[petsc-users] Algorithms to remove null spaces in a singular system

[Fande Kong]

On Wed, Oct 12, 2016 at 9:12 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:

>
> > On Oct 12, 2016, at 10:00 PM, Amneet Bhalla <mail2amneet at gmail.com>
> wrote:
> >
> >
> >
> > On Monday, October 10, 2016, Barry Smith <bsmith at mcs.anl.gov> wrote:
> >
> > > On Oct 10, 2016, at 4:01 PM, Kong, Fande <fande.kong at inl.gov> wrote:
> > >
> > > Hi All,
> > >
> > > I know how to remove the null spaces from a singular system using
> creating a MatNullSpace and attaching it to Mat.
> > >
> > > I was really wondering what is the philosophy behind this? The exact
> algorithms we are using in PETSc right now?  Where we are dealing with
> this, preconditioner, linear solver, or nonlinear solver?
> >
> >    It is in the Krylov solver.
> >
> >    The idea is very simple. Say you have   a singular A with null space
> N (that all values Ny are in the null space of A. So N is tall and skinny)
> and you want to solve A x = b where b is in the range of A. This problem
> has an infinite number of solutions     Ny + x*  since A (Ny + x*) = ANy +
> Ax* = Ax* = b where x* is the "minimum norm solution; that is Ax* = b and
> x* has the smallest norm of all solutions.
> >
> >       With left preconditioning   B A x = B b GMRES, for example,
> normally computes the solution in the as alpha_1 Bb   + alpha_2 BABb +
> alpha_3 BABABAb + ....  but the B operator will likely introduce some
> component into the direction of the null space so as GMRES continues the
> "solution" computed will grow larger and larger with a large component in
> the null space of A. Hence we simply modify GMRES a tiny bit by building
> the solution from alpha_1 (I-N)Bb   + alpha_2 (I-N)BABb + alpha_3
> (I-N)BABABAb + ....  that is we remove from each new direction anything in
> the direction of the null space. Hence the null space doesn't directly
> appear in the preconditioner, just in the KSP method.   If you attach a
> null space to the matrix, the KSP just automatically uses it to do the
> removal above.
> >
> > Barry, if identity matrix I is of size M x M (which is also the size of
> A) then are you augmenting N (size M x R; R < M) by zero colums to make I
> - N possible? If so it means that only first R values of vector Bb are used
> for scaling zero Eigenvectors of A. Does this choice affect iteration
> count, meaning one can arbitrarily choose any R values of the vector Bb to
> scale zero eigenvectors of A?
>
>   Yes I wasn't very precise here. I should have written it as as the
> projection of the vector onto the complement of the null space which I
> think can be written as I - N(N'N)^-1N'
> This is done with the routine MatNullSpaceRemove(). The basis of the null
> space you provide to MatNullSpaceCreate() should not effect the convergence
> of the Krylov method at all since the same null space is removed regardless
> of the basis.
>

I think we need to make sure that the basis vectors are orthogonal to each
other and they are normalized. Right?

Fande,


>
>   Barry
>
>
> >
> >     With right preconditioning the solution is built from alpha_1 b   +
> alpha_2 ABb + alpha_3 ABABb + .... and again we apply (I-N) to each term to
> remove any part that is in the null space of A.
> >
> >    Now consider the case   A y = b where b is NOT in the range of A. So
> the problem has no "true" solution, but one can find a least squares
> solution by rewriting b = b_par + b_perp where b_par is in the range of A
> and b_perp is orthogonal to the range of A and solve instead    A x =
> b_perp. If you provide a MatSetTransposeNullSpace() then KSP automatically
> uses it to remove b_perp from the right hand side before starting the KSP
> iterations.
> >
> >   The manual pages for MatNullSpaceAttach() and
> MatTranposeNullSpaceAttach() discuss this an explain how it relates to the
> fundamental theorem of linear algebra.
> >
> >   Note that for symmetric matrices the two null spaces are the same.
> >
> >   Barry
> >
> >
> >    A different note: This "trick" is not a "cure all" for a totally
> inappropriate preconditioner. For example if one uses for a preconditioner
> a direct (sparse or dense) solver or an ILU(k) one can end up with a very
> bad solver because the direct solver will likely produce a very small pivot
> at some point thus the triangular solver applied in the precondition can
> produce HUGE changes in the solution (that are not physical) and so the
> preconditioner basically produces garbage. On the other hand sometimes it
> works out ok.
> >
> >
> > >
> > >
> > > Fande Kong,
> >
> >
> >
> > --
> > --Amneet
> >
> >
> >
> >
>
>
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