On Techniques to Improve the Robustness and Scalability of the Schur Complement Method
Ichitaro Yamazaki (Lawrence Berkeley National Laboratory)Xiaoye Li (Lawrence Berkeley National Laboratory)
Abstract:
A hybrid linear solver based on the Schur complement method
has great potential to be a general purpose solver scalable on
tens of thousands of processors. It is imperative to exploit two levels
of parallelism; namely, solving independent subdomains in parallel
and using multiple processors per subdomain.
This hierarchical parallelism can lead to a scalable implementation
which maintains numerical stability at the same time. In this framework,
load imbalance and excessive communication, which can lead to
performance bottlenecks, occur at two levels:
in an intra-processor group assigned to the same subdomain
and among inter-processor groups assigned to different subdomains.
We developed several
techniques to address these issues, such as taking advantage of
the sparsity of right-hand-side columns
during sparse triangular solutions with interfaces,
load balancing sparse matrix-matrix multiplication to form update matrices, and
designing an effective asynchronous point-to-point
communication of the update matrices.
We present numerical results to demonstrate that
with the help of these techniques,
our hybrid solver can efficiently solve large-scale highly-indefinite
linear systems on thousands of processors.
Keywords:
Parallel and Distributed Computing, ,
