P. Concus, G. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 220-252.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....small computational cost. Finally, we chose to experiment the behavior in our framework of GMRES(m) originally presented by Saad and Schultz [90] For all these methods, it is known that preconditioning can greatly influence the convergence. Therefore, following some authors (e.g. Concus et al. [24], Axelsson [6] and Bruaset [21] we applied a preconditioner based upon the block structure of our problem. The block preconditioner we used is built on the LU factorization of the first block of our stacked system. If we dropped the leads and lags of the model, the 0 1000 2000 3000 4000 0 500 ....

P. Concus, G. H. Golub, and G. Meurant. Block Preconditioning for the Conjugate Gradient Method. SIAM J. Sci. Stat. Comput., 6:220--252, 1985.

....and their fast matrix vector multiplications. The conjugate gradient type methods [2, 4, 37] are possible choices. However, their convergence rates are slow in general. To speed up the convergence rate, we consider preconditioned conjugate gradient methods. We note that the MILU [31, 38] and MINV [25, 26] based preconditioners are not appropriate due to their expensive construction costs. One of the early applications of preconditioned conjugate gradient methods in solving queueing networks was done by Chan [9, 10] For Markovian overflow networks with traffic density close to 1, the generator ....

P. Concus, G. Golub and G. Meurant, Block Preconditioning for Conjugate Gradient Method, SIAM J. Statist. Comput., 6 (1985), pp. 220--252.

.... Lk 1P j 0 for j k j with j 1 being the starting homogenization level. Hence, the decay rate is preserved after homogenization. The decay estimate in [9] for A j 0 is uniform in k and may not be sharp for a xed k. There is, for example, a general result by Concus, Golub and Meurant, [12], for diagonal dominant, symmetric and tridiagonal matrices. For those cases, which include A j corresponding to the discretization in (115) of the one dimensional elliptic operator, the inverse has exponential decay, A k C jk j ; 0 1: 124) This holds also when the elliptic ....

C. Concus, G. H. Golub, and G. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comp., 6:220-252, 1985.

....and small computational cost. Finally, we chose to experiment the behavior in our framework of GMRES(m) originally presented by Saad and Schultz [24] For all these methods, it is known that preconditioning can greatly influence the convergence. Therefore, following some authors (e.g. Concus [7], Axelsson [2] and Bruaset [6] we applied a preconditioner based upon the block structure of our problem. The block preconditioner we used is built on the LU factorization of the first block of our stacked system. If we dropped the leads and lags of the model, the Jacobian matrix would be block ....

P. Concus, G. Golub, and G. Meurant. Block Preconditioning for the Conjugate Gradient Method. SIAM J. Sci. Stat. Comput., 6:220--252, 1985.

....A 1 ; D i = A i Gamma B i Gamma1 D i Gamma1 : In the complete block LU factorization, all the D i s are full matrices. The main idea in the incomplete version is to modify the above recurrence in order to keep the D i s sparse. For example, in the INV method due to Concus, Golub and Meurant [39], the term D i Gamma1 is replaced by its main tridiagonal part (it turns out this can be computed efficiently if D i Gamma1 is itself tridiagonal) Since for 5 point operators, the B i s are diagonal and the A i s are tridiagonal, it follows from the recurrence for D i that all the D i s ....

....part (it turns out this can be computed efficiently if D i Gamma1 is itself tridiagonal) Since for 5 point operators, the B i s are diagonal and the A i s are tridiagonal, it follows from the recurrence for D i that all the D i s generated by the incomplete factorization remain tridiagonal. In [39], the authors also proposed the MINV method, which is anaologous to the MILU method, by modifying the diagonal entries of the INV tridiagonal approximation to D i Gamma1 so that the row sum is preserved. A Fourier analysis [34] shows that the condition number bounds for INV and MINV are ....

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P. Concus, G.H. Golub, and G. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comput., 6:220--252, 1985.

....of A [3] Pi = L v .Pp. L. stands for the normalized point LU factoriza ion of P, Each P, is pointwise ridiagonal. tridiag(P7 ) may be cheaply computed from he normalized poin LU facorizaion of Pij, see, e.g. 1] Oher choices for approximate inverse 7of pivo block submatrices are discussed in [8, 15]. Algorkhms ha handle more general matrices may be found in [8, 24] w is he relaxation parameter ( 1 w 1) Wkh w = 0 one obtains he sandard block incomplete LU facorizaion (LU) while wkh = 1 one ges he sandard modified varian (MBILU) whose condkioning properties are heoreically investigated ....

....of ) iA, which slows down the convergence of PCG process [33] DRBILU, DMBILU and DRBILU successfully break with the dependence upon ) iA, which reflects in the number of PCG iterations. As predicted, DRBILU is a little bit more efficient than DRBILU. 4. In accordance with previous works, [15] (2D) 2, 28] 3D) blockwise (linewise) methods turn out to be more efficient than pointwise counterparts. In the case of 2D problems, the reduction in the number of iterations, from point methods to block methods, is at least about 50 , while it is around 30 in the 3D cases. The gain is even ....

