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, > >