Nonstationary parallel relaxed multisplitting methods

Mas, José; Migallón, Violeta; Penadés, José; Szyld, Daniel B.

doi:10.1016/0024-3795(95)00583-8

Cited by 25 publications

(19 citation statements)

References 20 publications

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“…. , K in Theorem 3.1, it reduces to Theorem 3.1 in [10]; If ω 1 = ω 2 = · · · = ω K and ω = 1 in Theorem 3.2, it reduces to theorem 3.4 in [11]; If ω k = 1 for all k = 1, 2, . .…”

Section: Remark 31mentioning

confidence: 99%

Convergence behaviors of multisplitting methods withK+1relaxed parameters

Cheng

Huang

Jing

et al. 2009

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper, two multisplitting methods with K + 1 relaxed parameters are established for solving a linear system whose coefficient matrix is a large sparse M-matrix or H-matrix and the corresponding convergence behaviors are studied. Then the implementation of these two methods with ILU factorizations as inner splittings is investigated. Finally, some numerical experiments are presented to illustrate the effectiveness of the preconditioners obtained from our methods when combined with BiCGSTAB.

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“…. , K in Theorem 3.1, it reduces to Theorem 3.1 in [10]; If ω 1 = ω 2 = · · · = ω K and ω = 1 in Theorem 3.2, it reduces to theorem 3.4 in [11]; If ω k = 1 for all k = 1, 2, . .…”

Section: Remark 31mentioning

confidence: 99%

Convergence behaviors of multisplitting methods withK+1relaxed parameters

Cheng

Huang

Jing

et al. 2009

Journal of Computational and Applied Mathematics

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“…The concept of multisplittings, first introduced in [34], provides a very general setting to study parallel block methods; see, e.g., [2], [3], [4], [13], [17], [19], [21], [28], [31], [39]. This general setting encompasses cases, e.g., where there is overlap, i.e., where more than one processor computes approximations to the same variable, and the weighting matrices E have positive entries smaller than 1, see e.g., [22], [29].…”

Section: Algorithm 2 (Nonlinear Two-stage Multisplitting)mentioning

confidence: 99%

“…Asynchronous methods have the potential of converging much faster than synchronous methods, especially when there is load imbalance, e.g., when one of the systems (2) takes much longer to solve than all the others; see e.g., [13], [24], [27], [31]. A block asynchronous method for the solution of (1) was analyzed in [9] using a different approach, and without considering the two-stage case.…”

Section: Introductionmentioning

confidence: 99%

Block and asynchronous two-stage methods for mildly nonlinear systems

Bai

Migallón

Penadés

et al. 1999

Numerische Mathematik

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Block parallel iterative methods for the solution of mildly nonlinear systems of equations of the form Ax = Φ(x) are studied. Two-stage methods, where the solution of each block is approximated by an inner iteration, are treated. Both synchronous and asynchronous versions are analyzed, and both pointwise and blockwise convergence theorems provided. The case where there are overlapping blocks is also considered. The analysis of the asynchronous method when applied to linear systems includes cases not treated before in the literature.

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“…In the case of ω = 1 we have a Relaxed Non-stationary Multisplitting Algorithm (Algorithm 3). The convergence of algorithms 2 and 3 when A is an H-matrix was studied by Mas, Migallón, Penadés and Szyld [20]. Furthermore, in [20] the authors report computational results of those algorithms on a multiprocessor system that show a better behavior of the non-stationary models than the stationary ones (Algorithm 1).…”

Section: Algorithm 2 (Non-stationary Multisplitting)mentioning

confidence: 99%

“…The convergence of algorithms 2 and 3 when A is an H-matrix was studied by Mas, Migallón, Penadés and Szyld [20]. Furthermore, in [20] the authors report computational results of those algorithms on a multiprocessor system that show a better behavior of the non-stationary models than the stationary ones (Algorithm 1). For a background on parallel non-stationary models see also [5], [6], [7], [13], [14] or [15].…”

Section: Algorithm 2 (Non-stationary Multisplitting)mentioning

confidence: 99%