Parallel algorithms for solving linear systems with block-tridiagonal matrices on multi-core CPU with GPU

Adding element tridiagonal periodic matrices have an important effect for the algorithms of solving linear systems，computing the inverses, the triangular factorization，the boundary value problems by finite difference methods, interpolation by cubic splines, three-term difference equations and so on. In this paper, we give a fast algorithm for the Inverse Matrices of periodic adding element tridiagonal matrices.

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Linear complexity parallel algorithms for linear systems of equations with recursive structure

Linear Algebra and its Applications ◽

10.1016/0024-3795(87)90113-3 ◽

1987 ◽

Vol 88-89 ◽

pp. 271-315 ◽

Cited By ~ 46

Author(s):

I. Gohberg ◽

T. Kailath ◽

I. Koltracht ◽

P. Lancaster

Keyword(s):

Parallel Algorithms ◽

Linear Systems ◽

Linear Complexity ◽

Systems Of Equations ◽

Recursive Structure ◽

Linear Systems Of Equations

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Book Reviews

Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems and Control Engineering ◽

10.1243/pime_proc_1994_208_318_02 ◽

1994 ◽

Vol 208 (2) ◽

pp. 135-136

Keyword(s):

Optimal Control ◽

The Netherlands ◽

Parallel Algorithms ◽

Linear Systems ◽

Large Scale

PRECISION SENSORS, ACTUATORS AND SYSTEMS. Edited by H S Tzou and T Fukuda, Kluwer Academic Publishers, Dordrecht, The Netherlands: 1992, pp 480, £105 (ISBN 0 79232 015 8) PARALLEL ALGORITHMS FOR OPTIMAL CONTROL OF LARGE SCALE LINEAR SYSTEMS. Zoran Gajic and Xuemin Shen, Springer-Verlag, Heidelberg: 1993, pp 455, DM 114 (ISBN 3 54019 825 3)

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Parallel Algorithms for Solving Linear Systems on Hybrid Computers

Cybernetics and Computer Technologies ◽

10.34229/2707-451x.20.2.6 ◽

2020 ◽

pp. 53-66

Author(s):

Alexander Khimich ◽

Victor Polyanko ◽

Tamara Chistyakova

Keyword(s):

Parallel Algorithms ◽

Computer Architecture ◽

Linear Systems ◽

High Performance ◽

Numerical Data ◽

Algebraic Equations ◽

Existing Problems ◽

Mathematical Properties ◽

Linear Algebraic Equations ◽

Computer Problem

Introduction. At present, in science and technology, new computational problems constantly arise with large volumes of data, the solution of which requires the use of powerful supercomputers. Most of these problems come down to solving systems of linear algebraic equations (SLAE). The main problem of solving problems on a computer is to obtain reliable solutions with minimal computing resources. However, the problem that is solved on a computer always contains approximate data regarding the original task (due to errors in the initial data, errors when entering numerical data into the computer, etc.). Thus, the mathematical properties of a computer problem can differ significantly from the properties of the original problem. It is necessary to solve problems taking into account approximate data and analyze computer results. Despite the significant results of research in the field of linear algebra, work in the direction of overcoming the existing problems of computer solving problems with approximate data is further aggravated by the use of contemporary supercomputers, do not lose their significance and require further development. Today, the most high-performance supercomputers are parallel ones with graphic processors. The architectural and technological features of these computers make it possible to significantly increase the efficiency of solving problems of large volumes at relatively low energy costs. The purpose of the article is to develop new parallel algorithms for solving systems of linear algebraic equations with approximate data on supercomputers with graphic processors that implement the automatic adjustment of the algorithms to the effective computer architecture and the mathematical properties of the problem, identified in the computer, as well with estimates of the reliability of the results. Results. A methodology for creating parallel algorithms for supercomputers with graphic processors that implement the study of the mathematical properties of linear systems with approximate data and the algorithms with the analysis of the reliability of the results are described. The results of computational experiments on the SKIT-4 supercomputer are presented. Conclusions. Parallel algorithms have been created for investigating and solving linear systems with approximate data on supercomputers with graphic processors. Numerical experiments with the new algorithms showed a significant acceleration of calculations with a guarantee of the reliability of the results. Keywords: systems of linear algebraic equations, hybrid algorithm, approximate data, reliability of the results, GPU computers.

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A parallel preconditioner based on the approximation of an inverse matrix by power series for solving sparse linear systems on graphics processors

Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie) ◽

10.26089/nummet.v20r439 ◽

2019 ◽

pp. 444-456

Author(s):

А.В. Юлдашев ◽

Н.В. Репин ◽

В.В. Спеле

Keyword(s):

Power Series ◽

Linear Systems ◽

Numerical Experiments ◽

Computing System ◽

Neumann Series ◽

Inverse Matrix ◽

Reservoir Modeling ◽

Sparse Linear Systems ◽

Graphics Processors ◽

Tridiagonal Matrices

Рассмотрена применимость метода AIPS, аппроксимирующего обратную матрицу на основе степенного разложения в ряд Неймана, в рамках двухступенчатого предобусловливателя CPR. Предложен ориентированный на архитектуру CUDA параллельный алгоритм решения линейных систем с трехдиагональной матрицей, состоящей из независимых блоков различного размера. Показано, что реализация предложенного алгоритма может более чем в 2 раза превосходить по быстродействию функции решения трехдиагональных систем из библиотеки cuSPARSE. Проведено тестирование метода BiCGStab с предобусловливателем CPRAIPS на современных GPU, в том числе на гибридной вычислительной системе с 4 GPU NVIDIA Tesla V100, показавшее приемлемую масштабируемость данного предобусловливателя, а также возможность ускорить решение линейных систем, характерных для задачи гидродинамического моделирования нефтегазовых месторождений, по сравнению с CPRAMG. The applicability of the AIPS method approximating an inverse matrix using Neumann series is considered in the framework of the CPR two stage preconditioner. A parallel CUDAoriented algorithm is proposed for solving linear systems with tridiagonal matrices consisting of independent blocks of different sizes. It is shown that the implementation of the proposed algorithm can be more than twice the speed of the similar functions from the cuSPARSE library. Experimental evaluation of the BiCGStab method with the CPRAIPS preconditioner on modern GPUs, including a hybrid computing system with 4 GPU NVIDIA Tesla V100, is performed. Numerical experiments show an adequate scalability of this preconditioner as well as the possibility (compared to the CPRAMG) to accelerate the solution of linear systems being typical for the reservoir modeling problems.

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