No abstract

Multigrid algorithms are widely used to solve large-scale sparse linear systems, which is essential for many high-performance workloads. The symmetric Gauss-Seidel (SYMGS) method is often responsible for the performance bottleneck of MG. This paper presents new methods to parallelize and enhance the computation and parallelization efficiency of the SYMGS and MG algorithms on multi-core CPUs. Our solution employs a matrix splitting strategy and a revised computation formula to decrease the computation operations and memory accesses in SYMGS. With this new SYMGS strategy, we can then merge the two most time-consuming components of MG. On top of these, we propose a new asynchronous parallelization scheme to reduce the synchronization overhead when parallelizing SYMGS. We demonstrate the benefit of our techniques by integrating them with the HPCG benchmark and two real-life applications. Evaluation conducted on four architectures, including three ARMv8 and one x86, shows that our techniques greatly surpass the performance of engineer-and vendor-tuned implementations across various workloads and platforms. CCS CONCEPTS• Mathematics of computing → Solvers; Mathematical software performance; • Computing methodologies → Massively parallel algorithms.

In this paper, a novel algorithm is proposed for reducing a banded symmetric generalized eigenvalue problem to a banded symmetric standard eigenvalue problem, based on the sequentially semiseparable (SSS) matrix techniques. It is the first time that the SSS matrix techniques are used in such eigenvalue problems. The newly proposed algorithm only requires linear storage cost and O(n2) computation cost for matrices with dimension n, and is also potentially good for parallelism. Some experiments have been performed by using Matlab, and the accuracy and stability of algorithm are verified.

Structured grid-based sparse matrix-vector multiplication and Gauss–Seidel iterations are very important kernel functions in scientific and engineering computations, both of which are memory intensive and bandwidth-limited. GPDSP is a general purpose digital signal processor, which is a very significant embedded processor that has been introduced into high-performance computing. In this paper, we designed various optimization methods, which included a blocking method to improve data locality and increase memory access efficiency, a multicolor reordering method to develop Gauss–Seidel fine-grained parallelism, a data partitioning method designed for GPDSP memory structures, and a double buffering method to overlap computation and access memory on structured grid-based SpMV and Gauss–Seidel iterations for GPDSP. At last, we combined the above optimization methods to design a multicore vectorization algorithm. We tested the matrices generated with structured grids of different sizes on the GPDSP platform and obtained speedups of up to 41× and 47× compared to the unoptimized SpMV and Gauss–Seidel iterations, with maximum bandwidth efficiencies of 72% and 81%, respectively. The experiment results show that our algorithms could fully utilize the external memory bandwidth. We also implemented the commonly used mixed precision algorithm on the GPDSP and obtained speedups of 1.60× and 1.45× for the SpMV and Gauss–Seidel iterations, respectively.