[petsc-users] KSPSolve convergence and norms

[Torquil Macdonald Sørensen]

On 25/10/13 18:59, Matthew Knepley wrote:
> On Fri, Oct 25, 2013 at 11:33 AM, Torquil Macdonald Sørensen 
> <torquil at gmail.com <mailto:torquil at gmail.com>> wrote:
>
>     Hi!
>
>     I'm using PETSc 3.4.3, with only default runtime settings, to
>     solve Ax=b, for complex numbers (I have used
>     --with-scalar-type=complex).
>
>     After reading sections 4.3.1, 4.3.2, 4.3.3 in the manual, I
>     decided to investigate the convergence in my program, since I'm
>     having some problems when the system is larger. So, I'm running it
>     with the options
>
>     -ksp_monitor_true_residual -ksp_converged_reason -ksp_view
>
>     The output shows that the relative tolerance required for
>     convergence is 1e-5, but none of the reported norms are that
>     small. Despite this, convergence is reported (CONVERGED_RTOL).
>
>     I though that CONVERGED_RTOL should indicate that ||r||_2/||b||_2
>     < 1e-5, where r is the preconditioned residual?
>
>     Here is the output of my program (which includes some code to
>     compute the l_2 norm of b, just before KSPSolve is run):
>
>     *************************************************************
>     Norm of b: 0.179829
>       0 KSP preconditioned resid norm 2.508789303280e+04 true resid
>     norm 1.798290175843e-01 ||r(i)||/||b|| 1.000000000000e+00
>       1 KSP preconditioned resid norm 1.856991132478e+00 true resid
>     norm 3.186462510386e-01 ||r(i)||/||b|| 1.771940120227e+00
>       2 KSP preconditioned resid norm 1.704678606994e-01 true resid
>     norm 4.004071321891e-02 ||r(i)||/||b|| 2.226599119363e-01
>
>
> So the relative preconditioned residual is:
>
>   1.7e-01 / 2.5e+04 < 1.0e-5
>
> while the relative true residual is 2.2e-1. This happens because you 
> are using ILU(0). It can be really
> crappy, and in this case the preconditioner is very ill-conditioned. I 
> can't imagine a problem where I
> would recommend using ILU, but its our default because its the only 
> black-box PC that exists.
>
>   Thanks
>
>      Matt
>

 From your explanation I can see that I had misunderstood the meaning of 
"relative preconditioned residual norm", but now I understand. Thanks 
for clearing it up!

Best regards
Torquil Sørensen
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