It sounds like you have a symmetric positive definite systems like du/dt - div(alpha(x) grad)u. The du/dt term makes the systems easier to solve. I'm guessing your hard system does not have this mass term and so is purely elliptic. Multigrid is well suited for this type of problem, but the vector nature requires some thought. You could use PETSc AMG -pc_type gamg but you need to tell it that you have a system of two dof/vertex. You can do that with something like: ierr = MatSetBlockSize( mat, 2 ); CHKERRQ(ierr); For the best results from GAMG you need to give it null space information but we can worry about that later. Mark On Nov 30, 2011, at 8:15 AM, Matthew Knepley wrote:

On Wed, Nov 30, 2011 at 12:41 AM, Dave Nystrom <[email protected]> wrote: I have a linear system in a code that I have interfaced to petsc that is taking about 80 percent of the run time per timestep. This linear system is a symmetric block banded matrix where the blocks are 2x2. The matrix looks as follows:

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1X X Y Y Y 2X X X Y Y Y 3 X X X Y Y Y 4 X X X Y Y Y 5 X X X Y Y Y 6 X X X Y Y Y 7 X X X Y Y Y 8 X X X Y Y Y 9 X X X Y Y Y 0 X X X Y Y Y 1 X X X Y Y Y 2 X X X Y Y Y 3Z X X X Y Y Y 4Z Z X X X Y Y Y 5Z Z Z X X X Y Y Y 6 Z Z Z X X X Y Y Y 7 Z Z Z X X X Y Y Y 8 Z Z Z X X X Y Y Y 9 Z Z Z X X X Y Y Y 0 Z Z Z X X X Y Y Y

So in my diagram above, X, Y and Z are 2x2 blocks. The symmetry of the matrix requires that X_ij = transpose(X_ji) and Y_ij = transpose(Z_ji). So far, I have just input this matrix to petsc without indicating that it was block banded with 2x2 blocks. I have also not told petsc that the matrix is symmetric. And I have allowed petsc to decide the best way to store the matrix.

I can solve this linear system over the course of a run using -ksp_type preonly -pc_type lu. But that will not scale very well to larger problems that I want to solve. I can also solve this system over the course of a run using -ksp_type cg -pc_type jacobi -vec_type cusp -mat_type aijcusp. However, over the course of a run, the iteration count ranges from 771 to 47300. I have also tried sacusp, ainvcusp, sacusppoly, ilu(k) and icc(k) with k=0. The sacusppoly preconditioner fails because of a thrust error related to an invalid device pointer, if I am remembering correctly. I reported this problem to petsc-maint a while back and have also reported it for the cusp bugtracker. But it does not appear that anyone has really looked into the bug. For the other preconditioners of sacusp, ilu(k) and icc(k), they do not result in convergence to a solution and the runs fail.

All preconditioners are custom. Have you done a literature search for PCs known to work for this problem? Can yu say anything about the spectrum of the operator? conditioning? what is the principal symbol (if its a PDE)? The pattern is not enough to recommend a PC.

Matt

I'm wondering if there are suggestions of other preconditioners in petsc that I should try. The only third party package that I have tried is the txpetscgpu package. I have not tried hypre or any of the multigrid preconditioners yet. I'm not sure how difficult it is to try those packages. Anyway, so far I have not found a preconditioner available in petsc that provides a robust solution to this problem and would be interested in any suggestions that anyone might have of things to try.

I'd be happy to provide additional info and am planning on packaging up a couple of examples of the matrix and rhs for people I am interacting with at Tech-X and EMPhotonics. So I'd be happy to provide the matrix examples for this forum as well if anyone wants a copy.

Thanks,

Dave

-- Dave Nystrom

phone: 505-661-9943 (home office) 505-662-6893 (home) skype: dave.nystrom76 email: [email protected] smail: 219 Loma del Escolar Los Alamos, NM 87544

-- 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