Sparse Matrix Inversion using PETSc

[Aron Ahmadia]

Dear Tim,

It is possible to carry out the explicit inversion of a sparse matrix
using the PETSc framework with the methodology you outlined below.  I
would encourage you to consider Cholesky/LU factorizations of the
matrix, which occassionally result in sparser triangular solve times
than an explicit inverse-matrix-vector multiply would.

As for the correct way to do this, I would start with the fastest
methods for multiple right hand sides and reasonably sized matrices, a
direct method such as LU.  I'm unaware of any functionality in PETSc
for handling multiple right hand sides, but PETSc will keep the
factorization from a previous direct solve, so A\b2 will be much
faster than A\b1.  I think the best bet is a naive for loop over each
of the vectors to assemble the matrix piece by piece.

The PETSc developers may have some more thoughts on this.

Good luck,
~Aron

On 8/15/07, Dr. Timothy Stitt <timothy.stitt at ichec.ie> wrote:
> Hi all,
>
> I am currently investigating the best way to perform the inversion of a
> large sparse matrix and came upon the idea of using PETSc as a framework
> for testing various strategies from direct to iterative methods on my
> sample matrices. In this setup for an NxN sparse matrix A I would have N
> rhs's representing the Identity matrix and then solve for X. I wanted to
> experiment with both parallel and serial strategies ranging from LU
> Decomposition using SuperLU, MUMPS etc. to iterative methods using GMRES
> etc. Am I right in thinking that all this can be done in PETSc by
> setting up a core framework and then varying the solver methods etc?
>
> I have looked over the sample KSP Solver codes although they only seem
> to suggest single vectors for x and b. Can this be changed to accept
> multiple vectors? Can anyone suggest a sample code that maybe
> demonstrates the sort of thing I want to achieve...if it is in fact
> possible.
>
> Thanks in advance for any assistance given,
>
> Regards,
>
> Tim.
>
>