P. CONCUS, G.H. GOLUB, AND G.A. MEURANT, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Cornput., 6 (1985), pp. 220 252.

....in this paper. To construct the preconditioner, compute the block ILU factorizations of the coefficient matrices derived from the discretization of the restriction of the PDE on the overlapping subdomains. The incomplete factorizations for these local matrices are their INV (1) factorizations [CGM85, CM86, Meu89]. The ordering is the natural order. No effort is made to select a particular ordering for the grid points or for the subdomains. The performance of the preconditioner Q IDD is investigated. Throughout, the BiCGSTAB and GMRES(50) used in conjunction with a preconditioning matrix C will 86 DIAZ, ....

Concus P., Golub G., and Meurant G. (1985) Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comput. 6: 220--252.

.... but is widely used to solve indefinite or nearly semi definite symmetric problems. The key to the success of these iterative methods is often the use of efficient preconditioners 3 aimed at accelerating their convergence and considerable work has also been published on these aspects (see [2, 7, 9, 21, 22] among many others) Extensions have been made on variable or flexible preconditioners allowing for preconditioners to be fine tuned through the course of the resolution [30] and even across separate closely coupled resolutions [6] For the case of the matrices arising in the GCV application ....

P. Concus, G. Golub, and G. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comp., 6:220--252, 1985.

.... L AK 1 R ) KRx) K 1 L b) 2) is converged faster than the given system (1) In this research, Splitting Correction (SC) preconditioner for linear systems that arise from PDEs with periodic boundary conditions was proposed [10] This preconditioner is based on block preconditioning [1][5]. The convergence of the iterative methods with this preconditioner is improved much more than the block incomplete factorization. Nevertheless, its calculating cost increases only a little for the incomplete one. About the numerical results in [10] the converging behavior is distinctive between ....

....in [10] the converging behavior is distinctive between the SC and the block incomplete factorization. The convergence rate of the CG method depends on spectral properties of the given matrix, such as the eigenvalue distribution and the condition number, and on the given right hand side [3] 4][5][12] 13] In this paper, the spectral properties by using the SC and block incomplete Cholesky factorization are evaluated. In section II, physical model, the linear system and typical solver for this system is presented. In section III, the SC preconditioner is detailed, and the e#ect of the SC ....

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P. Concus, G. H. Golub and G. Meurant, Block Preconditioning for the Conjugate Gradient Method, SIAM J. Sci. Stat. Comput. , 6, pp. 220--252, 1985.

....than the scalar entries, for instance deriving blocks from lines of grid points in the physical domain of the PDE. The scalar factorisation problem often appears in these methods, since they may require an approximation to the inverse of the pivot blocks. The interested reader is referred to [1, 6, 5, 10, 12, 31, 38, 42]. 4 Conclusion While ideally an incomplete factorisation should be judged solely on its accuracy properties, in practice there is rst the hurdle of guaranteeing its existence. Various strategies have been proposed to ascertain this Some of them, such as the methods of Gustafsson and of Kershaw, ....

P. Concus, G.H. Golub, and G. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comput., 6:220-252, 1985.

....techniques was originally developed for M matrices by Meijerink and van der Vorst (1977) and then was extended to H matrices and block H matrices, for which theoretical properties such as existence, stability and accuracy can be established. For details one can refer to Axelsson (1994, 1985) Concus, Golub and Meurant (1985), Donato and Chan (1992) Elman(1986, 1989) Gustafsson (1978) Manteuffel (1980) Saad (1996) and references therein. For general unsymmetric matrices, although a number of efficient incomplete LU factorization techniques have been presented (see Axelsson (1994) and Saad (1996) it is more ....

Concus, P., Golub, G. and Meurant, G. (1985), `Block preconditioning for the conjugate gradient method', SIAM J. Scientific and Statistical Computing 6, 220--252.

....convection dominated problems. This approach was further improved by including a threshold technique for fill in as is done in the ILUT algorithm, see [148, p. 287] Another major step forward, for important classes of problems, was the introduction of block variants of incomplete factorizations [167, 44, 5], and modified variants of them [44, 5, 112] It was observed, by Meurant, that these block variants were more successful for discretized 2 dimensional problems than for 3 dimensional problems, unless the 2 dimensional blocks in the latter case were solved accurately. For discussions and ....

....was further improved by including a threshold technique for fill in as is done in the ILUT algorithm, see [148, p. 287] Another major step forward, for important classes of problems, was the introduction of block variants of incomplete factorizations [167, 44, 5] and modified variants of them [44, 5, 112]. It was observed, by Meurant, that these block variants were more successful for discretized 2 dimensional problems than for 3 dimensional problems, unless the 2 dimensional blocks in the latter case were solved accurately. For discussions and analysis on ordering strategies, in relation to ....

