[petsc-users] MatSetNullSpace

[Bikash Kanungo]

Thanks again Matthew. I'll try to use MatSetNullSpaceFunction and see if it
resolves the issue.

On Thu, Jun 1, 2017 at 6:00 PM, Matthew Knepley <knepley at gmail.com> wrote:

> 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/
>



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