[petsc-users] fieldsplit-preconditioner for block-system of linear equations

[Matthew Knepley]

On Wed, Sep 30, 2015 at 3:00 AM, Eike Mueller <E.Mueller at bath.ac.uk> wrote:

> Dear PETSc,
>
> I am solving a linear system of equations
>
> (1) A.X = B
>
> where the vector X is divided into chunks, such that X= (x_1,x_2,…,x_k)
> and each of the vectors x_i has length n_i (same for the vector
> B=(b_1,b_2,…,b_k)). k=5 in my case, and n_i >> 1. This partitioning implies
> that the matrix has a block-structure, where the submatrices A_{ij} are of
> size n_i x n_j. So explicitly for k=3:
>
>         A_{1,1}.x_1 + A_{1,2}.x_2 + A_{1,3}.x_3 = b_1
> (2)    A_{2,1}.x_1 + A_{2,2}.x_2 + A_{2,3}.x_3 = b_2
>         A_{3,1}.x_1 + A_{3,2}.x_2 + A_{3,3}.x_3 = b_3
>
> I now want to solve this system with a Krylov-method (say, GMRES) and
> precondition it with a field-split preconditioner, such that the
> Schur-complement is formed in the 1-block. I know how to do this if I have
> assembled the big matrix A, and I store all x_i in one big vector X (I
> construct the index-sets corresponding to the vectors x_1 and (x_2,x_3),
> and then call PCFieldSplitSetIS()). This all works without any problems.
>
> However, I now want to do this for an existing code which
>
> 1. Assembles the matrices A_{ij} separately (i.e. they are not stored as
> part of a big matrix A, but are stored in independent memory locations)
> 2. Stores the vectors x_i separately, i.e. they are not stored as parts of
> one big chunk of memory of size n_1+…+n_k
>
> I can of course implement the matrix application via a matrix shell, but
> is there still an easy way of using the fieldsplit preconditioner?
>

>From your description, it sounds like you could use


http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatGetLocalSubMatrix.html

to assemble your separate matrices directly into a large matrix.
Furthermore, if it really benefits you
to have separate memory, you can change the matrix type from MATIJ to
MATNEST dynamically,
but none of your code would change. This would let you check things with
LU, but still optimize when
necessary.

  Thanks,

      Matt


> The naive way I can think of is to allocate a new big matrix A, and copy
> the A_{ij} into the corresponding blocks. Then allocate big vectors x and
> b, and also copy in/out the data before and after the solve. This, however,
> seems to be wasteful, so before I go down this route I wanted to double
> check if there is a way around it, since this seems to be a very common
> problem?
>
> Thanks a lot,
>
> Eike
>
>


-- 
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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