A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method
TL;DR: This study develops a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides and proposes a parallel realization of the domain decomposition method (the Schur complement method).
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Abstract: In this study, we develop a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides. The method is a generalization of the parallel dichotomy algorithm for solving systems of linear equations with tridiagonal matrices \cite{terekhov:Dichotomy}. Using this approach, we propose a parallel realization of the domain decomposition method (\mbox{the Schur} complement method). The calculation of acoustic wave fields using the spectral-difference technique improves the efficiency of the parallel algorithms. A near-linear dependence of the speedup with the number of processors is attained using both several and several thousands of processors. This study is innovative because the parallel algorithm developed for solving block-tridiagonal systems of equations is an effective and simple set of procedures for solving engineering tasks on a supercomputer.
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Figures
Figure 2: Dependence of the speedup value on the number of processors for M = 60 (a) and M = 150 (b) on various N . 
Figure 4: Snapshots for the wave field at t = 3s (a) for model pic.3.a and same model with additional low-velocity layer (b). Nz ×Nr = 12134 × 65536. 
Table 4: P is the total time of the preliminaries, T is the time of computing one harmonic from (9), S is the speedup value, d is the bandwidth of the precondition matrix, NP is the number of processors, Nr,Nz is the number of mesh size towards R and Z, respectively. 
Figure 3: Medium model (a) and solution mesh (b). 
Table 3: (Gen) is the time of calculation of the preconditioning matrix B. (Total) is the time needed for solution to one problem of the form of (13). In this case it is necessary to carry out (Iter) iterations. (d) is the bandwidth of the preconditioning matrix, the number of processes being constant p = 256. 
Figure 1: Components of the solution vector to be calculated for dividing the original equations system into subproblems.
Citations
Tridiagonal Matrix Algorithm for Real-Time Simulation of a Two-Dimensional PEM Fuel Cell Model
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Semi-supervised learning for hierarchically structured networks
TL;DR: Two versions of an algorithm, based on semi-supervised learning, for a hierarchically structured network are proposed and the experimental results show that the proposed algorithms perform well with hierarchic structured data, and, outperform an ordinary semi- supervised learning algorithm.
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A parallel solving method for block-tridiagonal equations on CPU---GPU heterogeneous computing systems
Yang Wangdong,Kenli Li,Keqin Li +2 more
TL;DR: A solving method which mixes direct and iterative methods for solving block-tridiagonal systems of linear equations, and an improved algorithm to solve the sub-equations by thread blocks on GPU, so as to significantly reduce the latency of memory access.
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The Laguerre finite difference one-way equation solver
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TL;DR: Numerical experiments and numerical comparisons show that the PML technique works better than the others in all cases; using it allows to obtain a higher accuracy in some problems and a release of computational requirements in some others.
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TL;DR: The Handbook of Mathematical Functions with Formulas (HOFF-formulas) as mentioned in this paper is the most widely used handbook for mathematical functions with formulas, which includes the following:
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