Re: [matrixprogramming] Which solver to use.

[Mark Hoemmen]

Greetings --

If your matrix is sparse, you have two algorithmic options for solving
your linear system: a sparse factorization, or an iterative solver.
Iterative solvers require preconditioners for solving ill-conditioned
problems (like yours) efficiently. Picking a good preconditioner
requires some knowledge about your problem. In this case, you would
need a good Helmholtz preconditioner. I would suggest for now that
you use a sparse factorization with iterative refinement to improve
accuracy.

I am a Trilinos developer. Trilinos does offer interfaces to existing
third-party sparse factorizations, but does not implement its own
sparse factorizations (this may have changed; I would need to check).
It has many iterative solvers and preconditioners.

mfh

On Wed, Jun 5, 2013 at 10:44 AM, Ganesh Diwan <gcd...@gmail.com> wrote:
> Dear Dr. Rudnyi,
>
> First of all, thank you for creating the matrixprogramming site and for
> sharing such valuable and useful information.
>
> I have been looking for some guidance on choosing the best solver for the
> system of linear equations for my case - MUMPS, TRILINOS or Pardiso. I am
> interested in these solvers as they are Opensource and either g95 or
> gfortran can be used for using the libraries.
>
> I intend to solve a system, Ax = b, where A is complex, sparse, unsymmetric
> and highly ill-conditioned (condition number ~ 1E+20) square or rectangular
> matrix. I expect the solver to solve a matrix of size at least 1 million by
> 1 million. The matrix is generated from the Finite Element formulation of
> the Helmholtz equation. I have been able to solve the system with ZGELSS in
> LAPACK accurately. But as the degrees of freedom in my system grow, it takes
> a long time to solve the system on a PC with ZGELSS as the sparsity is not
> exploited. Recently I tried SuperLU (using Harwell-Boeing storage) for the
> same system but the results were inaccurate for condition number > 1E+12 (I
> am not sure if this is a numerical issue with the pivoting).
>
> I am more inclined towards using already developed solvers. Is there a
> robust solver which can solve the system I mentioned quickly (i.e.
> exploiting the sparsity) and reliably (in view of the condition numbers)?
>
> I was able to use the optimized BLAS from AMD (ACML) but again the time
> needed is too long.
> Do you have any suggestions as to which solver I should use for my case?
>
> Thanks in advance for your help.
>
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[Evgenii Rudnyi]

I personally like MUMPS. I do not know how it work for your problem
though. I have little experience with Pardiso.

The simplest way is just to try it out.

You may also want to ask this question at the MUMPS discussion list, I
bieleve that your will find there the discussion about using MUMPS for
the Helmholtz equation.

Evgenii


On 05.06.2013 18:44 Ganesh Diwan said the following: