A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method

Terekhov, Andrew V.

doi:10.1016/j.parco.2013.03.003

Cited by 24 publications

(18 citation statements)

References 43 publications

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“…It has been shown that this dichotomy yields almost a linear speedup on a high performance system with many processors [19]. We expect the same speedup for our proposed methods.…”

Section: Parallel Algorithm For the Proposed Adi Schemessupporting

confidence: 70%

“…It is to be noted that the implementation of the ADI schemes requires solving the same block-tridiagonal matrix with different right-hand sides. This can be done efficiently by using the fast parallel algorithm given in [19]. This algorithm [19] is a generalization of the parallel dichotomy algorithm for solving tridiagonal liner system of equations [20].…”

Section: Parallel Algorithm For the Proposed Adi Schemesmentioning

confidence: 99%

“…The exact solution of the nonlinear system can be derived as follows: we assume [19] ( , , ) = sech ( )…”

Section: Appendixmentioning

confidence: 99%

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Parallel Methods and Higher Dimensional NLS Equations

Ismail

Taha

2013

Abstract and Applied Analysis

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Alternating direction implicit (ADI) schemes are proposed for the solution of the two-dimensional coupled nonlinear Schrödinger equation. These schemes are of second- and fourth-order accuracy in space and second order in time. The resulting schemes in each ADI computation step correspond to a block tridiagonal system which can be solved by using one-dimensional block tridiagonal algorithm with a considerable saving in computational time. These schemes are very well suited for parallel implementation on a high performance system with many processors due to the nature of the computation that involves solving the same block tridiagonal systems with many right hand sides. Numerical experiments on one processor system are conducted to demonstrate the efficiency and accuracy of these schemes by comparing them with the analytic solutions. The results show that the proposed schemes give highly accurate results.

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“…It has been shown that this dichotomy yields almost a linear speedup on a high performance system with many processors [19]. We expect the same speedup for our proposed methods.…”

Section: Parallel Algorithm For the Proposed Adi Schemessupporting

confidence: 70%

Section: Parallel Algorithm For the Proposed Adi Schemesmentioning

confidence: 99%

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Parallel Methods and Higher Dimensional NLS Equations

Ismail

Taha

2013

Abstract and Applied Analysis

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“…. To solve this system of algebraic equations, we will use generalized method of fast parallelization in complex case [9].…”

Section: Numerical Schemementioning

confidence: 99%

A Fully Conservative Parallel Numerical Algorithm with Adaptive Spatial Grid for Solving Nonlinear Diffusion Equations in Image Processing

Bulygin

Vrazhnov

2019

JSFI

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In this paper we present simple yet efficient parallel program implementation of grid-difference method for solving nonlinear parabolic equations, which satisfies both fully conservative property and second order of approximation on non-uniform spatial grid according to geometrical sanity of a task. The proposed algorithm was tested on Perona-Malik method for image noise filtering task based on differential equations. Also in this work we propose generalization of the Perona-Malik equation, which is a one of diffusion in complex-valued region type. This corresponds to the conversion to such types of nonlinear equations like Leontovich-Fock equation with a dependent on the gradient field according to the nonlinear law coefficient of diffraction. This is a special case of generalization of the Perona-Malik equation to the multicomponent case. This approach makes noise removal process more flexible by increasing its capabilities, which allows achieving better results for the task of image denoising.

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“…There are many effective parallel algorithm for block-tridiagonal system of linear equations, such as block cyclic reduction algorithm (Hirshman et al, 2010) and partitioned Thomas algorithm (Wang, 1981). And some block-tridiagonal parallel algorithms have already been used for wavefield modeling (Terekhov, 2011). But they ignored that the block matrix for 2D frequency-domain modeling is usually banded.…”

Section: Introductionmentioning

confidence: 99%

A new parallel direct solver for 2D frequency-domain modeling

Gao

2016

SEG Technical Program Expanded Abstracts 2016

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The parallel direct solver is important for 2D frequencydomain modeling. It can effectively reduce the memory requirement and computing time. And multi-shot seismic data can be simulated efficiently in this way. However the traditional parallel sparse LU direct solvers, such as MUMPS and superLU, have poor speed-up performance. In this paper, we introduce a new parallel direct solver called banded-block cyclic reduction(BBCR), which is developed to get a better speed-up performance and reduce the computing time. Numerical experiments for Marmousi model show that the speed-up ratio of BBCR is obviously superior to the speed-up ratio of MUMPS, and the frequency-domain wavefields got by BBCR and MUMPS are almost the same. Besides, since BBCR makes full use of banded matrix operation, it can almost reduce the computing time of traditional block cyclic reduction(BCR) solver by half.

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A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method

Cited by 24 publications

References 43 publications

Parallel Methods and Higher Dimensional NLS Equations

Parallel Methods and Higher Dimensional NLS Equations

A Fully Conservative Parallel Numerical Algorithm with Adaptive Spatial Grid for Solving Nonlinear Diffusion Equations in Image Processing

A new parallel direct solver for 2D frequency-domain modeling

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