An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling

We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination, and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to 7 fold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including the Intel Xeon Phi (MIC). The code is part of a software package called STRUMPACK -- STRUctured Matrices PACKage, which also has a distributed memory component for dense rank-structured matrices

The Reverse Cuthill-McKee Algorithm in Distributed-Memory

Ordering vertices of a graph is key to minimize fill-in and data structure size in sparse direct solvers, maximize locality in iterative solvers, and improve performance in graph algorithms. Except for naturally parallelizable ordering methods such as nested dissection, many important ordering methods have not been efficiently mapped to distributed-memory architectures. In this paper, we present the first-ever distributed-memory implementation of the reverse Cuthill-McKee (RCM) algorithm for reducing the profile of a sparse matrix. Our parallelization uses a two-dimensional sparse matrix decomposition. We achieve high performance by decomposing the problem into a small number of primitives and utilizing optimized implementations of these primitives. Our implementation shows strong scaling up to 1024 cores for smaller matrices and up to 4096 cores for larger matrices

Evaluation of OpenMP for the Cyclops multithreaded architecture

Multithreaded architectures have the potential of tolerating large memory and functional unit latencies and increase resource utilization. The Blue Gene/Cyclops architecture, being developed at the IBM T. J. Watson Research Center, is one such systems that offers massive intra-chip parallelism. Although the BG/C architecture was initially designed to execute specific applications, we believe that it can be effectively used on a broad range of parallel numerical applications. Programming such applications for this unconventional design requires a significant porting effort when using the basic built-in mechanisms for thread management and synchronization. In this paper, we describe the implementation of an OpenMP environment for parallelizing applications, currently under development at the CEPBA-IBM Research Institute, targeting BG/C. The environment is evaluated with a set of simple numerical kernels and a subset of the NAS OpenMP benchmarks. We identify issues that were not initially considered in the design of the BG/C architecture to support a programming model such as OpenMP. We also evaluate features currently offered by the BG/C architecture that should be considered in the implementation of an efficient OpenMP layer for massive intra-chip parallel architectures.Peer ReviewedPostprint (author's final draft

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

Dynamic Multiple Work Stealing Strategy for Flexible Load Balancing

Lazy-task creation is an efficient method of overcoming the overhead of the grain-size problem in parallel computing. Work stealing is an effective load balancing strategy for parallel computing. In this paper, we present dynamic work stealing strategies in a lazy-task creation technique for efficient fine-grain task scheduling. The basic idea is to control load balancing granularity depending on the number of task parents in a stack. The dynamic-length strategy of work stealing uses run-time information, which is information on the load of the victim, to determine the number of tasks that a thief is allowed to steal. We compare it with the bottommost first work stealing strategy used in StackThread/MP, and the fixed-length strategy of work stealing, where a thief requests to steal a fixed number of tasks, as well as other multithreaded frameworks such as Cilk and OpenMP task implementations. The experiments show that the dynamic-length strategy of work stealing performs well in irregular workloads such as in UTS benchmarks, as well as in regular workloads such as Fibonacci, Strassen\u27s matrix multiplication, FFT, and Sparse-LU factorization. The dynamic-length strategy works better than the fixed-length strategy because it is more flexible than the latter; this strategy can avoid load imbalance due to overstealing

Speculative Segmented Sum for Sparse Matrix-Vector Multiplication on Heterogeneous Processors

Sparse matrix-vector multiplication (SpMV) is a central building block for scientific software and graph applications. Recently, heterogeneous processors composed of different types of cores attracted much attention because of their flexible core configuration and high energy efficiency. In this paper, we propose a compressed sparse row (CSR) format based SpMV algorithm utilizing both types of cores in a CPU-GPU heterogeneous processor. We first speculatively execute segmented sum operations on the GPU part of a heterogeneous processor and generate a possibly incorrect results. Then the CPU part of the same chip is triggered to re-arrange the predicted partial sums for a correct resulting vector. On three heterogeneous processors from Intel, AMD and nVidia, using 20 sparse matrices as a benchmark suite, the experimental results show that our method obtains significant performance improvement over the best existing CSR-based SpMV algorithms. The source code of this work is downloadable at https://github.com/bhSPARSE/Benchmark_SpMV_using_CSRComment: 22 pages, 8 figures, Published at Parallel Computing (PARCO

Parallel structurally-symmetric sparse matrix-vector products on multi-core processors

We consider the problem of developing an efficient multi-threaded implementation of the matrix-vector multiplication algorithm for sparse matrices with structural symmetry. Matrices are stored using the compressed sparse row-column format (CSRC), designed for profiting from the symmetric non-zero pattern observed in global finite element matrices. Unlike classical compressed storage formats, performing the sparse matrix-vector product using the CSRC requires thread-safe access to the destination vector. To avoid race conditions, we have implemented two partitioning strategies. In the first one, each thread allocates an array for storing its contributions, which are later combined in an accumulation step. We analyze how to perform this accumulation in four different ways. The second strategy employs a coloring algorithm for grouping rows that can be concurrently processed by threads. Our results indicate that, although incurring an increase in the working set size, the former approach leads to the best performance improvements for most matrices.Comment: 17 pages, 17 figures, reviewed related work section, fixed typo