Abstract
The results of a parallel implementation of a randomized vector algorithm for solving systems of linear equations are presented in the paper. The solution is represented in the form of a Neumann series. The stochastic method computes this series by sampling only random columns, avoiding multiplication of matrix by matrix and matrix by vector. We consider the case when the matrix is too large to fit in random-access memory (RAM). We use two approaches to solve this problem. In the first approach, the matrix is divided into parts that are distributed among MPI processes and stored in the available RAM of the cluster nodes. In the second approach, the entire matrix is stored on each node’s hard drive, loaded into RAM, and processed in parts. Independent Monte Carlo experiments for random column indices are distributed among MPI processes or OpenMP threads for both approaches to matrix storage. The efficiency of parallel implementations is analyzed. Results are given for a system governed by dense matrices of size \(10^4\) and \(10^5\).
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All data generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The work is supported by the Russian Science Foundation, Grant 19-11-00019 in the part of randomized algorithms implementation, and Russian Fund of Basic Research, under Grant 20-51-18009, in the part of stochastic simulation theory development
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KKS proposed and developed the randomized vector algorithm for solving large systems of linear equations; designed and directed the project. SK proposed the computational scheme for the implementations of the algorithm; contributed to the developing of the algorithm implementation, analysis of the results and writing of the manuscript. AK implemented a parallel version of the algorithm; wrote the manuscript; contributed to the analysis of the results. All authors discussed the results and commented on the manuscript.
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Sabelfeld, K.K., Kireev, S. & Kireeva, A. Parallel implementations of randomized vector algorithm for solving large systems of linear equations. J Supercomput 79, 10555–10569 (2023). https://doi.org/10.1007/s11227-023-05079-5
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DOI: https://doi.org/10.1007/s11227-023-05079-5