[petsc-dev] [petsc-users] Any suggestion for this kinds of matrix?

[Jed Brown]

On Fri, Nov 11, 2011 at 11:17, Mark F. Adams <mark.adams at columbia.edu>wrote:

> All I recall was that I was confused because the residual dropped 10
> orders of magnitude, apparently, in the first iteration and Barry explained
> the issue, I added 'unpreconditioned' residual and it was fine.


My recollection with that problem was that the initial guess "almost"
satisfied the PDE in the interior, but did not satisfy the boundary
conditions. The preconditioned residual made the boundaries and interior of
roughly equal importance, so the initial preconditioned residual looked
large due to the big disagreement near boundaries. In unpreconditioned
residuals, you never see the boundary equations in the norm, but they
happen to be satisfied because the preconditioner tends to fix them. Or
something like this, but with ksp_diagonal_scale, the residual drop was
similar for preconditioned and unpreconditioned residuals.

I want methods to be scale invariant, but I don't think it's practical to
ask algorithms to behave the same way if you scale some equations very
differently from others. That changes the conditioning of the operator and
makes dot products with the operator meaningless. I think it's achievable
to ask people to scale their equations properly if they want good
performance. The questions is which causes less confusion when people mess
up. I'm not sold either way regarding preconditioned or unpreconditioned
residuals.
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