Benchmarking mixed-mode PETSc performance on high-performance architectures

The trend towards highly parallel multi-processing is ubiquitous in all modern computer architectures, ranging from handheld devices to large-scale HPC systems; yet many applications are struggling to fully utilise the multiple levels of parallelism exposed in modern high-performance platforms. In order to realise the full potential of recent hardware advances, a mixed-mode between shared-memory programming techniques and inter-node message passing can be adopted which provides high-levels of parallelism with minimal overheads. For scientific applications this entails that not only the simulation code itself, but the whole software stack needs to evolve. In this paper, we evaluate the mixed-mode performance of PETSc, a widely used scientific library for the scalable solution of partial differential equations. We describe the addition of OpenMP threaded functionality to the library, focusing on sparse matrix-vector multiplication. We highlight key challenges in achieving good parallel performance, such as explicit communication overlap using task-based parallelism, and show how to further improve performance by explicitly load balancing threads within MPI processes. Using a set of matrices extracted from Fluidity, a CFD application code which uses the library as its linear solver engine, we then benchmark the parallel performance of mixed-mode PETSc across multiple nodes on several modern HPC architectures. We evaluate the parallel scalability on Uniform Memory Access (UMA) systems, such as the Fujitsu PRIMEHPC FX10 and IBM BlueGene/Q, as well as a Non-Uniform Memory Access (NUMA) Cray XE6 platform. A detailed comparison is performed which highlights the characteristics of each particular architecture, before demonstrating efficient strong scalability of sparse matrix-vector multiplication with significant speedups over the pure-MPI mode

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

Preparing sparse solvers for exascale computing.

Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'