scholarly journals Parallel Algorithms for Solving Linear Systems on Hybrid Computers

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.

scholarly journals Performance Analysis of Effective Symbolic Methods for Solving Band Matrix SLAEs

2019 ◽  
Vol 214 ◽  
pp. 05004
Author(s):  
Milena Veneva ◽  
Alexander Ayriyan
Keyword(s):  
Performance Analysis ◽  
High Performance ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Performance Study ◽  
Band Matrix ◽  
Linear Algebraic Equations ◽  
The Individual ◽  
Performance Computing ◽  
Symbolic Algorithms

This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pentadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms “HybriLIT” and “Avitohol”. The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed.

METHOD OF SOLVING SYSTEM OF LINEAR ALGEBRAIC EQUATIONS WITH USE OF HIGH PERFORMANCE COMPUTING ON GPU

2019 ◽  
pp. 112-115
Author(s):  
M. Z. Benenson
Keyword(s):  
High Performance ◽  
Operation Time ◽  
General Purpose ◽  
Algebraic Equations ◽  
Graphics Processors ◽  
Computing Platform ◽  
Software Interface ◽  
Linear Algebraic Equations ◽  
Gauss Method ◽  
Performance Computing

The  article  discusses  the  use  of  graphics  processing  units  for  solving  large  system  of  linear  algebraic  equations  (SLAE).  A heterogeneous multiprocessor computing platform produced by the NIIVK, whose architecture allows the integration of general‑ purpose microprocessor modules with graphics processor modules was used as an equipment for solving SLAEs. The description  of the SLAE solution program, developed on the basis of the CUBLAS CUDA software interface library, is given. A method is proposed for increasing the accuracy of calculations of linear systems based on the use of a modified Gauss method. It has been  established that the use of the modified Gauss method practically does not increase the program operation time with a significant  increase in the accuracy of calculations. It is concluded that the use of graphics processors for solving SLAEs allows processing  matrices of a larger size compared to the use of general‑purpose microprocessors.

Modern computer technologies for modeling the life cycle of buildings based on parallel calculations

2019 ◽  
Keyword(s):  
Life Cycle ◽  
High Performance ◽  
Strain Analysis ◽  
Positive Definite Matrix ◽  
Algebraic Equations ◽  
Software Complex ◽  
Parallel Calculations ◽  
Modern Computer ◽  
Linear Algebraic Equations ◽  
Geological Conditions

The article deals with high-performance information technology (HPC) for the problems of stress-strain analysis at all stages of the life cycle of buildings and structures: construction, operation and reconstruction. The results of numerical simulation of high buildings using software as a processor component based on a new hybrid algorithm for solving systems of linear algebraic equations [1] with a symmetric positive-definite matrix that combines computation on multi-core processors and graphs. It has been found that to accelerate the calculations, hybrid systems that combine multi-core CPUs with accelerator coprocessors, including GPUs, are promising [5]. To test the effectiveness of the proposed parallel algorithm for solving systems of linear algebraic equations [1], numerical experiments were carried out at the most dangerous loads of a 27-story building. Results of numerical researches with use for preprocessor (input of initial data) and postprocessor (output of results of calculations) of processing of the LIRA-SAPR software complex are presented [2, 4, 6]. The results of numerical studies of the behavior of structures of high buildings have shown a multiple reduction in the time of solving systems of linear algebraic equations with symmetric matrices on multiprocessor (multi-core) computers with graphical accelerators using the proposed hybrid algorithms [1]. High-performance technologies based on parallel calculations give more effect than more complex processes: modeling of life cycle of high buildings, bridges, especially complex structures of NPPs, etc. for static and dynamic loads, including emergencies in normal and difficult geological conditions, which make up 70% of Ukraine's territories.

Calculation of Correlation Matrices for Linear Systems Subjected to Nonwhite Excitation

1972 ◽  
Vol 39 (2) ◽  
pp. 559-562 ◽  
Author(s):  
I-Min Yang ◽  
W. D. Iwan
Keyword(s):  
Spectral Density ◽  
Linear Systems ◽  
Random Response ◽  
Algebraic Equations ◽  
Correlation Matrices ◽  
Linear Algebraic Equations ◽  
Special Case

This paper presents an approach which provides a particularly simple and direct way of determining the instantaneous correlation matrices for the stationary random response of multidegree-of-freedom linear systems subjected to excitations of nearly arbitrary spectral density. In the special case of white excitation, the instantaneous correlation matrices are determined directly from a set of linear algebraic equations. When the excitation is nonwhite, some integrals must be evaluated before solving a system of linear algebraic equations. However, the form of these integrals is considerably simpler than that encountered in other common approaches.

