Solving Large Sparse Linear Systems using Asynchronous Multisplitting

Iterative methods for solving large sparse systems of linear equations are widely used in many HPC applications. Extreme
scaling of these methods can be difficult, however, since global communication to form dot products is typically required at
every iteration.
To try to overcome this limitation we propose a hybrid approach, where the matrix is partitioned into blocks. Within each
block, we use a highly optimised (parallel) conventional solver, but we then couple the blocks together using block Jacobi or
some other multisplitting technique that can be implemented in either a synchronous or an asynchronous fashion. This allows us
to limit the block size to the point where the conventional iterative methods no longer scale, and to avoid global communication
(and possibly synchronisation) across all processes.
Our block framework has been built to use PETSc, a popular scientific suite for solving sparse linear systems, as the
synchronous intra-block solver, and we demonstrate results on up to 32768 cores of a Cray XE6 system. At this scale, the
conventional solvers are still more efficient, though trends suggest that the hybrid approach may be beneficial at higher core
counts.