scholarly journals PARALLEL SOLUTION METHODS AND PRECONDITIONERS FOR EVOLUTION EQUATIONS

2018 ◽  
Vol 23 (2) ◽  
pp. 287-308 ◽  
Author(s):  
Owe Axelsson ◽  
Maya Neytcheva ◽  
Zhao-Zheng Liang
Keyword(s):  
Computer Architecture ◽  
High Performance ◽  
Krylov Subspace ◽  
Evolution Equations ◽  
Linear Equations ◽  
Nonlinear Problems ◽  
Iterative Refinement ◽  
Clear Trend ◽  
Eigenvalue Bounds ◽  
Solution Methods

The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow efficient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization. The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very efficient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative refinement method. The analytical background is shown as well as some illustrating numerical examples.

scholarly journals Krylov subspace methods for estimating operator-vector multiplications in Hilbert spaces

2021 ◽  
Author(s):  
Yuka Hashimoto ◽  
Takashi Nodera
Keyword(s):  
Krylov Subspace ◽  
Time Series Data ◽  
Linear Equations ◽  
Transfer Operator ◽  
Krylov Subspace Methods ◽  
Krylov Subspace Method ◽  
Linear Operators ◽  
Series Data ◽  
Subspace Method ◽  
Subspace Methods

AbstractThe Krylov subspace method has been investigated and refined for approximating the behaviors of finite or infinite dimensional linear operators. It has been used for approximating eigenvalues, solutions of linear equations, and operator functions acting on vectors. Recently, for time-series data analysis, much attention is being paid to the Krylov subspace method as a viable method for estimating the multiplications of a vector by an unknown linear operator referred to as a transfer operator. In this paper, we investigate a convergence analysis for Krylov subspace methods for estimating operator-vector multiplications.

Application of High-Performance Computing to Numerical Simulation of Human Movement

1995 ◽  
Vol 117 (1) ◽  
pp. 155-157 ◽  
Author(s):  
F. C. Anderson ◽  
J. M. Ziegler ◽  
M. G. Pandy ◽  
R. T. Whalen
Keyword(s):  
Computer Architecture ◽  
High Performance ◽  
Large Scale ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Human Movement ◽  
Parallel Machine ◽  
Optimal Controls ◽  
Processing Machine ◽  
Vector Processing

We have examined the feasibility of using massively-parallel and vector-processing supercomputers to solve large-scale optimization problems for human movement. Specifically, we compared the computational expense of determining the optimal controls for the single support phase of gait using a conventional serial machine (SGI Iris 4D25), a MIMD parallel machine (Intel iPSC/860), and a parallel-vector-processing machine (Cray Y-MP 8/864). With the human body modeled as a 14 degree-of-freedom linkage actuated by 46 musculotendinous units, computation of the optimal controls for gait could take up to 3 months of CPU time on the Iris. Both the Cray and the Intel are able to reduce this time to practical levels. The optimal solution for gait can be found with about 77 hours of CPU on the Cray and with about 88 hours of CPU on the Intel. Although the overall speeds of the Cray and the Intel were found to be similar, the unique capabilities of each machine are better suited to different portions of the computational algorithm used. The Intel was best suited to computing the derivatives of the performance criterion and the constraints whereas the Cray was best suited to parameter optimization of the controls. These results suggest that the ideal computer architecture for solving very large-scale optimal control problems is a hybrid system in which a vector-processing machine is integrated into the communication network of a MIMD parallel machine.

Nonlinear problems of shell theory and their solution methods (review)

1991 ◽  
Vol 27 (10) ◽  
pp. 929-947 ◽  
Author(s):  
Ya. M. Grigorenko ◽  
V. I. Gulyaev
Keyword(s):  
Shell Theory ◽  
Nonlinear Problems ◽  
Solution Methods

scholarly journals Changing Trends in Computer Architecture : A Comprehensive Analysis of ARM and x86 Processors

2021 ◽  
pp. 619-631
Author(s):  
Khushi Gupta ◽  
Tushar Sharma
Keyword(s):  
Computer Architecture ◽  
High Performance ◽  
Comprehensive Analysis ◽  
Low Power Consumption ◽  
Modern World ◽  
Development Environment ◽  
Processor Architectures ◽  
Software Development Environment ◽  
Changing Trends ◽  
Microprocessor Industry

