### Citations

291 |
der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix
- Meijerink, van
- 1977
(Show Context)
Citation Context ...neral 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)... |

251 |
Finite element solution of boundary value problems, in: Theory and Computation
- Axelsson, Baker
- 1984
(Show Context)
Citation Context ...m, 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 moder... |

102 | Block preconditioning for the conjugate gradient method
- Concus, Golub, et al.
- 1985
(Show Context)
Citation Context ...m, 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 moder... |

72 |
On the eigenvalue distribution of a class of preconditioning methods
- Axelsson, Lindskog
- 1986
(Show Context)
Citation Context ...1] 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 ssi = ~a(p... |

29 |
Meurant, The effect of ordering on preconditioned conjugate gradients
- Duff, A
- 1989
(Show Context)
Citation Context ...s to local triangular solves which may be carried out independently. Some degradation is expected in the convergence rate but, as will be seen, limited compared with the gain in parallelism, see also =-=[9, 10, 20]-=-. Here, we consider the use of this principle within the framework of the results of [19]. In the latter paper, a parallelization technique is proposed which is based on the equivalence between any it... |

28 |
Parallel conjugate gradient-like algorithms for solving sparse non-symmetric systems on a vector multiprocessor
- Brozolo, Robert
- 1989
(Show Context)
Citation Context ... (p) inp (1 \Gammas!i) + !i oei) ffl set ssnp \GammasX ^np 2 E`(np) ss^np Remark It is relevant to compare here briefly our approach with the interesting method proposed by Radicati di Brozolo et al. =-=[23, 12]-=-. In the latter papers, it is suggested to split the unknowns into overlapping blocks, preferably in such a way that all nonzero entries in the system matrix appear in at least one block. Then, to eac... |

27 | Analysis of parallel incomplete point factorizations
- Eijkhout
- 1991
(Show Context)
Citation Context ...s to local triangular solves which may be carried out independently. Some degradation is expected in the convergence rate but, as will be seen, limited compared with the gain in parallelism, see also =-=[9, 10, 20]-=-. Here, we consider the use of this principle within the framework of the results of [19]. In the latter paper, a parallelization technique is proposed which is based on the equivalence between any it... |

22 | The role of the inner product in stopping criteria for conjugate gradient iterations
- Ashby, Holst, et al.
- 1992
(Show Context)
Citation Context ...akes the test dependent of the preconditioner, but avoids the computation of any additional scalar product, reason for which this test serves as basis for the "natural" stopping criterion proposed in =-=[1]-=-. In practice, we observed that this dependence has only a limited influence on the number of iterations. Another advantage of this criterion in our context is that k r(k)g kB\Gamma 1g may be directly... |

21 |
Ordering techniques for the preconditioned conjugate gradient method on parallel computers
- Elman, Argon
- 1989
(Show Context)
Citation Context ...approach has given satisfying results for parallel computers with moderated number of processors (see for instance [4, 14]), while the second one seems more adapted to massively parallel systems (see =-=[6, 11]-=-, among many others ). The solver presented here is based on the conjugate gradient algorithm and belongs to the latter category. As usual with this method, the tricky point is the choice of the preco... |

19 |
Parallization of robust multi-grid methods: ILU factorization and frequency decomposition method
- Bastian, Horton
- 1991
(Show Context)
Citation Context ... of the total number of arithmetic operations is kept minimal. Up to now, the former approach has given satisfying results for parallel computers with moderated number of processors (see for instance =-=[4, 14]-=-), while the second one seems more adapted to massively parallel systems (see [6, 11], among many others ). The solver presented here is based on the conjugate gradient algorithm and belongs to the la... |

18 |
Parallel elliptic preconditioners: Fourier analysis and performance
- CHAN, Kuo, et al.
- 1989
(Show Context)
Citation Context ...approach has given satisfying results for parallel computers with moderated number of processors (see for instance [4, 14]), while the second one seems more adapted to massively parallel systems (see =-=[6, 11]-=-, among many others ). The solver presented here is based on the conjugate gradient algorithm and belongs to the latter category. As usual with this method, the tricky point is the choice of the preco... |

