Hi Barry,

Thanks so much for this work. I will checkout your branch, and take a look.

Thanks again!

Fande Kong,

On Thu, Oct 13, 2016 at 8:10 PM, Barry Smith <bsm...@mcs.anl.gov> wrote:

>
>   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,
> >
> >
>
>

Reply via email to