A numerical method for solving two-dimensional diffusion equations

Buleev, N. I.

doi:10.1007/bf01481461

Cited by 5 publications

(10 citation statements)

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“…Incomplete factorization methods were introduced for the first time by Buleev in the then-Soviet Union in the late 1950s, and independently by Varga (see [72,73,179,281]; see also [231]). However, Meijerink and van der Vorst deserve credit for recognizing the potential of incomplete factorizations as preconditioners for the conjugate gradient method.…”

Section: Preconditionersmentioning

confidence: 99%

Preconditioning Techniques for Large Linear Systems: A Survey

Benzi¹

2002

Journal of Computational Physics

1,011

659

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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper. c 2002 Elsevier Science (USA)

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Section: Preconditionersmentioning

confidence: 99%

Preconditioning Techniques for Large Linear Systems: A Survey

Benzi¹

2002

Journal of Computational Physics

1,011

659

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“…The resulting system expressed what we call now the preconditioned system of equations, in which the preconditioner was directly combined with the system matrix. The method was generalized to stencils for three dimensional problems in [13]. An independent derivation and its interpretation as an incomplete factorization (that is, a factorization in which some of the fill entries are ignored) for a matrix from a simple 5-point stencil was given by Varga [42] (see also [3,34]).…”

Section: Introductionmentioning

confidence: 99%

“…At that time, the sparsity structure essentially expressed the stencils for discretized partial differential equations on structured grids. In particular, the EBM-2 method of Buleev [12] interpolated values of the function at a grid point using a combination of the function values at neighbouring grid points. The solution process was accelerated by additional parametrization derived from smoothness assumptions.…”

Section: Introductionmentioning

confidence: 99%

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The importance of structure in incomplete factorization preconditioners

Scott

Tůma

2010

Bit Numer Math

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In this paper, we consider level-based preconditioning, which is one of the basic approaches to incomplete factorization preconditioning of iterative methods. It is well-known that while structure-based preconditioners can be very useful, excessive memory demands can limit their usefulness. Here we present an improved strategy that considers the individual entries of the system matrix and restricts small entries to contributing to fewer levels of fill than the largest entries. Using symmetric positive-definite problems arising from a wide range of practical applications, we show that the use of variable levels of fill can yield incomplete Cholesky factorization preconditioners that are more efficient than those resulting from the standard level-based approach. The concept of level-based preconditioning, which is based on the structural properties of the system matrix, is then transferred to the numerical incomplete decomposition. In particular, the structure of the incomplete factorization determined in the symbolic factorization phase is explicitly used in the numerical factorization phase. Further numerical results demonstrate that our level-based approach can lead to much sparser but efficient incomplete factorization preconditioners.

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“…Their development started by the work of Buleev at the end of fifties [17], [18]. Throughout the time, the algorithms have achieved a considerable degree of efficiency and robustness.…”

Section: Introduction We Consider the Linear Systemmentioning

confidence: 99%

Balanced Incomplete Factorization

Bru¹,

Uring²,

Ma³

2008

SIAM J. Sci. Comput.

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Abstract. In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU/LDL T factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the Sherman-Morrison formula [16]. In contrast to the RIF algorithm [9], the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. Although we describe the theory behind the factorization for general nonsymmetric matrices, in implementation and experiments we restrict for clarity and conciseness only to the case when the system matrix is symmetric and positive definite. In this case, we call the new approximate LDL T factorization Balanced Incomplete Factorization (BIF). Our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill-conditioned problems by preconditioned iterative methods. Moreover, the internal coupling of computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness.

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A numerical method for solving two-dimensional diffusion equations

Cited by 5 publications

References 0 publications

Preconditioning Techniques for Large Linear Systems: A Survey

Preconditioning Techniques for Large Linear Systems: A Survey

The importance of structure in incomplete factorization preconditioners

Balanced Incomplete Factorization

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