[petsc-users] Symmetric non-positive definite system

[hong at aspiritech.org]

You may try MINRES (-ksp_type minres)
See http://web.stanford.edu/group/SOL/reports/SOL-2011-2R.pdf

Hong

On Fri, Jul 18, 2014 at 1:22 PM, Matthew Knepley <knepley at gmail.com> wrote:
> On Fri, Jul 18, 2014 at 1:19 PM, Chetan Jhurani <chetan.jhurani at gmail.com>
> wrote:
>>
>> > From: Jed Brown <jed at jedbrown.org>
>> >
>> > Jozsef Bakosi <jbakosi at lanl.gov> writes:
>> >
>> > > Hi folks,
>> > >
>> > > I have a matrix as a result of a finite element discretization of the
>> > > Poisson
>> > > operator and associated right hand side. As it turns out, the matrix
>> > > is
>> > > symmetric but not positive definite since it has at least two negative
>> > > small
>> > > eigenvalues. I have been solving this system without problem using the
>> > > conjugate
>> > > gradients (CG) algorithm with ML as a preconditioner, but I'm
>> > > wondering why it
>> > > works.
>> >
>> > Is the preconditioned matrix
>> >
>> >   P^{-1/2} A P^{-T/2}
>> >
>> > positive definite?
>>
>> This does not answer the original question regarding CG/ML, but
>
>
> It is possible for CG to convergence with an indefinite matrix, but it is
> also
> possible for it to fail.
>
>   Matt
>
>>
>> will reduce some debugging work.  P^{-1/2} A P^{-T/2} cannot be
>> positive definite since A is not positive definite.  This is due
>> to Sylvester's inertia theorem.
>>
>> http://books.google.com/books?id=P3bPAgAAQBAJ&pg=PA202
>> Applied Numerical Linear Algebra By James W. Demmel, p 202
>>
>> Chetan
>>
>> > > Shouldn't CG fail for a non-positive-definite matrix? Does PETSc do
>> > > some
>> > > additional magic if it detects that the dot-product in the CG
>> > > algorithm is
>> > > negative? Does it solve the system using the normal equations, A'A,
>> > > instead?
>> >
>> > CG will report divergence in case a direction of negative curvature is
>> > found.
>>
>
>
>
> --
> What most experimenters take for granted before they begin their experiments
> is infinitely more interesting than any results to which their experiments
> lead.
> -- Norbert Wiener