 2.232483529951e-05
> 3 KSP Residual norm 1.575733222126e-05
> 4 KSP Residual norm 1.286925271415e-05
> ...
> 27 KSP Residual norm 4.369261660031e-06
> 28 KSP Residual norm 4.287577212381e-06
> 29 KSP Residual norm 4.210308816255e-06
> 30 KSP Residual norm 2.235170904896e-05
> 31 KSP Residual norm 1.709007477757e-07
> Residual norm 8.45584e-07
> Error norm 0.000370784
> Error norm 9.54664e-05
> Error norm 1.38935e-07
>
> thanks,
> Chris Cooper
> Animal Logic
>
> Animal Logic
> http://www.animallogic.com
>
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