Increasing the Efficiency of Sparse Matrix-Matrix Multiplication with a 2.5D Algorithm and One-Sided MPI

Matrix-matrix multiplication is a basic operation in linear algebra and an essential building block for a wide range of algorithms in various scientific fields. Theory and implementation for the dense, square matrix case are well-developed. If matrices are sparse, with application-specific sparsity patterns, the optimal implementation remains an open question. Here, we explore the performance of communication reducing 2.5D algorithms and one-sided MPI communication in the context of linear scaling electronic structure theory. In particular, we extend the DBCSR sparse matrix library, which is the basic building block for linear scaling electronic structure theory and low scaling correlated methods in CP2K. The library is specifically designed to efficiently perform block-sparse matrix-matrix multiplication of matrices with a relatively large occupation. Here, we compare the performance of the original implementation based on Cannon's algorithm and MPI point-to-point communication, with an implementation based on MPI one-sided communications (RMA), in both a 2D and a 2.5D approach. The 2.5D approach trades memory and auxiliary operations for reduced communication, which can lead to a speedup if communication is dominant. The 2.5D algorithm is somewhat easier to implement with one-sided communications. A detailed description of the implementation is provided, also for non ideal processor topologies, since this is important for actual applications. Given the importance of the precise sparsity pattern, and even the actual matrix data, which decides the effective fill-in upon multiplication, the tests are performed within the CP2K package with application benchmarks. Results show a substantial boost in performance for the RMA based 2.5D algorithm, up to 1.80x, which is observed to increase with the number of involved processes in the parallelization.Comment: In Proceedings of PASC '17, Lugano, Switzerland, June 26-28, 2017, 10 pages, 4 figure

Achieving Efficient Strong Scaling with PETSc using Hybrid MPI/OpenMP Optimisation

The increasing number of processing elements and decreas- ing memory to core ratio in modern high-performance platforms makes efficient strong scaling a key requirement for numerical algorithms. In order to achieve efficient scalability on massively parallel systems scientific software must evolve across the entire stack to exploit the multiple levels of parallelism exposed in modern architectures. In this paper we demonstrate the use of hybrid MPI/OpenMP parallelisation to optimise parallel sparse matrix-vector multiplication in PETSc, a widely used scientific library for the scalable solution of partial differential equations. Using large matrices generated by Fluidity, an open source CFD application code which uses PETSc as its linear solver engine, we evaluate the effect of explicit communication overlap using task-based parallelism and show how to further improve performance by explicitly load balancing threads within MPI processes. We demonstrate a significant speedup over the pure-MPI mode and efficient strong scaling of sparse matrix-vector multiplication on Fujitsu PRIMEHPC FX10 and Cray XE6 systems

Scalable Task-Based Algorithm for Multiplication of Block-Rank-Sparse Matrices

A task-based formulation of Scalable Universal Matrix Multiplication Algorithm (SUMMA), a popular algorithm for matrix multiplication (MM), is applied to the multiplication of hierarchy-free, rank-structured matrices that appear in the domain of quantum chemistry (QC). The novel features of our formulation are: (1) concurrent scheduling of multiple SUMMA iterations, and (2) fine-grained task-based composition. These features make it tolerant of the load imbalance due to the irregular matrix structure and eliminate all artifactual sources of global synchronization.Scalability of iterative computation of square-root inverse of block-rank-sparse QC matrices is demonstrated; for full-rank (dense) matrices the performance of our SUMMA formulation usually exceeds that of the state-of-the-art dense MM implementations (ScaLAPACK and Cyclops Tensor Framework).Comment: 8 pages, 6 figures, accepted to IA3 2015. arXiv admin note: text overlap with arXiv:1504.0504

A Novel Partitioning Method for Accelerating the Block Cimmino Algorithm

We propose a novel block-row partitioning method in order to improve the convergence rate of the block Cimmino algorithm for solving general sparse linear systems of equations. The convergence rate of the block Cimmino algorithm depends on the orthogonality among the block rows obtained by the partitioning method. The proposed method takes numerical orthogonality among block rows into account by proposing a row inner-product graph model of the coefficient matrix. In the graph partitioning formulation defined on this graph model, the partitioning objective of minimizing the cutsize directly corresponds to minimizing the sum of inter-block inner products between block rows thus leading to an improvement in the eigenvalue spectrum of the iteration matrix. This in turn leads to a significant reduction in the number of iterations required for convergence. Extensive experiments conducted on a large set of matrices confirm the validity of the proposed method against a state-of-the-art method

The Parallelism Motifs of Genomic Data Analysis

Genomic data sets are growing dramatically as the cost of sequencing continues to decline and small sequencing devices become available. Enormous community databases store and share this data with the research community, but some of these genomic data analysis problems require large scale computational platforms to meet both the memory and computational requirements. These applications differ from scientific simulations that dominate the workload on high end parallel systems today and place different requirements on programming support, software libraries, and parallel architectural design. For example, they involve irregular communication patterns such as asynchronous updates to shared data structures. We consider several problems in high performance genomics analysis, including alignment, profiling, clustering, and assembly for both single genomes and metagenomes. We identify some of the common computational patterns or motifs that help inform parallelization strategies and compare our motifs to some of the established lists, arguing that at least two key patterns, sorting and hashing, are missing