Abstract
We describe our recent work on designing algorithms and software for solving sparse systems using direct methods on parallel computers. This work has been conducted within an EU Horizon 2020 Project called NLAFET. We first discuss the solution of large sparse symmetric positive definite systems. We use a runtime system to express and execute a DAG-based Cholesky factorization. The runtime system plays the role of a software layer between the application and the architecture and handles the management of task dependencies as well as task scheduling and maintaining data coherency. Although runtime systems are widely used in dense linear algebra, this approach is challenging for sparse algorithms because of the irregularity and variable granularity of the DAGs arising in these systems. We have implemented our software using the OpenMP standard and the runtime systems StarPU and PaRSEC. We compare these implementations to HSL_MA87, a state-of-the-art DAG-based solver for positive definite systems. We demonstrate comparable performance on a multicore architecture. We also consider the case when the matrix is symmetric indefinite. For highly unsymmetric systems, we use a completely different approach based on developing a parallel version of a Markowitz threshold ordering. This work is less advanced but we discuss some of the algorithmic challenges involved. Finally, we briefly discuss using a hybrid direct-iterative solver that combines the best of the two approaches and enables the solution of even larger problems in parallel.
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Notes
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An entry that is zero in A but is nonzero in the corresponding entry of the factors is termed fill-in.
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Acknowledgements
This work is supported by the NLAFET Project funded by the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement 671633. We would like to thank Philippe Gambron for doing the experiments reported in Sect. 8, Jonathan Hogg for his earlier work in the project developing the first versions of some of the kernels used in Sects. 5 and 6, and Tim Davis (Texas A&M) for discussions on parallel Markowitz. We also thank Jennifer Scott and Bo Kågström (Umeå) for their comments on a draft of this manuscript.
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Appendix
Test matrices used in experiments.
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Duff, I., Lopez, F., Nakov, S. (2018). Sparse Direct Solution on Parallel Computers. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. NAO 2017. Springer Proceedings in Mathematics & Statistics, vol 235. Springer, Cham. https://doi.org/10.1007/978-3-319-90026-1_4
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