Scaling Block Conjugate Gradient Variants Orthomin and Orthodir

Iterative solvers based on Krylov subspace methods are widely used for the solution of the problems that appear in large-
scale parallel scienti c simulations. These solvers, when parallelized, often su er from global synchronization overheads
due to the collective communication operations. Block CG variants have the advantage of reduced communication
overheads at the expense of increased computation per iteration. The aim of this project is the scalable parallelization of
two such block CG variants, Orthomin and Orthodir, proposed by FET-HPC project NLAFET to enable the use of these
methods on future exascale systems through reducing number of the synchronization points. We investigate 1D- and
2D-partitioning of the sparse coecient matrix for encapsulating the minimization of the communication overhead as well
as one- and two-constraint partitioning for computational load balancing. Two di erent parallel codes for Orthomin and
Orthodir variants are developed. The relative performance of di erent partitioning techniques are evaluated by running
the parallel Orthomin and Orthodir codes on two di erent HPC systems up to 1024 processors with 26 symmetric positive
de nite matrices. The number of rows in these matrices ranges between 13,681 and 1,391,349, whereas the number of
nonzero entries ranges between 340,431 and 77,651,847.