Parallel Tridiagonal Equation Solvers

Stone, Harold S.

doi:10.1145/355656.355657

Cited by 139 publications

(57 citation statements)

References 5 publications

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“…Most probably we can rule out the recursive doubling method since it has a scalar count with order higher than the other two methods. Stone [8] devised a modified recursive doubling procedure with a lower scalar count that is consistent. But the count is still about twice as big as either of the two competing methods.…”

Section: Comparison With Existing Parallel Methodsmentioning

confidence: 99%

“…With the recent development and availability of various parallel and vector computers, new algorithms have appeared for solving tridiagonal systems of equations suitable for these machines. Notable among these methods are the recursive doubling method {Stone [8]) and the cyclic reduction method (Lambiotte and Voigt [5]). The recursive doubling method is designed for a parallel computer such as the Illiac IV.…”

Section: Introductionmentioning

confidence: 99%

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A Parallel Method for Tridiagonal Equations

Wang

1981

ACM Trans. Math. Softw.

245

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A new (partition) method for solving a tndiagonal system of lmear equations is presented in this paper The method is suitable for both parallel and vector computers. Although the partition method has a shghtly higher vector operatmn count than those of the two competing methods (the recursive doubling method and the cychc reduction method), it has a scalar count much smaller than that of the recursive doubling. The scalar counts between the partition method and the cyclic reduction method are so close as to make a timing evaluation inconclusive without considering the data management problem, especmlly when large systems are solved. Various situations under which the partmon method can be preferable are described.

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Section: Comparison With Existing Parallel Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Parallel Method for Tridiagonal Equations

Wang

1981

ACM Trans. Math. Softw.

245

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“…= P ! ), because the algorithm, whether it is Partitioned Thomas method [7] or odd--even cyclic reduction [6], is easily applicable to a 1--D processor grid, and the usual block size n ! is much smaller than the number of blocks n !…”

Section: Selection Of the Parallel Algorithmmentioning

confidence: 99%

“…The "fill--in" blocks are managed to find a solution through the communications between the subsystems [7]. Conversely, the cyclic odd--even reduction algorithm [6] has no matrix fill--in step that requires significant additional time in the partitioned Thomas method. The logarithmic cyclic reductions of the algorithm are the most efficient when both P !…”

Section: Selection Of the Parallel Algorithmmentioning

confidence: 99%

“…The "tridiagonal" system is parallelized in the block row dimension, and the "block" matrix system is parallelized by distributing the blocks. In other words, in some cases the parallel algorithm for a block tridiagonal system is adopted in which the rows of the master matrix are divided among a one--dimensional (1--D) processor grid and they are calculated with a "cyclic reduction method" [6] or "partitioned Thomas method" [7--8] as in the simple tridiagonal problem. The "cyclic reduction method" of solution uses the simultaneous odd row elimination with logarithmic recursive steps, O(log(n !…”

Section: Introductionmentioning

confidence: 99%

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A block-tridiagonal solver with two-level parallelization for finite element-spectral codes

Lee¹,

Wright²

2014

Computer Physics Communications

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Abstract:Two--level parallelization is introduced to solve a massive block--tridiagonal matrix system. One--level is used for distributing blocks whose size is as large as the number of block rows due to the spectral basis, and the other level is used for parallelizing in the block row dimension. The purpose of the added parallelization dimension is to retard the saturation of the scaling due to communication overhead and inefficiencies in the single--level parallelization only distributing blocks. As a technique for parallelizing the tridiagonal matrix, the combined method of "Partitioned Thomas method" and "Cyclic Odd--Even Reduction" is implemented in an MPI--Fortran90 based finite element--spectral code (TORIC) that calculates the propagation of electromagnetic waves in a tokamak. The two--level parallel solver using thousands of processors shows more than 5 times improved computation speed with the optimized processor grid compared to the single--level parallel solver under the same conditions. Three--dimensional RF field reconstructions in a tokamak are shown as examples of the physics simulations that have been enabled by this algorithmic advance.

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Parallel block iterative method for multiaquifer flow models

Pini

Teatini

1997

Numer. Methods Partial Differential Eq.

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Flow in a multiaquifer porous system can be simulated by the so-called "quasi three-dimensional" models. When heterogeneous and/or aquitards with nonlinear hydrogeologic behavior are considered, a fully numerical approach is required for the model solution. If the finite element method is used to integrate the partial differential flow equations, the final solution of large systems is required. In the present article, an original iterative solution strategy is developed. The global system is decoupled into a number of smaller subsystems consistent with the geologic structure (aquitards and aquifers) of the multiaquifer system. The aquifer and the aquitard equations are solved separately with the modified conjugate gradient and the Thomas algorithms, respectively, while the final coupled solution is obtained with an iterative procedure equivalent to a Block Jacobi scheme. The procedure can be efficiently implemented on a parallel super-computer distributing the computational load, so that two successive blocks (related to an aquifer and the underlying aquitard) are solved on the same processor. The procedure is analyzed with linear porous media, where the convergence is theoretically ensured. The results obtained with a realistic linear multiaquifer system, employing a massively parallel computer like the CRAY T3D, have pointed out the high degree of parallelization of the algorithm. Comparison with the parallel implementation of the Block SOR and Block Gauss-Seidel schemes shows that parallel Block Jacobi performs significantly better with a reduction of the elapsed times, which depends on the rate of leakage between neighboring aquifers.

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Parallel Tridiagonal Equation Solvers

Abstract: Parallel Tridiagonal Equation Solversby Harold S. Stone

Cited by 139 publications

References 5 publications

A Parallel Method for Tridiagonal Equations

A Parallel Method for Tridiagonal Equations

A block-tridiagonal solver with two-level parallelization for finite element-spectral codes

Parallel block iterative method for multiaquifer flow models

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