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DOE OSTI2018

Fast and effective reordering of columns within supernodes using partition refinement

In this paper, we consider the problem of computing a triangular factorization of a sparse symmetric matrix using Gaussian elimination. We assume that the sparse matrix has been permuted using a fill-reducing ordering algorithm. When the matrix is symmetric positive definite, the sparsity structure of the triangular factor can be determined once the fill-reducing ordering has been computed. Thus, an efficient numerical factorization scheme can be designed so that only the nonzero entries are stored and operated on. On modern architectures, the positions of the nonzero entries in the triangular factor play a big role in determining the efficiency. It is desirable to have dense blocks in the factor so that the computation can be cast in terms of level-3 BLAS as much as possible. On architectures with GPUs, for example, it is also desirable for these dense blocks to be as large as possible in order to reduce the times to transfer data between the main CPU and the GPUs. We address the problem of locally refining the ordering so that the number of dense blocks is reduced and the sizes of these dense blocks are increased in the triangular factor.

Jacquelin, Mathias↗

DOE OSTI2016

An Efficient Multicore Implementation of a Novel HSS-Structured Multifrontal Solver Using Randomized Sampling

Here, 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 factoriz ation 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.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

A supernodal all-pairs shortest path algorithm

We show how to exploit graph sparsity in the Floyd-Warshall algorithm for the all-pairs shortest path (Apsp) problem. Floyd-Warshall is an attractive choice for Apsp on high-performing systems due to its structural similarity to solving dense linear systems and matrix multiplication. However, if sparsity of the input graph is not properly exploited, Floyd-Warshall will perform unnecessary asymptotic work and thus may not be a suitable choice for many input graphs. To overcome this limitation, the key idea in our approach is to use the known algebraic relationship between Floyd-Warshall and Gaussian elimination, and import several algorithmic techniques from sparse Cholesky factorization, namely, fill-in reducing ordering, symbolic analysis, supernodal traversal, and elimination tree parallelism. When combined, these techniques reduce computation, improve locality and enhance parallelism. We implement these ideas in an efficient shared memory parallel prototype that is orders of magnitude faster than an efficient multi-threaded baseline Floyd-Warshall that does not exploit sparsity. Our experiments suggest that the Floyd-Warshall algorithm can compete with Dijkstra's algorithm (the algorithmic core of Johnson's algorithm) for several classes sparse graphs.

Sao, Piyush↗