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## on massively parallel machines

### Citations

1136 |
Methods of conjugate gradients for solving linear systems
- Hestenes, Stiefel
- 1952
(Show Context)
Citation Context ... with sparse matrices from various FE applications. 2 Theoretical background 2.1 The conjugate gradient method The method of conjugate gradients — originally developed in 1952 by Hestenes and Stiefel =-=[10]-=- — is an algorithm for solving systems of linear equations Ax = b, particularly when A is a sparse coefficient matrix. The method applies to symmetric positive definite (spd) matrices A ∈ IR n×n . Ayk... |

622 |
der Vorst. Templates for the Solution of Linear Systems
- Barret, Berry, et al.
- 1993
(Show Context)
Citation Context ...l blocks. 3 Storage scheme Efficient storage schemes for large sparse matrices depend on the sparsity pattern of the matrix, the considered algorithm, and the architecture of the computer system used =-=[4]-=-. In this article, we apply the CRS format (compressed row storage). This format is often used in FE programs and is suited for matrices with regular as well as irregular structure. The principle of t... |

176 | Chebyshev polynomials - Rivlin - 1974 |

165 |
Introduction to Parallel and Vector Solutions of Linear Systems
- Ortega
- 1988
(Show Context)
Citation Context ..., but does not affect the more advantageous parallelization properties of Algorithm 2.1 compared with the original method. The convergence of the CG method depends on the eigenvalue distribution of A =-=[11]-=-. A criterion for the width of the spectrum is the euclidean condition number that is for spd matrices With γ := κ := λmax(A) λmin(A) (≥ 1). √ κ−1 √ κ+1 , the distance to the exact solution x ∗ in the... |

48 |
S-step iterative methods for symmetric linear systems,
- Chronopoulos, Gear
- 1989
(Show Context)
Citation Context ...r equations Ax = b, particularly when A is a sparse coefficient matrix. The method applies to symmetric positive definite (spd) matrices A ∈ IR n×n . Aykanat e.a. [3] as well as Chronopoulos and Gear =-=[7]-=- suggested a modified CG iteration (Algorithm 2.1) that has better parallelization properties than the original method. Algorithm 2.1. The modified CG method Init.: x0, r0 = b − Ax0, d0 = r0 i = 0,1,.... |

26 |
Iterative Solution Methods (Cambridge:
- Axelsson
- 1994
(Show Context)
Citation Context ...−1 − wi−2 + 1 η vi−1) + wi−2 2.3 Incomplete Cholesky preconditioning Incomplete Cholesky factorization methods have been shown to be very efficient preconditioners for structural engineering problems =-=[2]-=- [9]. Axelsson [2] proves the existence of incomplete factorizations for M-matrices, positive definite matrices on block tridiagonal form, and block H-matrices. We describe a simple but efficient meth... |

18 |
Minimax polynomial preconditioning for Hermitian linear systems
- Ashby
- 1991
(Show Context)
Citation Context ...ons of the simply diagonally scaled CG and are shown to be well suited for massively parallel machines. The basic operations of the pure CG iteration as well as the polynomially preconditioned method =-=[1]-=- are matrix-vector products with the coefficient matrix and vector-vector computations. For incomplete Cholesky preconditioning, the factorization of the matrix before the CG iteration and a forward/b... |

9 |
Is a simple diagonal scaling the best preconditioner for conjugate gradients on supercomputers
- PINI, GAMBOLATI
- 1990
(Show Context)
Citation Context ... ∗ �2 . The right hand side increases with growing condition number. Hence lower condition numbers usually accelerate convergence. A simple but often very efficient preconditioner is diagonal scaling =-=[5, 6, 11, 12]-=-. The CG method is applied to the scaled system D −1 AD −1 ˜x = D −1 b with dj,j = √ aj,j, j = 1,... ,n, as the elements of the diagonal matrix D and ˜x = Dx. The original solution is given by x = D −... |

9 |
3DFEMWATER--A three-dimensional finite element model of WATER flow through saturated-unsaturated porous
- Yeh
- 1987
(Show Context)
Citation Context ...arried out with spd matrices from various application problems. The matrix FLOW3D comes from a FE model of environmental science in which the behavior of pollutants in geological systems is simulated =-=[15, 16]-=-. The matrix STRUCT stems from a structural mechanics model in which stresses in materials induced by thermal expansion are calculated by applying the FE program SMART [20]. The matrices PRESS, BLOCK,... |

