[petsc-users] MatSetNullSpace

[Matthew Knepley]

On Thu, Jun 1, 2017 at 4:58 PM, Bikash Kanungo <bikash at umich.edu> wrote:

> Thank you Matthew for the quick response.
>
> I should provide some more details.
>
> 1) This implies that A is symmetric
>
>
> Yes A is symmetric
>
> 2) I think you mean the b is orthogonal to Q
>
>
> No. I've an extra condition on the solution x - that it has to be
> orthogonal to Q. Without this condition the problem will have multiple
> solutions. It's same as finding the minimum norm solution
>
>
>> 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.
>
>
> I was unable to get convergence without using MatSetNullSpace(). So it
> seems that the Krlov solvers are inadequate just by themselves.
>
> Second, we remove components in the nullspace from each iterate, which is
>> not what you are doing above.
>>
>
> This might be the reason why without MatSetNullSpace() I'm unable to
> attain convergence.
>
>
>> It seems easier to just give your Q as the nullspace.
>
> Yeah using Q as nullspace is the easiest option. But my basis is
> non-orthogonal and has an overlap matrix S. So in order to provide
> orthonormal nullvectors as Q to Petsc, I need to evaluate S^{-1/2}. As of
> now, for small problems I can evalaute S^{-1/2}, however, it's a show
> stopper for large ones. I can possibly use MatNullSpaceSetFunction() to
> circumvent the requirement of orthonormal nullvectors in MatSetNullSpace().
> I would appreciate your input on this.
>

You do not need this. Just use


http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatNullSpaceSetFunction.html#MatNullSpaceSetFunction

to implement your (I - Q^T Q).

  Thanks,

     Matt


> Lastly, I would like to know what operations do MatSetNullSpace enforce
> within a KSP solve.
>
> Thanks,
> Bikash
>
> On Thu, Jun 1, 2017 at 5:25 PM, Matthew Knepley <knepley at gmail.com> wrote:
>
>> 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/
>>
>
>
>
> --
> 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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