P. Concus, G. H. Golub, and G. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Statist. Comput., 6:220--252, 1985.

....on which relies both the numerical efficiency of the method and its intrinsic parallelism, the remaining of the algorithm being not difficult to parallelize on most architectures. Incomplete factorization preconditioners allow an efficient reduction of the number of iterations (see e.g. [2, 5, 8, 21]) but require two triangular solves per iterations, which, when using classical ordering schemes, prevents large scale parallelization, even if interesting implementations where developed for a moderate number of processors thanks to a careful dependency analysis of the involved recursions [4, ....

.... the communications required in the general algorithm given in [19] We give the algorithm for the DRIC method [18, 21] which we chosed for the numerical tests in the next section; it is easily converted in an algorithm for the IC method [15] the MIC method (without perturbations) see e.g. [8]) or the RIC method [3] by using respectively i j 0, i j 1 or i j for some 0 1. ff is a parameter such that 0 ff 1 (see Section 5) and oe an auxiliary vector. FOR ALL p : ffl initialize, for all i i = a (p) ii oe i = X j i a (p) ij 8 ffl set Gamma Sigma f ( ....

P. Concus, G. H. Golub, and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 220--252. 20

....the multigrid program MGD9V [31] In Section 5.6 the prospects of vectorization and parallelization are briefly contemplated. In Section 5.7 conclusions are summarized. 5. 2 ILLU for discretized coupled PDEs The ILLU decomposition originates from Underwood [26] and has been investigated e.g. in [4, 5, 8, 20]. This ILLU method may serve both as a smoother [16, 17, 18, 19, 20, 25, 31] in a multigrid context and as a preconditioner for conjugate gradient type methods, e.g. in [25] In [18, 25] an extensive description of the method can be found. The method is of interest because it is more robust than ....

P. Concus, G.H. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput. 6 (1985) 220--252.

....grids the Incomplete Line LU (ILLU) or Incomplete Block LU) appears to be the most robust choice (see [8, 9, 12, 21] By this method full advantage is taken of the matrix structure. Here we repeat the general outline of this method, which has been originated by Underwood [14] see also [3] for an overview on block type methods) We want to solve the linear system Ax = b (4.31) that we assume to have a block tridiagonal form, so A = 0 B B B B B B D 1 U 1 L 2 D 2 U 2 L 3 D 3 Delta Delta Delta Delta Delta Delta Delta Delta Dny 1 C C C C C C A (4.32) where n y is the ....

P. Concus, G.H. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput. 6 (1985) 220--252.

....4 1 1 4 . of order 1 N . 3. GAUSS JACOBI METHOD FOR BLOCK TRIDIAGONAL SYSTEMS The model problem is then solved via Block Gauss Jacobi Method with blocks of dimension 15 n . Some methods of resolution for these systems are given in [2] Borges et al. ,1998) 3] Borges, 1992) [ 4] (Concus, et al. 1985) and [5] Golub, et al. 1983) Iterative methods for the model problem (1) are considered. Take the linear system on the form: Au f = 4) where A is a matrix of order ( N M 1 1 , u u u u M T = 1 2 1 , # , f f f f M T = 1 2 1 , # as ....

Concus, P., Golub, G.H. and Meurant, G. - Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6, pp. 220-252, 1985.

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P. Concus, G. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 220-252.

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P. Concus, G. Golub, G. Meurant, \Block preconditioning for the conjugate gradient method", SIAM J. Sci. Stat. Comp., 6 (1985), pp. 220-252.

No context found.

P. Concus, G. Golub, G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 220-252.

....which appears to have some advantages for convection dominated problems. This approach was further improved by includ ing a threshold technique for fill in [82] Another major step forward, for important classes of problems, was the introduction of block variants of incomplete factorizations [92, 20, 4], and modified variants of them [4, 66] It was observed, by Meurant, that these block variants were more successful for discretized 2 dimensional problems than for 3 dimensional problems, unless the 2 dimensional blocks in the latter case were solved accurately. For discussions and analysis on ....

P. Concus, G. H. Golub, and G. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comput., 6:220--252, 1985.

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P. Concus, G.H. Golub and G. Meurant, 1985. Block preconditioning for the conjugate gradient method. SIAM J. Scientific and Statistical Computing, 6: 220-- 252.

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P. Concus, G. H. Golub, and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 220--252. 35

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P. Concus, G. Golub, and G. Meurant, Block Preconditioning for the Conjugate Gradient Method, SIAM J. Sci. Statist. Comput., 6 (1985) 220--252.

No context found.

P. Concus, G. H. Golub, and G. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Statist. Comput., 6:220-- 252, 1985.

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P. Concus, G. H. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput. 6, 187--209 (1983). 24

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