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

```  Fande,

I have done some work, mostly understanding and documentation, on handling
singular systems with KSP in the branch barry/improve-matnullspace-usage. This
also includes a new example that solves both a symmetric example and an example
where nullspace(A) != nullspace(A') src/ksp/ksp/examples/tutorials/ex67.c  ```
```
My understanding is now documented in the manual page for KSPSolve(), part
of this is quoted below:

-------
If you provide a matrix that has a MatSetNullSpace() and
MatSetTransposeNullSpace() this will use that information to solve singular
systems
in the least squares sense with a norm minimizing solution.
\$
\$                   A x = b   where b = b_p + b_t where b_t is not in the range
of A (and hence by the fundamental theorem of linear algebra is in the
nullspace(A') see MatSetNullSpace()
\$
\$    KSP first removes b_t producing the linear system  A x = b_p (which has
multiple solutions) and solves this to find the ||x|| minimizing solution (and
hence
\$    it finds the solution x orthogonal to the nullspace(A). The algorithm is
simply in each iteration of the Krylov method we remove the nullspace(A) from
the search
\$    direction thus the solution which is a linear combination of the search
directions has no component in the nullspace(A).
\$
\$    We recommend always using GMRES for such singular systems.
\$    If nullspace(A) = nullspace(A') (note symmetric matrices always satisfy
this property) then both left and right preconditioning will work
\$    If nullspace(A) != nullspace(A') then left preconditioning will work but
right preconditioning may not work (or it may).

Developer Note: The reason we cannot always solve  nullspace(A) !=
nullspace(A') systems with right preconditioning is because we need to remove
at each iteration
the nullspace(AB) from the search direction. While we know the
nullspace(A) the nullspace(AB) equals B^-1 times the nullspace(A) but except
for trivial preconditioners
such as diagonal scaling we cannot apply the inverse of the
preconditioner to a vector and thus cannot compute the nullspace(AB).
------

Any feed back on the correctness or clarity of the material is appreciated. The
punch line is that right preconditioning cannot be trusted with nullspace(A) !=
nullspace(A') I don't see any fix for this.

Barry

> On Oct 11, 2016, at 3:04 PM, Kong, Fande <fande.k...@inl.gov> wrote:
>
>
>
> On Tue, Oct 11, 2016 at 12:18 PM, Barry Smith <bsm...@mcs.anl.gov> wrote:
>
> > On Oct 11, 2016, at 12:01 PM, Kong, Fande <fande.k...@inl.gov> wrote:
> >
> >
> >
> > On Tue, Oct 11, 2016 at 10:39 AM, Barry Smith <bsm...@mcs.anl.gov> wrote:
> >
> > > On Oct 11, 2016, at 9:33 AM, Kong, Fande <fande.k...@inl.gov> wrote:
> > >
> > > Barry, Thanks so much for your explanation. It helps me a lot.
> > >
> > > On Mon, Oct 10, 2016 at 4:00 PM, Barry Smith <bsm...@mcs.anl.gov> wrote:
> > >
> > > > On Oct 10, 2016, at 4:01 PM, Kong, Fande <fande.k...@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
> > >
> > >  Does "I" mean an identity matrix? Could you possibly send me a link for
> > > this GMRES implementation, that is, how PETSc does this in the actual
> > > code?
> >
> >    Yes.
> >
> >     It is in the helper routine KSP_PCApplyBAorAB()
> > #undef __FUNCT__
> > #define __FUNCT__ "KSP_PCApplyBAorAB"
> > PETSC_STATIC_INLINE PetscErrorCode KSP_PCApplyBAorAB(KSP ksp,Vec x,Vec
> > y,Vec w)
> > {
> >   PetscErrorCode ierr;
> >   PetscFunctionBegin;
> >   if (!ksp->transpose_solve) {
> >     ierr = PCApplyBAorAB(ksp->pc,ksp->pc_side,x,y,w);CHKERRQ(ierr);
> >     ierr = KSP_RemoveNullSpace(ksp,y);CHKERRQ(ierr);
> >   } else {
> >     ierr = PCApplyBAorABTranspose(ksp->pc,ksp->pc_side,x,y,w);CHKERRQ(ierr);
> >   }
> >   PetscFunctionReturn(0);
> > }
> >
> >
> > PETSC_STATIC_INLINE PetscErrorCode KSP_RemoveNullSpace(KSP ksp,Vec y)
> > {
> >   PetscErrorCode ierr;
> >   PetscFunctionBegin;
> >   if (ksp->pc_side == PC_LEFT) {
> >     Mat          A;
> >     MatNullSpace nullsp;
> >     ierr = PCGetOperators(ksp->pc,&A,NULL);CHKERRQ(ierr);
> >     ierr = MatGetNullSpace(A,&nullsp);CHKERRQ(ierr);
> >     if (nullsp) {
> >       ierr = MatNullSpaceRemove(nullsp,y);CHKERRQ(ierr);
> >     }
> >   }
> >   PetscFunctionReturn(0);
> > }
> >
> > "ksp->pc_side == PC_LEFT" deals with the left preconditioning Krylov
> > methods only? How about the right preconditioning ones? Are  they just
> > magically right for the right preconditioning Krylov methods?
>
>    This is a good question. I am working on a branch now where I will add
> some more comprehensive testing of the various cases and fix anything that
> comes up.
>
>    Were you having trouble with ASM and bjacobi only for right
> preconditioning?
>
>
> Yes. ASM and bjacobi works fine for left preconditioning NOT for RIGHT
> preconditioning. bjacobi converges, but produces a wrong solution. ASM needs
> more iterations, however the solution is right.
>
>
>
>     Note that when A is symmetric the range of A is orthogonal to null space
> of A so yes I think in that case it is just "magically right" but if A is not
> symmetric then I don't think it is "magically right". I'll work on it.
>
>
>    Barry
>
> >
> > Fande Kong,
> >
> >
> > There is no code directly in the GMRES or other methods.
> >
> > >
> > > (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.
> > >
> > >     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.
> > >
> > > What preconditioners  are appropriate? asm, bjacobi, amg? I have an
> > > example which shows  lu and ilu indeed work, but asm and bjacobi do not
> > > at all. That is why I am asking questions about algorithms. I am trying
> > > to figure out a default preconditioner for several singular systems.
> >
> >    Hmm, normally asm and bjacobi would be fine with this unless one or more
> > of the subblocks are themselves singular (which normally won't happen). AMG
> > can also work find sometimes.
> >
> >    Can you send a sample code?
> >
> >   Barry
> >
> > >
> > > Thanks again.
> > >
> > >
> > > Fande Kong,
> > >
> > >
> > >
> > > >
> > > >
> > > > Fande Kong,
>
>

```