[petsc-users] Solving singular systems with petsc

[zakaryah]

Hi Nidish,

I may not fully understand your problem, but it sounds like you could
benefit from continuation methods. Have you looked into this? If it's
helpful, I have some experience with this and I can discuss with you by
email.

Cheers, Zak

On Mon, Aug 17, 2020 at 2:31 PM Nidish <nb25 at rice.edu> wrote:

> Thankfully for this step, the matrix is not dense. But thank you.
>
> Nidish
>
> On 8/17/20 8:52 AM, Jose E. Roman wrote:
> >
> >> El 17 ago 2020, a las 14:27, Barry Smith <bsmith at petsc.dev> escribió:
> >>
> >>
> >>   Nidish,
> >>
> >>      Your matrix is dense, correct? MUMPS is for sparse matrices.
> >>
> >>      Then I guess you could use Scalapack
> http://netlib.org/scalapack/slug/node48.html#SECTION04323200000000000000
> to do the SVD. The work is order N^3 and parallel efficiency may not be
> great but it might help you solve your problem.
> >>
> >>       I don't know if SLEPc has an interface to Scalapack for SVD or
> not.
> > Yes, SLEPc (master) has interfaces for ScaLAPACK and Elemental for both
> SVD and (symmetric) eigenvalues.
> >
> > Jose
> >
> >>     Barry
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>> On Aug 17, 2020, at 2:51 AM, Jose E. Roman <jroman at dsic.upv.es> wrote:
> >>>
> >>> You can use SLEPc's SVD to compute the nullspace, but it has pitfalls:
> make sure you use an absolute convergence test (not relative); for the
> particular case of zero singular vectors, accuracy may not be very good and
> convergence may be slow (with the corresponding high computational cost).
> >>>
> >>> MUMPS has functionality to get a basis of the nullspace, once you have
> computed the factorization. But I don't know if this is easily accessible
> via PETSc.
> >>>
> >>> Jose
> >>>
> >>>
> >>>
> >>>> El 17 ago 2020, a las 3:10, Nidish <nb25 at rice.edu> escribió:
> >>>>
> >>>> Oh damn. Alright, I'll keep trying out the different options.
> >>>>
> >>>> Thank you,
> >>>> Nidish
> >>>>
> >>>> On 8/16/20 8:05 PM, Barry Smith wrote:
> >>>>> SVD is enormously expensive, needs to be done on a full dense matrix
> so completely impractical. You need the best tuned iterative method, Jose
> is the by far the most knowledgeable about that.
> >>>>>
> >>>>>   Barry
> >>>>>
> >>>>>
> >>>>>> On Aug 16, 2020, at 7:46 PM, Nidish <nb25 at rice.edu> wrote:
> >>>>>>
> >>>>>> Thank you for the suggestions.
> >>>>>>
> >>>>>> I'm getting a zero pivot error for the LU in slepc while
> calculating the rest of the modes.
> >>>>>>
> >>>>>> Would conducting an SVD for just the stiffness matrix and then
> using the singular vectors as bases for the nullspace work? I haven't tried
> this out just yet, but I'm wondering if you could provide me insights into
> whether this will.
> >>>>>>
> >>>>>> Thanks,
> >>>>>> Nidish
> >>>>>>
> >>>>>> On 8/16/20 2:50 PM, Barry Smith wrote:
> >>>>>>> If you know part of your null space explicitly (for example the
> rigid body modes) I would recommend you always use that information
> explicitly since it is extremely expensive numerically to obtain. Thus
> rather than numerically computing the entire null space compute the part
> orthogonal to the part you already know. Presumably SLEPc has tools to help
> do this, naively I would just orthogonalized against the know subspace
> during the computational process but there are probably better ways.
> >>>>>>>
> >>>>>>>   Barry
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>>> On Aug 16, 2020, at 11:26 AM, Nidish <nb25 at rice.edu> wrote:
> >>>>>>>>
> >>>>>>>> Well some of the zero eigenvectors are rigid body modes, but
> there are some more which are introduced by lagrange-multiplier based
> constraint enforcement, which are non trivial.
> >>>>>>>>
> >>>>>>>> My final application is for a nonlinear simulation, so I don't
> mind the extra computational effort initially. Could you have me the
> suggested solver configurations to get this type of eigenvectors in slepc?
> >>>>>>>>
> >>>>>>>> Nidish
> >>>>>>>> On Aug 16, 2020, at 00:17, Jed Brown <jed at jedbrown.org> wrote:
> >>>>>>>> It's possible to use this or a similar algorithm in SLEPc, but
> keep in mind that it's more expensive to compute these eigenvectors than to
> solve a linear system.  Do you have a sequence of systems with the same
> null space?
> >>>>>>>>
> >>>>>>>> You referred to the null space as "rigid body modes".  Why can't
> those be written down?  Note that PETSc has convenience routines for
> computing rigid body modes from coordinates.
> >>>>>>>>
> >>>>>>>> Nidish <
> >>>>>>>> nb25 at rice.edu
> >>>>>>>>> writes:
> >>>>>>>>
> >>>>>>>> I just use the standard eigs function (
> https://www.mathworks.com/help/matlab/ref/eigs.html
> >>>>>>>> ) as a black box. I think it uses a lanczos type method under the
> hood.
> >>>>>>>>
> >>>>>>>> Nidish
> >>>>>>>>
> >>>>>>>> On Aug 15, 2020, 21:42, at 21:42, Barry Smith <
> >>>>>>>> bsmith at petsc.dev
> >>>>>>>>> wrote:
> >>>>>>>>
> >>>>>>>> Exactly what algorithm are you using in Matlab to get the 10
> smallest
> >>>>>>>> eigenvalues and their corresponding eigenvectors?
> >>>>>>>>
> >>>>>>>> Barry
> >>>>>>>>
> >>>>>>>>
> >>>>>>>>
> >>>>>>>> On Aug 15, 2020, at 8:53 PM, Nidish <nb25 at rice.edu
> >>>>>>>>> wrote:
> >>>>>>>> The section on solving singular systems in the manual starts with
> >>>>>>>>
> >>>>>>>> assuming that the singular eigenvectors are already known.
> >>>>>>>>
> >>>>>>>>
> >>>>>>>> I have a large system where finding the singular eigenvectors is
> not
> >>>>>>>>
> >>>>>>>> trivially written down. How would you recommend I proceed with
> making
> >>>>>>>> initial estimates? In MATLAB (with MUCH smaller matrices), I
> conduct an
> >>>>>>>> eigensolve for the first 10 smallest eigenvalues and take the
> >>>>>>>> eigenvectors corresponding to the zero eigenvalues from this. This
> >>>>>>>> approach doesn't work here since I'm unable to use SLEPc for
> solving
> >>>>>>>>
> >>>>>>>>
> >>>>>>>> K.v = lam*M.v
> >>>>>>>>
> >>>>>>>> for cases where K is positive semi-definite (contains a few "rigid
> >>>>>>>>
> >>>>>>>> body modes") and M is strictly positive definite.
> >>>>>>>>
> >>>>>>>>
> >>>>>>>> I'd appreciate any assistance you may provide with this.
> >>>>>>>>
> >>>>>>>> Thank you,
> >>>>>>>> Nidish
> >>>>>>>>
> >>>>>> --
> >>>>>> Nidish
> >>>> --
> >>>> Nidish
> --
> Nidish
>
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