> On Oct 12, 2016, at 10:18 PM, Fande Kong <fdkong...@gmail.com> wrote: > > > > On Wed, Oct 12, 2016 at 9:12 PM, Barry Smith <bsm...@mcs.anl.gov> wrote: > > > On Oct 12, 2016, at 10:00 PM, Amneet Bhalla <mail2amn...@gmail.com> wrote: > > > > > > > > On Monday, October 10, 2016, 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 (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?

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Yes, MatNullSpaceCreate() should report an error if all the vectors are not orthonormal. #if defined(PETSC_USE_DEBUG) if (n) { PetscScalar *dots; for (i=0; i<n; i++) { PetscReal norm; ierr = VecNorm(vecs[i],NORM_2,&norm);CHKERRQ(ierr); if (PetscAbsReal(norm - 1.0) > PETSC_SQRT_MACHINE_EPSILON) SETERRQ2(PetscObjectComm((PetscObject)vecs[i]),PETSC_ERR_ARG_WRONG,"Vector %D must have 2-norm of 1.0, it is %g",i,(double)norm); } if (has_cnst) { for (i=0; i<n; i++) { PetscScalar sum; ierr = VecSum(vecs[i],&sum);CHKERRQ(ierr); if (PetscAbsScalar(sum) > PETSC_SQRT_MACHINE_EPSILON) SETERRQ2(PetscObjectComm((PetscObject)vecs[i]),PETSC_ERR_ARG_WRONG,"Vector %D must be orthogonal to constant vector, inner product is %g",i,(double)PetscAbsScalar(sum)); } } ierr = PetscMalloc1(n-1,&dots);CHKERRQ(ierr); for (i=0; i<n-1; i++) { PetscInt j; ierr = VecMDot(vecs[i],n-i-1,vecs+i+1,dots);CHKERRQ(ierr); for (j=0;j<n-i-1;j++) { if (PetscAbsScalar(dots[j]) > PETSC_SQRT_MACHINE_EPSILON) SETERRQ3(PetscObjectComm((PetscObject)vecs[i]),PETSC_ERR_ARG_WRONG,"Vector %D must be orthogonal to vector %D, inner product is %g",i,i+j+1,(double)PetscAbsScalar(dots[j])); } } PetscFree(dots);CHKERRQ(ierr); } #endif > > 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 > > > > > > > > > >