[petsc-users] Iterative solution for schur complement

[Jed Brown]

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

> Dear all,
>
> I am looking at a system where I am trying to investigate this 
> ill-conditioned problem with some iterative tricks or not. Namely, the 
> system that I try to solve is
>
> (B - C^T A^{-1}C) x2 = b2
>
> which results from block symmetric representation
>
> A  C
> C^T B

What physics do you have here?

> Unfortunately, B is indefinite, I tried some tries in MATLAB but none of 
> them gave convergence. What I tried is listed below:
>
> + Even if B is indefinite, it is symmetric, and I am trying to use its 
> LDLT decomposition as a preconditioner for the above system. Besides, I 
> am also using the LDLT for the matrix vector multiplications which comes 
> from A^{-1} related terms. I am not forming the operator matrix here but 
> it is a function handle that represents the matrix-vector 
> multiplication. Also the preconditioner solve related to B2 is also a 
> function handle. Of course in this case, my preconditioner B is not SPD 
> however it is the direct factor of B. Due to this reason, I was 
> expecting to get better results while using also the direct 
> factorization of A2.
>
> + The strange thing in MATLAB is that CG fails on the very first 
> iteration due to the reason that some parameters are too small to 
> continue, I can understand this since that implicitly boils down to the 
> Cholesky decomposition of the projected system. But minres also fails 
> with the same error. I am hesitating whether I shall program minres or 
> gmres myself to detect the source of the problem, any ideas on where the 
> problem might come from, especially while using minres?

Both CG and MINRES require an SPD preconditioner.  It sounds like B is a
poor approximation to the Schur complement S = B - C^T A^{-1} C.
Depending on your application area, there are a few classes of
preconditioners that you might consider.  These include the
least-squares commutator, physics-based approximate commutator,
SIMPLE(R), and DD and multigrid methods applied directly to the
indefinite problem.
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