[petsc-users] Number of Blocks in the Block Jacobi preconditionr

[Jed Brown]

On Sun, Sep 23, 2012 at 4:58 PM, Wenlei Xie <wenleix at cs.cornell.edu> wrote:

> I am using PETSc to solve linear systems with Block Jacobi preconditioner,
> and I am currently studying the trade off between the number of blocks.
> In my experiments I tried a very sparse 1M*1M matrix and try different
> number of blocks. I expect the result would be, if I use less blocks (thus
> each block has larger size), then the convergence rate would increase but
> the time spend on each iteration would would also increase; while for more
> blocks (and each block has smaller size), both the convergence rate and the
> time on each iteration will be decreased. However, the result turns out to
> that the time on each iteration also decreased as the block size increased.
> Is it because I misused it?
>
> I am use Richardson method as the Krylov method, and ILU as the solver
> inside each block matrix. The matrix contains around 3.5M non-zeros entries
> and I use METIS to partition it into blocks.
>

The work required for ILU is essentially linear in the subdomain size so
the only possible speedup comes from potentially better cache reuse by
small blocks. With the default -sub_ksp_type preonly, that isn't
significant yet.

To see the effect you expect, use -sub_pc_type lu. The cost of the direct
solve (especially in 3D) grows noticeably superlinearly, thus large
subdomains become much more expensive than the overhead of decomposing the
problem at that granularity.
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