12 |
Computational Methods in Engineering and Science,”
- Nakamura
- 1977
(Show Context)
Citation Context ...matrices Ap and right hand sides bp, whose assembly gives the global system matrix and right hand side; when using a finite element discretization or the finite difference mesh box integration scheme =-=[16]-=-, it suffices indeed to perform a local discretization on the subdomains, treating internal boundaries as if they were external with natural boundary conditions. To manage (2) while minimizing the con... |

12 | Ordering methods for approximate factorization preconditioning. Report IT/IF14-11, Université Libre de Bruxelles
- Notay
- 1993
(Show Context)
Citation Context ..., 21]. Moreover, from our tests, it appears that this number of iterations do generally not increases when #proc: increases from 1 to 4 (in accordance with the theoretical and experimental results in =-=[9, 20]-=-). Then, it increases slightly less then O((#proc:)1=4), and, for 256 processors, remains not greater than about twice that required by the single or four processor version, which represents a quite r... |

7 |
Conditioning analysis of positive definite matrices by approximate factorizations
- Beauwens, Wilmet
- 1989
(Show Context)
Citation Context ...m, 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 moder... |

4 |
Algorithmic on a distributed memory mimd computer: a case study
- Manneback, Qin
- 1992
(Show Context)
Citation Context ... of the total number of arithmetic operations is kept minimal. Up to now, the former approach has given satisfying results for parallel computers with moderated number of processors (see for instance =-=[4, 14]-=-), while the second one seems more adapted to massively parallel systems (see [6, 11], among many others ). The solver presented here is based on the conjugate gradient algorithm and belongs to the la... |

4 |
On the parallelization of approximate factorization preconditioning
- Notay
- 1993
(Show Context)
Citation Context ...d in the convergence rate but, as will be seen, limited compared with the gain in parallelism, see also [9, 10, 20]. Here, we consider the use of this principle within the framework of the results of =-=[19]-=-. In the latter paper, a parallelization technique is proposed which is based on the equivalence between any iterative scheme applied to the global system with the same 1siterative scheme applied to a... |

3 |
di Brozolo, Parallel preconditioned conjugate-gradient type algorithms for general sparsity structures
- Filippone, Marrone, et al.
- 1992
(Show Context)
Citation Context ... (p) inp (1 \Gammas!i) + !i oei) ffl set ssnp \GammasX ^np 2 E`(np) ss^np Remark It is relevant to compare here briefly our approach with the interesting method proposed by Radicati di Brozolo et al. =-=[23, 12]-=-. In the latter papers, it is suggested to split the unknowns into overlapping blocks, preferably in such a way that all nonzero entries in the system matrix appear in at least one block. Then, to eac... |

3 |
Implementation of parallel block preconditionings on a transputerbased multiprocessor
- Magolu
- 1993
(Show Context)
Citation Context ...actually 513 when both left and right sides of the domain have been specified Neumann boundary conditions. The same remark holds for the implementation of block preconditionings like that proposed in =-=[13]-=-. Comparing now our solver with a standard Jacobi preconditioned conjugate gradient solution, it is clear that the latter is oumatched by far, even taking into account the factorization cost on the on... |

3 |
R'esolution it'erative de syst`emes lin'eaires par factorisations approch'ees, PhD thesis, Service de M'etrologie Nucl'eaire, Universit'e Libre de Bruxelles
- Notay
- 1991
(Show Context)
Citation Context ...t finite difference or linear finite element matrix, see e.g. [2]. Other techniques used to derive Stieltjes approximations may also be used to discard some selected negative offdiagonal entries, see =-=[17]-=-. Hence, we may assume here, without loss of generality, that, together with the local system matrices Ap, it is furnished local approximations ~Ap whose connections are all compatible with (4) and (5... |

2 |
A new incomplete factorization method, in Incomplete Decomposition
- Notay
- 1993
(Show Context)
Citation Context ...below, where we take advantage of some features of the present implementation to simplify 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... |

1 |
DRIC : a dynamic version of the RIC method. to appear
- Notay
- 1994
(Show Context)
Citation Context |