[petsc-users] MatSetNullSpace

[Matthew Knepley]

On Thu, Jun 1, 2017 at 2:07 PM, Bikash Kanungo <bikash at umich.edu> wrote:

> Hi,
>
> I'm trying to solve a linear system of equations  Ax=b, where A has a null
> space (say Q) and x is known to be orthogonal to Q. In order to avoid
> ill-conditioning, I was trying to do the following:
>

1) This implies that A is symmetric

2) I think you mean the b is orthogonal to Q


>
>    1.  Create A as a shell matrix
>    2.  Overload the MATOP_MULT operation for with my own function which
>    returns
>    y = A*(I - QQ^T)x instead of y = Ax
>    3. Upon convergence, solution = (I-QQ^T)x instead of x.
>
> However, I realized that the linear solver can make x have any arbitrary
> component along Q and still y = A*(I-QQ^T)x will remain unaffected, and
> hence can cause convergence issues. Indeed, I saw such convergence
> problems. What fixed the problem was using MatSetNullSpace for A with Q as
> the nullspace, in addition to the above three steps.
>
> So my question is what exactly is MatSetNullSpace doing? And since the
> full A information is not present and A is only accessed through
> MAT_OP_MULT, I'm confused as how MatSetNullSpace might be fixing the
> convergence issue.
>

Its not MatSetNullSpace() that fixes this issue, it is the linear solver
itself. Many Krylov solvers can converge to the
minimum norm solution of this rank deficient problem. Second, we remove
components in the nullspace from each
iterate, which is not what you are doing above. It seems easier to just
give your Q as the nullspace.

  Thanks,

     Matt


> Thanks,
> Bikash
>
> --
> Bikash S. Kanungo
> PhD Student
> Computational Materials Physics Group
> Mechanical Engineering
> University of Michigan
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

http://www.caam.rice.edu/~mk51/
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