A Parallel Prefix Algorithm for Almost Toeplitz Tridiagonal Systems
Sun Xian-He,D Joslin Ronald +1 more
TL;DR: Experimental results show that the simple parallel prefix algorithm is a good algorithm for symmetric, almost symmetric Toeplitz tridiagonal systems and for the compact scheme on high-performance computers.
read more
Abstract: A compact scheme is a discretization scheme that is advantageous in obtaining highly accurate solutions. However, the resulting systems from compact schemes are tridiagonal systems that are difficult to solve efficiently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel prefix (SPP), is proposed. The SPP algorithm requires less memory than the conventional LU decomposition and is efficient on parallel machines. It consists of a prefix communication pattern and AXPY operations. Both the computation and the communication can be truncated without degrading the accuracy when the system is diagonally dominant. A formal accuracy study has been conducted to provide a simple truncation formula. Experimental results have been measured on a MasPar MP-1 SIMD machine and on a Cray 2 vector machine. Experimental results show that the simple parallel prefix algorithm is a good algorithm for symmetric, almost symmetric Toeplitz tridiagonal systems and for the compact scheme on high-performance computers.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Figures
Figure 9. Measured and predicted accuracy for solving matrix A2. 
Figure 4. Communication of Fast sine transform method. 
Figure 12. Timing of SPP and LU algorithms: single system. 
Figure 13. Timing of SPP and LU algorithms: multiple right sides. 
Table 1. Computation and communication count of the Simple Pre x Algorithm 
Figure 8. Measured and predicted accuracy for solving matrix A1.
Citations
•Proceedings Article
A high-order direct solver for Helmholtz equations with Neumann boundary conditions
He Xian-Sun,Yu Zhuang +1 more
- 01 Feb 1997
TL;DR: Analytical and experimental results show this newly proposed fourth order Helmholtz solver is comparable to the conventional second-order elliptic solver when accuracy is not a primary concern and is significantly faster than that of the conventional solver if a highly accurate solution is required.
19
Discussion of DNS: Past, Present, and Future
Ronald D. Joslin
- 01 Jan 1997
TL;DR: This paper covers the review, status, and projected future of direct numerical simulation methodology relative to the state-of-the-art in computer technology, numerical methods, and the trends in fundamental research programs.
10
Adaptive wavelet ADI method: application and parallelization
Wei Cai,Xian-He Sun +1 more
- 21 Aug 2000
TL;DR: It is shown that the adaptive wavelet ADI method is highly parallel and scalable, a method which is fundamentally more appropriate for highly parallel computing than existing methods.
5
•Book
A scalable parallel algorithm for periodic symmetric Toeplitz tridiagonal systems
Xian-He Sun
- 01 Jan 2001
TL;DR: A scalable parallel algorithm is proposed for solving periodic symmetric Toeplitz tridiagonal systems in this study that has the same parallel computation count as that of the SPP algorithm for non-periodic systems, and it requires only shift communication.
3
Automatic performance modeling of multithreaded programs
Alexander Tarvo
- 31 May 2014
TL;DR: A novel methodology for automatically building performance models of industrial multithreaded programs with complex non-linear dependency between their configuration and the performance is presented.
References
Compact finite difference schemes with spectral-like resolution
TL;DR: In this article, the authors present finite-difference schemes for the evaluation of first-order, second-order and higher-order derivatives yield improved representation of a range of scales and may be used on nonuniform meshes.
6.3K
Direct methods for sparse matrices
TL;DR: This book aims to be suitable also for a student course, probably at MSc level, and the subject is intensely practical and this book is written with practicalities ever in mind.
2K
A Fast Direct Solution of Poisson's Equation Using Fourier Analysis
TL;DR: This work has developed a direct method of solution involving Fourier analysis which can solve Poisson''s equation in a square region covered by a 48 x 48 mesh in 0.9 seconds on the IBM 7090.
715
Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique
TL;DR: In this paper, a general fourth order differencing scheme was developed and applied to three viscous test problems to verify the accuracy and applicability of the technique, which is atypical since only three nodes are necessary to obtain the desired fourth order accuracy.
492
•Book
Solution of Partial Differential Equations on Vector and Parallel Computers
James M. Ortega,Robert G. Voigt +1 more
- 01 Jan 1987
TL;DR: The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms.