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 -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20111228/1349450f/attachment.html>
