[petsc-dev] Options for Solving Separate Preconditioning Matrix

[Matthew Knepley]

On Wed, Dec 28, 2011 at 12:56 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> On Wed, Dec 28, 2011 at 12:51, Dave Nystrom <dnystrom1 at comcast.net> wrote:
>
>> I am experimenting now with solving my various linear systems with petsc
>> using a separate preconditioning matrix.  These linear systems are all
>> banded
>> systems arising from discretization of pdes on a 2d structured grid.  My
>> preconditioning matrix is the inner band of diagonals about the main
>> diagonal.  So, my preconditioning matrices are all band systems with a
>> narrow
>> bandwidth.  On 4 of them, they are tridiagonal.  Another is
>> heptadiagonal.  I
>> am using lu as my preconditioner.
>>
>> So, I am wondering if there are better ways to solve the preconditioning
>> matrix with petsc than using a general sparse direct lu solver such as
>> petsc
>> lu.  I don't know if petsc lu does anything special when it encounters a
>> sparse matrix that is tridiagonal or banded with narrow bandwidth.  I'm
>> guessing not but it would be nice if it did.  I assume that I could solve
>> such a tridiagonal preconditioning matrix more efficiently using something
>> like the lapack tridiagonal solver that would interface to a vendor
>> library
>> like MKL.  I also wonder if there is much difference between the
>> performance
>> of the triangular solves for a tridiagonal system with vendor lapack
>> versus
>> normal lapack.  That would seem to be the main issue since I am using the
>> tridiagonal solve as a preconditioner that takes many iterations.
>>
>
> Profile. This sparse direct solve is likely a very small fraction of run
> time, so it doesn't matter that it could be done marginally faster using a
> tridiagonal solver.
>
> The much more important issue is that your matrices are discarding
> coupling in all but one spatial direction. That is generally quite bad
> algorithmically (except for very strongly anisotropic problems, such that
> the Green's function decays very fast in the transverse direction), leading
> to slow convergence.
>

Yes, there is an ENORMOUS preconditioning literature on these systems. This
is a time to get a bunch
of papers and stat reading. I would look at ADI.

   Matt

-- 
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
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