Unrestricted Harmonic Balance

1980 ◽  
Vol 35 (10) ◽  
pp. 1054-1061 ◽  
Author(s):  
Friedrich Franz Seelig
Keyword(s):  
Linear Systems ◽  
Harmonic Balance ◽  
Open Systems ◽  
Periodic Structures ◽  
Simulation Method ◽  
Algebraic Equations ◽  
Good Convergence ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
The Stability

Abstract Periodic structures in chemical kinetic systems can be evaluated by an extension of the well-known method of harmonic balance, which yields very simple expressions in the case of linear systems containing only zero and first order reactions. The far more interesting non-linear systems containing e.g. second order reactions which in case of open systems far from thermodynamic equilibrium give rise to non-classical phenomena like oscillations, chemical waves, excitability, hysteresis, multistability, dissipative structures etc. can be treated in a similar way by introducing new pseudo-linear quantities utilizing certain group properties of harmonic expansions. The resulting complicated implicit non-linear algebraic equations are solved by a method developed by Powell and show good convergence. Since this method - in contrast to the conventional method of simulation - is independent from the stability of the periodic structure to be evaluated it can even be applied to unstable cases where the simulation method necessarily fails. An evaluation of the stability is included in the developed computer program.

scholarly journals The inclusion of additional topics to the program of linear algebra course for economists and managers

2019 ◽  
pp. 21-25
Author(s):  
Sergej Yurevich Shashkin
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Algebraic Equations ◽  
Unified Approach ◽  
Regularization Algorithm ◽  
Unstable Systems ◽  
Linear Algebraic Equations

The paper generalizes the concept of “solving a system of linear algebraic equations in order to formulate a unified approach to the analysis of incompatible, indefinite and unstable systems”. Examples of unstable systems of linear algebraic equations are considered, which solutions depend on small changes in the numerical coefficients in the equations. The reasons for the instability of linear systems and the regularization algorithm for finding the solution of any system of linear algebraic equations are discussed. As the author notes, the Tikhonov regulatory algorithm is the most popular and practically convenient for solving unstable SLAES.

scholarly journals Residual-Based Simpler Block GMRES for Nonsymmetric Linear Systems with Multiple Right-Hand Sides

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Qinghua Wu ◽  
Liang Bao ◽  
Yiqin Lin
Keyword(s):  
Linear Systems ◽  
Linear Dependence ◽  
Numerical Experiments ◽  
Gmres Method ◽  
Algebraic Equations ◽  
Rank Deficiency ◽  
Nonsymmetric Linear Systems ◽  
Right Hand ◽  
Linear Algebraic Equations ◽  
Block Arnoldi

We propose in this paper a residual-based simpler block GMRES method for solving a system of linear algebraic equations with multiple right-hand sides. We show that this method is mathematically equivalent to the block GMRES method and thus equivalent to the simpler block GMRES method. Moreover, it is shown that the residual-based method is numerically more stable than the simpler block GMRES method. Based on the deflation strategy proposed by Calandra et al. (2013), we derive a deflation strategy to detect the possible linear dependence of the residuals and a near rank deficiency occurring in the block Arnoldi procedure. Numerical experiments are conducted to illustrate the performance of the new method.

scholarly journals On a method for calculating generalized normal solutions of underdetermined linear systems

2020 ◽  
Vol 44 (1) ◽  
pp. 133-136
Author(s):  
A.I. Zhdanov ◽  
Y.V. Sidorov
Keyword(s):  
Linear Systems ◽  
Iterative Refinement ◽  
Algebraic Equations ◽  
Underdetermined Systems ◽  
Underdetermined Linear Systems ◽  
Refinement Method ◽  
Linear Algebraic Equations ◽  
Chemical Equations ◽  
Minimum Residual ◽  
Novel Algorithm

The article presents a novel algorithm for calculating generalized normal solutions of underdetermined systems of linear algebraic equations based on special extended systems. The advantage of this method is the ability to solve very poorly conditioned (possibly sparse) underdetermined linear systems of large dimension using modern versions of the iterative refinement method based on the generalized minimum residual method (GMRES - IT). Results of applying the considered algorithm to solve the problem of balancing chemical equations (mass balance) are presented.

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