A parallel solving method for block-tridiagonal equations on CPU–GPU heterogeneous computing systems

Wu, Yang; Li, Kenli; Li, Keqin

doi:10.1007/s11227-016-1881-x

Cited by 20 publications

(8 citation statements)

References 39 publications

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“…Then the numerical solution of such a set of inner boundary points can be calculated quickly only by solving the fourth order sparse linear Equation (15). In the same way, the points on the other five groups of inner boundary lines are also calculated by Group Explicit method, and we will not represent them one by one here.…”

Section: Implementation Of Even Level Iterationmentioning

confidence: 99%

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Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

Liu

2020

Mathematics

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Poisson equation is a widely used partial differential equation. It is very important to study its numerical solution. Based on the strategy of domain decomposition, the alternating asymmetric iterative algorithm for 3D Poisson equation is provided. The solution domain is divided into several sub-domains, and eight asymmetric iterative schemes with the relaxation factor for 3D Poisson equation are constructed. When the numbers of iteration are odd or even, the computational process of the presented iterative algorithm are proposed respectively. In the calculation of the inner interfaces, the group explicit method is used, which makes the algorithm to be performed fast and in parallel, and avoids the difficulty of solving large-scale linear equations. Furthermore, the convergence of the algorithm is analyzed theoretically. Finally, by comparing with the numerical experimental results of Jacobi and Gauss Seidel iterative algorithms, it is shown that the alternating asymmetric iterative algorithm based on domain decomposition has shorter computation time, fewer iteration numbers and good parallelism.

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Section: Implementation Of Even Level Iterationmentioning

confidence: 99%

“…C. P. Stone et al [14] analyze the performance of a block tridiagonal benchmark. Many authors have given the implementation of scalar tridiagonal solver on GPUs [15][16][17]. Y. Zhang [16] also illustrates several methods to solve tridiagonal systems on GPUs.…”

Section: Introductionmentioning

confidence: 99%

Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

Liu

2020

Mathematics

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“…On the other hand, some parallel computing algorithms are also designed for solving tridiagonal systems on graphics processing unit (GPU), which are parallel cyclic reduction [9] and partition methods [10]. Recently, Yang et al [11] presented a parallel solving method that mixes direct and iterative methods for block-tridiagonal equations on CPU-GPU heterogeneous computing systems, whereas Myllykoski et al [12] proposed a generalized graphics processing unit implementation of partial solution variant of the cyclic reduction (PSCR) method to solve certain types of separable block tridiagonal linear systems. Compared to an equivalent CPU implementation that utilizes a single CPU core, PSCR method indicated up to 24-fold speedups.…”

Section: Introductionmentioning

confidence: 99%

Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

Wei

Jiang

et al. 2019

Adv Differ Equ

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In this paper, we deal mainly with a class of periodic tridiagonal Toeplitz matrices with perturbed corners. By matrix decomposition with the Sherman–Morrison–Woodbury formula and constructing the corresponding displacement of matrices we derive the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type I in the form of products of Fermat numbers and some initial values. Furthermore, the properties of type II matrix can be also obtained, which benefits from the relation between type I and II matrices. Finally, we propose two algorithms for computing these properties and make some analysis about them to illustrate our theoretical results.

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“…On the other hand, some parallel computing algorithms are also designed for solving tridiagonal systems on graphics processing unit (GPU), which are parallel cyclic reduction [18] and partition methods [19]. Recently, Yang et al [20] presented a parallel solving method which mixes direct and iterative methods for block-tridiagonal equations on CPU-GPU heterogeneous computing systems, while Myllykoski et al [21] proposed a generalized graphics processing unit implementation of partial solution variant of the cyclic reduction (PSCR) method to solve certain types of separable block tridiagonal linear systems. Compared to an equivalent CPU implementation that utilizes a single CPU core, PSCR method indicated up to 24-fold speedups.…”

Section: Introductionmentioning

confidence: 99%

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

et al. 2019

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In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

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A parallel solving method for block-tridiagonal equations on CPU–GPU heterogeneous computing systems

Cited by 20 publications

References 39 publications

Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

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