[petsc-dev] KSP convergence test based on error on residual

[Mark F. Adams]

On Jul 16, 2011, at 5:17 PM, Jed Brown wrote:

> On Sat, Jul 16, 2011 at 16:11, Barry Smith <bsmith at mcs.anl.gov> wrote:
> 
>   Jed,
> 
>     I just remembered something I should have said when we talked about the rewrite of the KSPSPECEST stuff.
> 
>     If one has the bound on the smallest eigenvalue of the (preconditioned) operator then ones convergence test can take that into account and know that the 2-norm of the error of the linear solver (as opposed to the 2 norm of the residual) is less than some tolerance. Of course the KSPSPECEST can give us this information, even though Mark doesn't believe it is accurate enough :-), with enough iterations it can be.
> 
> I would expect it to become a good approximation when the KSP has converged on the low-frequency modes. I think Mark's

I would too, but this would be an experiment that I'd like to see.  For instance, solve a largish 5 point stencil Laplacian (or whatever) with a simple solver (and multilevel solver if you get ambitious) and plot the history of the lowest (and highest) eigen estimates vs. iteration number.  

> objection is that a few iterations are usually nowhere near actually converging, so you can only use the estimate in a meaningful way to estimate the high end of the spectrum.

Yes, thats right.  My comments on accuracy were in the context of eigen estimates for smoothers, where getting the lowest bound would be tantamount to solving the systems with the smoother (ie, not practical), an its not really needed for smoothers (but it would be nice if it were around).

Mark

>  
> So ideally we'd have options that allowed convergence tests to use the estimate of the error norm instead of just the residual norm.
> 
> How would you suggest handling restarts?

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