[petsc-dev] Deflated Krylov solvers for PETSc

[Jie Chen]

Mark, Barry, Jed,

You guys raised quite a few questions, meanwhile I found a mistake in my note. I think I should rewrite the note to incorporate a more comprehensive view. Hopefully the new note will answer you questions (at least to a certain degree). I have something very important to finish before Wednesday, so my participation in the discussion may delay. See the new note in the old place:

http://www.mcs.anl.gov/~jiechen/tmp/deflate.pdf

Jie




----- Original Message -----
From: "Mark F. Adams" <mark.adams at columbia.edu>
To: "For users of the development version of PETSc" <petsc-dev at mcs.anl.gov>
Sent: Saturday, March 2, 2013 12:54:34 PM
Subject: Re: [petsc-dev] Deflated Krylov solvers for PETSc


On Mar 2, 2013, at 12:45 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:

> 
> On Mar 2, 2013, at 11:16 AM, "Mark F. Adams" <mark.adams at columbia.edu> wrote:
> 
>>> 
>>> An alternative description for the method in the paper is "two-level unsmoothed aggregation applied to the ASM-preconditioned operator".
>>> 
>> 
>> Isn't this paper just doing two level unsmoothed aggregation with a V(0,k) cycle, where k (and the smoother) is K in the paper?
>> 
>> It looks to me like you are moving the multigrid coarse grid correction (projection) from the PC into the operator, in which case this method is identical to two level MG with the V(0,k) cycle.
> 
>  Well, its not "identical" is it?

My thinking is that with Q = P (RAP)^-1 R, and smoother S, then a two level V(0,1) PC (M) looks like: S (I - Q). (I think I'm missing an A in here, before the Q maybe) The preconditioned system being solved looks like:

MAx = Mb

with MG PC we have:

(S (I-Q)) A x = …

Move the brackets around

S ((I-Q)A)  x = ..

S is K in the paper and (I-Q)A is the new operator.  The preconditioned system does not have a null space and thats all that matters.

Anyway, I'm sure its not identical, I'm just not seeing it in this paper and don't have the patients work through it … and maybe I'm having a bad arithmetic day but on page 935, the definition of P (first equation on page) and Pw (last equation on page) don't look consistent.


> In the deflation approach it introduces a null space into the operator, in the multigrid approach it does not. So identical only in analogy? So what I am interested in EXACTLY how are they related, relationship between eigenvalues or convergence rate.
> 
>   Barry
> 
>> 
>> I'm sure I'm missing something.  Jie's writeup has an orthogonality condition on the restriction operator, which I don't see in the Vuik paper.
> 
>