8 |
Vectorization and parallelization of the conjugate gradient algorithm on hypercubeconnected vector processors,
- Aykanat, Ozguner, et al.
- 1990
(Show Context)
Citation Context ...lgorithm for solving systems of linear equations Ax = b, particularly when A is a sparse coefficient matrix. The method applies to symmetric positive definite (spd) matrices A ∈ IR n×n . Aykanat e.a. =-=[3]-=- as well as Chronopoulos and Gear [7] suggested a modified CG iteration (Algorithm 2.1) that has better parallelization properties than the original method. Algorithm 2.1. The modified CG method Init.... |

6 |
Numerical modeling of field scale transport in heterogeneous variably saturated porous media
- Vereecken, Lindenmayr, et al.
- 1993
(Show Context)
Citation Context ...arried out with spd matrices from various application problems. The matrix FLOW3D comes from a FE model of environmental science in which the behavior of pollutants in geological systems is simulated =-=[15, 16]-=-. The matrix STRUCT stems from a structural mechanics model in which stresses in materials induced by thermal expansion are calculated by applying the FE program SMART [20]. The matrices PRESS, BLOCK,... |

4 |
Conjugate Gradients Parallelized on the Hypercube
- Basermann
- 1993
(Show Context)
Citation Context ... ∗ �2 . The right hand side increases with growing condition number. Hence lower condition numbers usually accelerate convergence. A simple but often very efficient preconditioner is diagonal scaling =-=[5, 6, 11, 12]-=-. The CG method is applied to the scaled system D −1 AD −1 ˜x = D −1 b with dj,j = √ aj,j, j = 1,... ,n, as the elements of the diagonal matrix D and ˜x = Dx. The original solution is given by x = D −... |

4 |
Parallel sparse matrix computations in iterative solvers on distributed memory machines
- Basermann
- 1995
(Show Context)
Citation Context ... ∗ �2 . The right hand side increases with growing condition number. Hence lower condition numbers usually accelerate convergence. A simple but often very efficient preconditioner is diagonal scaling =-=[5, 6, 11, 12]-=-. The CG method is applied to the scaled system D −1 AD −1 ˜x = D −1 b with dj,j = √ aj,j, j = 1,... ,n, as the elements of the diagonal matrix D and ˜x = Dx. The original solution is given by x = D −... |

3 |
Numerische Mathematik. de Gruyter
- Deuflhard, Hohmann
- 1991
(Show Context)
Citation Context ...vector products, but decreases the total number of global synchronizations [14]. Finally we should remark that calculating the coefficients of the polynomial is not necessary. The Chebyshev iteration =-=[8]-=-, a 3-term recursion, can be exploited. Algorithm 2.2 displays the iteration 3sfor the calculation of C(A)z where C(A) is a polynomial of degree m. The value of P(A)di from the preconditioned CG itera... |

3 |
Institut fur Statik und Dynamik der Luft- und Raumfahrtkonstruktionen der Universitat Stuttgart. ISD-Berichte
- SMART
(Show Context)
Citation Context ...al systems is simulated [15, 16]. The matrix STRUCT stems from a structural mechanics model in which stresses in materials induced by thermal expansion are calculated by applying the FE program SMART =-=[20]-=-. The matrices PRESS, BLOCK, TURB, and CHEM originate in simulation models of automobile industry (pressing hoods), structural and mechanical loads, as well as chemistry (conductivity of sintered mate... |

2 |
Polynomial preconditioning for the conjugate gradient method on massively parallel systems
- Schelthoff, Basermann
- 1995
(Show Context)
Citation Context ...olynomial preconditioning by replacing the initial residual in Algorithm 2.1 by r0 = C(A)(b − Ax0) and yi = Adi by yi = P(A)di. To construct C(A), we apply scaled and translated Chebyshev polynomials =-=[14]-=- that require estimations ˜ λmin and ˜ λmax for the smallest and the largest eigenvalue of A. The connection between the Lanczos algorithm for symmetric eigenproblems and the CG method allows to impro... |