In the modern world, we use microprocessors which are either based on ARM or x86 architecture which are the most common processor architectures. ARM originally stood for ‘Acorn RISC Machines’ but over the years changed to ‘Advanced RISC Machines’. It was started as just an experiment but showed promising results and now it is omnipresent in our modern devices. Unlike x86 which is designed for high performance, ARM focuses on low power consumption with considerable performance. Because of the advancements in the ARM technology, they are becoming more powerful than their x86 counterparts. In this analysis we will collate the two architectures briefly and conclude which microprocessor will dominate the microprocessor industry. The processor which will perform better in different tests will be more suitable for the reader to use in their application. The shift in the industry towards ARM processors can change how we write softwares which in turn will affect the whole software development environment.

scholarly journals A Comparative Study on Power Flow Methods for Direct-Current Networks Considering Processing Time and Numerical Convergence Errors

Electronics ◽  
2020 ◽  
Vol 9 (12) ◽  
pp. 2062
Author(s):  
Luis Fernando Grisales-Noreña ◽  
Oscar Danilo Montoya ◽  
Walter Julian Gil-González ◽  
Alberto-Jesus Perea-Moreno ◽  
Miguel-Angel Perea-Moreno
Keyword(s):  
Direct Current ◽  
Power Flow ◽  
Processing Time ◽  
Linear Equations ◽  
Triangular Matrix ◽  
Taylor Series Expansion ◽  
Distribution Networks ◽  
Matrix Formulation ◽  
Numerical Convergence ◽  
Solution Methods

This study analyzes the numerical convergence and processing time required by several classical and new solution methods proposed in the literature to solve the power-flow problem (PF) in direct-current (DC) networks considering radial and mesh topologies. Three classical numerical methods were studied: Gauss–Jacobi, Gauss–Seidel, and Newton–Raphson. In addition, two unconventional methods were selected. They are iterative and allow solving the DC PF in radial and mesh configurations. The first method uses a Taylor series expansion and a set of decoupling equations to linearize around the desired operating point. The second method manipulates the set of non-linear equations of the DC PF to transform it into a conventional fixed-point form. Moreover, this method is used to develop a successive approximation methodology. For the particular case of radial topology, three methods based on triangular matrix formulation, graph theory, and scanning algorithms were analyzed. The main objective of this study was to identify the methods with the best performance in terms of quality of solution (i.e., numerical convergence) and processing time to solve the DC power flow in mesh and radial distribution networks. We aimed at offering to the reader a set of PF methodologies to analyze electrical DC grids. The PF performance of the analyzed solution methods was evaluated through six test feeders; all of them were employed in prior studies for the same application. The simulation results show the adequate performance of the power-flow methods reviewed in this study, and they permit the selection of the best solution method for radial and mesh structures.

scholarly journals Krylov Subspace Solvers and Preconditioners

2018 ◽  
Vol 63 ◽  
pp. 1-43
Author(s):  
C. Vuik
Keyword(s):  
Iterative Methods ◽  
Krylov Subspace ◽  
State Of The Art ◽  
Linear Equations ◽  
Iterative Solution ◽  
Krylov Subspace Methods ◽  
Numerical Linear Algebra ◽  
Subspace Methods ◽  
Starting Point ◽  
Lecture Notes

In these lecture notes an introduction to Krylov subspace solvers and preconditioners is presented. After a discretization of partial differential equations large, sparse systems of linear equations have to be solved. Fast solution of these systems is very urgent nowadays. The size of the problems can be 1013 unknowns and 1013 equations. Iterative solution methods are the methods of choice for these large linear systems. We start with a short introduction of Basic Iterative Methods. Thereafter preconditioned Krylov subspace methods, which are state of the art, are describeed. A distinction is made between various classes of matrices. At the end of the lecture notes many references are given to state of the art Scientific Computing methods. Here, we will discuss a number of books which are nice to use for an overview of background material. First of all the books of Golub and Van Loan [19] and Horn and Johnson [26] are classical works on all aspects of numerical linear algebra. These books also contain most of the material, which is used for direct solvers. Varga [50] is a good starting point to study the theory of basic iterative methods. Krylov subspace methods and multigrid are discussed in Saad [38] and Trottenberg, Oosterlee and Schüller [42]. Other books on Krylov subspace methods are [1, 6, 21, 34, 39].

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