[petsc-users] Solving a Singular System with PETSc

[Matthew Knepley]

On Sun, Feb 27, 2022 at 2:36 AM Bojan Niceno <
bojan.niceno.scientist at gmail.com> wrote:

> Dear all,
>
> I have coupled PETSc with my computational fluid dynamics (CFD) solver for
> incompressible flows where the most computationally intensive part is a
> solution of the linear system for pressure - which is singular.
>
> A simple call to PETSc solvers resulted in divergence, as expected, but
> things work when I set the null space for the pressure matrix as
> demonstrated in src/ksp/ksp/tutorials/ex29.c:
>     MatNullSpace nullspace;
>     ierr =
> MatNullSpaceCreate(PETSC_COMM_WORLD,PETSC_TRUE,0,0,&nullspace);CHKERRQ(ierr);
>     ierr = MatSetNullSpace(J,nullspace);CHKERRQ(ierr);
>     ierr = MatNullSpaceDestroy(&nullspace);CHKERRQ(ierr);
>
> However, the effect of setting the null space as described above, has
> almost the same effect (convergence history is almost the same) as if when
> I multiply each diagonal of the system matrix with (1.0 + 1.0e-6), i.e.,
> desingularize the matrix by making it slightly diagonally dominant.
>
> I prefer the former solution as the latter one seems a bit like an ad-hoc
> patch and I am not sure how general it is, but I wonder, from a
> mathematical point of view, is it the same thing?  Any thoughts on that?
>

I will give a slightly different explanation than Jose. When you set a
nullspace N, it tells us what space the solution must be orthogonal to

   N^T x = 0

We use this to project out these components at each step of the iterative
method. At the end, we get a solution

  A x = b

which _also_ satisfies the uniqueness condition

  N^T x = 0

When you perturb the matrix, you the the solution to a _different_ linear
system

  (A + sigma I) x = b

It is close, but not the same, and there is no guarantee (unless you know
something about A) that this is close
to the other solution in a normwise sense.

  Thanks,

     Matt


>     Cheers,
>
>     Bojan Niceno
>


-- 
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

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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