[petsc-users] Iterative solution for schur complement

[Umut Tabak]

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

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?

+ C matrix can be relatively sparse in comparison to A and B which is a 
coupling matrix, I am not sure if I can make use of this information or not?

+ Are there any SPD preconditioners that I can try with this system? I 
guess the above mentioned break down is also related to the selection of 
the preconditioner although I am not sure on this.

Any other pointers and ideas are appreciated on this problem.

Best regards,
Umut