[petsc-users] [PATCH 2/2] Call KSPSetFromOptions before setting fieldsplit-specific options

[Jed Brown]

Umut Tabak <u.tabak at tudelft.nl> writes:

> On 09/08/2013 06:17 PM, Jed Brown wrote:
>> block is indefinite, even worse.
>> A11 indefinite could lead to the Schur complement being singular.  The
>> approach breaks down if that's the case.  A11 singular or negative
>> semi-definite is the standard case.
> Even not singular, an already ill-conditioned problem becomes worse with 
> a Schur complement approach.

I don't know what you're trying to say, but using factorization makes
sense when you set the problem up so that the Schur complement is better
behaved than the original problem (either because it's better
conditioned or because you have a practical preconditioner for it).  Of
course you can choose bad splittings so that the Schur complement is
worse (or singular).

Note that the Schur complement can be completely different from what was
in that block of the matrix before factorization.

> And it is almost impossible to define a SPD preconditioner that could 
> work with S to result in decent iteration counts.
>
> This was a question I asked a couple of weeks ago but did not have the 
> time to go into details.
>
> I guess these kinds of separation ideas and use of independent 
> factorizations or combined preconditioners for different blocks will not 
> work for this problem...

Possible, or you're doing it wrong.

>> You need to read the literature for problems of your type.  There are a
>> few possible approaches to approximating the Schur complement.  The
>> splitting into blocks also might not be very good.
> well I took a look at the literature but could not really find something 
> useful up until now. The main difference was one of the blocks was 
> either SPD or well-conditioned(for A00 or A11).
>
> More  specifically, my problem is a kind of shifted problem for 
> eigenvalue solutions without going into too much detail. The symmetric 
> operator matrix is written as
>
> AA
> =
> (A-\sigmaB) C
> C^T (D-\sigmaE)
>
> and C is a rather sparse coupling matrix
>
> A00 = A-\sigmaB block is ill conditioned due to the shift
>
> and
>
> A11 = (D-\sigmaE)
>
> block is indefinite where D is a singular matrix with one zero 
> eigenvalue with 1 vector in the null space. E is rather well conditioned 
> but in combination it is not attackable by iterative methods. Moreover, 
> Schur complements are defined on this problematic system, AA.

Take a small problem size and use SVD to compute the extreme singular
values of S = A11 - C^T A00^{-1} C.  Also get an estimate of the
spectrum so you can tell whether it's indefinite.

> There is some literature on these shifted problems but I am not sure if 
> I should dive into that field or not at this point.
>
> Trying the 'fieldsplit' approach was some kind of a 'what-if' for me...
>
> But thanks for the help and comments anyway.
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