# [petsc-dev] Deflated Krylov solvers for PETSc

Matthew Knepley knepley at gmail.com
Sun Mar 3 17:42:50 CST 2013

```On Sun, Mar 3, 2013 at 5:15 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> This paper acknowledges the MG terminology and includes some numerical
> examples.
>
> http://dx.doi.org/10.1007/s10915-009-9272-6
>
> Unfortunately, they only solve heterogenous Poisson, for which all the
> deflation algorithms look like crude hacks next to MG (which they don't
> show results for).
>
> Note that in this paper, all the methods use the coarse operator E = Z^T A
> Z where A is the original operator, not a preconditioned operator. That
> makes these deflation methods merely V(0,1) or V(1,0) cycles. In
> particular, I don't see anything with a coarse operator E = Z^T (M^{-1/2} A
> M^{-1/2}) Z or E = W^T (M^{-1} A) W. If this is indeed true, then I think
> it's clear that deflation is something that should be implemented as a PC,
> perhaps with Z updated by KSP (if we intend to iteratively compute
> approximate low eigenvectors).
>

Form the explanation in this paper, as far as I understand it, deflation is
just used to augment the Krylov space,
so why not just use LGMRES, sticking in your approximations to the
eigenspace?

Matt

> On Sun, Mar 3, 2013 at 12:52 AM, Jie Chen <jiechen at mcs.anl.gov> wrote:
>
>> Barry,
>>
>> Putting P_{A} either to the left or to the right of A means the same
>> thing: we avoid touching the subspace spanned by W. This is why the
>> orthogonality condition b-Ax_0 \perp W is needed. What the condition says
>> is that the initial residual should have no components on W, presumably the
>> troublesome part of A. Then in a Krylov method, all later residuals have no
>> components on W. In other words, the part of the solution on the subspace W
>> is already fully computed even in the 0-th step. Getting such an x_0 is not
>> difficult; the difficult part is to define/compute W.
>>
>> When one adds another preconditioner M to the system, presumably W should
>> be the troublesome part of AM instead of A (I am always confused about the
>> notation M and M^{-1} but it does not affect my reasoning here). In the
>> aggressive way W can consist of eigenvectors of the pencil (A,M)
>> corresponding to the smallest eigenvalues in magnitude. On the other hand,
>> if one already found a good W for A and he/she is lazy and does not want to
>> re-figuring out W, then just use the old W. I guess the beauty of the
>> deflation theory is that W can be arbitrary but does not depend on A or M.
>>
>> I sense that you want a preconditioner appearing in (4) and the formula
>> for x_0? I will add something there later.
>>
>
>
>

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