[petsc-users] Solving dense rectangular linear systems

[Jed Brown]

BV holds a tall-skinny matrix or collection of vectors with equivalent semantics. If you start with a Mat, you can use:

https://slepc.upv.es/documentation/current/docs/manualpages/BV/BVCreateFromMat.html

BVOrthogonalize() computes a QR factorization in blocks or incrementally.

https://slepc.upv.es/documentation/current/docs/manualpages/BV/BVOrthogonalize.html

The BVOrthogBlockType can be used to select Cholesky QR or TSQR, among others. These have lower communication requirements than Gram-Schmidt. 

https://slepc.upv.es/documentation/current/docs/manualpages/BV/BVSetOrthogonalization.html#BVSetOrthogonalization

For more on BV:
https://slepc.upv.es/documentation/current/docs/manualpages/BV/index.html

Rakesh Halder <rhalder at umich.edu> writes:

> Hi Jed,
>
> I can’t find much on computing the QR decomposition in PETSc, are there any
> methods in particular?
>
> I looked at the documentation for BVOrthogonalize() and it seems to be only
> for square sequential matrices.
>
> On Sun, Nov 8, 2020 at 11:34 PM Jed Brown <jed at jedbrown.org> wrote:
>
>> Rakesh Halder <rhalder at umich.edu> writes:
>>
>> > Hi all,
>> >
>> > I'm wondering what the recommended method is to solve linear systems
>> Ax=b,
>> > where A is an N by n matrix (N >> n) and dense. I've used the CGLS and
>> LSQR
>> > algorithms, but have issues applying any kind of preconditioner. Looking
>> at
>> > the PETSc documentation, it doesn't look like there are any direct
>> solvers
>> > for rectangular systems, but only iterative ones which aren't very useful
>> > for dense systems.
>>
>> QR is the standard technique. For a highly parallel implementation, I
>> would check out BVOrthogonalize() from SLEPc, which has several good
>> options. Arguably, this feature should be migrated to PETSc.
>>
>> > I'm also wondering if it's best to use a dense or sparse (AIJ) matrix
>> > format in general when working with dense matrices. I've just been using
>> > sparse matrices, due to being able to preallocate memory.
>> >
>> > Thanks,
>> >
>> > Rakesh Halder
>>