[petsc-users] schur complement and condition number

[Jed Brown]

On Thu, Mar 31, 2011 at 13:44, Gong Ding <gdiso at ustc.edu> wrote:

> This afternoon, I installed slepc for computing the singualr value
> (condition number) of the jacobian matrix.
> However, slepc SVD solver broken for smallest singualr value.
> Fortunately, superlu gives an approximate condition number, which in the
> order of 1e22.
> If I remove the connection, condition number is in the order of 1e10.
>

A condition number larger than 1e12 or so usually indicates that the matrix
is singular. You could compile PETSc (dev) using gcc-4.6 and configuring
--with-precision=__float128. External packages won't work this way, but you
can use KSP to estimate your singular values more accurately and to solve a
nearly singular system.


>
> Is it possible to use Tikhonov regularization (A^TA + sigma I)x = A^Tb with
> this problem?
> Of course, I need to do it during the nonlinear iteration. That is a
> Newton-Tikhonov method.
>

Sure, you can regularize the Jacobian however you like when using SNES.


> Or I have to use direct solver without other choice?
>

Was the direct solver producing a good solution? With such an
ill-conditioned system, the solution could be incorrect in the first digit.
I would at least check with __float128.
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