A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems

Japan J. Indust. Appl. Math.

2019

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Many problems in science and engineering often require to solve a long sequence of large-scale non-Hermitian linear systems with different Right-hand sides (RHSs) but a unique operator. Efficiently solving such problems on extreme-scale platforms requires the minimization of global communications, reduction of synchronization and promotion of asynchronous communications. Unite and Conquer GMRES/LS-ERAM (UCGLE) method [30] is a suitable candidate with the reduction of global communications and the synchronization points of all computing units. It consists of three computing algorithms with asynchronous communications that allow the use of approximate eigenvalues to accelerate the convergence of solving linear systems and to improve the fault tolerance. In this paper, we extend both the mathematical model and the implementation of UCGLE method to adapt to solve sequences of linear systems. The eigenvalues obtained in solving previous linear systems by UCGLE can be recycled, improved on the fly and applied to construct a new initial guess vector for subsequent linear systems, which can achieve a continuous acceleration to solve linear systems in sequence. Numerical experiments using different test matrices to solve sequences of linear systems on supercomputer Tianhe-2 indicate a substantial decrease in both computation time and iteration steps when the approximate eigenvalues are recycled to generate the initial guess vectors.

Section: Workflow and Component Implementationmentioning

confidence: 99%

Section: Relation Between Ls Residual and Approximate Eigenvaluesmentioning

confidence: 99%

Section: Experiments Of Ucgle For Sequences Of Linear Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A distributed and parallel unite and conquer method to solve sequences of non-Hermitian linear systems

Japan J. Indust. Appl. Math.

2019

Self Cite

“…The spectral distribution of the preconditioned matrix might speed up the convergence. As an example, X. Wu [14] et al implemented a Unite and Conquer hybrid method for solving linear systems with the combination of a Krylov linear system solver, an eigenvalue solver, and a Least Square polynomial method proposed by Y. Saad [11] in 1987. In the preconditioning part of this method, the dominant eigenvalues are used to accelerate the convergence.…”

Section: Introductionmentioning

confidence: 99%

A Parallel Generator of Non-Hermitian Matrices Computed from Given Spectra

High Performance Computing for Computational Science – VECPAR 2018

2019

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Iterative linear algebra methods are the important parts of the overall computing time of applications in various fields since decades. Recent research related to social networking, big data, machine learning and artificial intelligence has increased the necessity for non-hermitian solvers associated with much larger sparse matrices and graphs. The analysis of the iterative method behaviors for such problems is complex, and it is necessary to evaluate their convergence to solve extremely large non-Hermitian eigenvalue and linear problems on parallel and/or distributed machines. This convergence depends on the properties of spectra. Then, it is necessary to generate large matrices with known spectra to benchmark the methods. These matrices should be non-Hermitian and non-trivial, with very high dimension. This paper highlights a scalable matrix generator that uses the user-defined spectrum to construct largescale sparse matrices and to ensure their eigenvalues as the given ones with high accuracy. This generator is implemented on CPUs and multi-GPU platforms. Good strong and weak scaling performance is obtained on several supercomputers. We also propose a method to verify its ability to guarantee the given spectra.

A parallel generator of non‐Hermitian matrices computed from given spectra

Concurrency and Computation

2020

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Summary Iterative linear algebra methods to solve linear systems and eigenvalue problems with non‐Hermitian matrices are important for both the simulation arising from diverse scientific fields and the applications related to big data, machine learning, and artificial intelligence. The spectral property of these matrices has impacts on the convergence of these solvers. Moreover, with the increase of the size of applications, iterative methods are implemented in parallel on clusters. Analysis of their behaviors with non‐Hermitian matrices on supercomputers is so complex that we need to generate large‐scale matrices with different given spectra for benchmarking. These test matrices should be non‐Hermitian and nontrivial, with high dimension. This paper highlights a scalable matrix generator that constructs large sparse matrices using the user‐defined spectrum, and the eigenvalues of generated matrices are ensured to be the same as the predefined spectrum. This generator is implemented on CPUs and multi‐GPUs platforms, with good strong and weak scaling performance on several supercomputers. We also propose a method to verify its ability to guarantee the given spectra. Finally, we give an example to evaluate the numerical properties and parallel performance of iterative methods using this matrix generator.