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- W2288591330 abstract "We consider in this paper the Incomplete Cholesky Conjugate Gradient (ICCG) method on the CDC Cyber 2031205 vector computers for the solution of an NxN system of linear equations Ax.=b. We assume that the matrix A is large, sparse, and symmetric positive definite with non-zero elements lying along a few diagonals of the matrix, such as arises in the solution of elliptic partial differential equations by finite difference or finite element discretizations. Results are given for two model problems, run on a Cyber 203 at NASA Langley Research Center. In the sequel, we shall refer to the Cyber 203 and 205 as the Cyber 200 unless there is a reason to differentiate between them. Since Meijerink and van der Vorst [1977], several authors have considered ICCG. Most of this work has been directed towards serial computers although Kershaw [1982] gives an implementation for the CRA Y-1 using a cyclic reduction technique applied to block tridiag onal matrices, and Lichnewsky [1984] discusses parallel and vector implementations for ICCG but, as in Kershaw, mainly gives algorithms with vector lengths better suited for the CRA Y computers. Efficient use of the Cyber 200 requires algorithms that consist mainly of operations on long vectors. In particular, our goal is vectors of length O(N/p), where N is the number of unknowns and p is a small constant independent of N. This is achieved in the basic conju gate gradient algorithm Cwith p = 1) and the problem is to achieve it also in the incomplete Cholesky preconditioning. To this end, we adopt and extend a suggestion of Schreiber and Tang [1982] to use a multicolor ordering of the grid points. This is combined with diagonal storage of the matrix and it is shown that by judicious choice of the multicolor ordering, vector lengths of OCN/p) are achieved, where p is the number of colors Cp=4 and 6 in the model problems). There are many incomplete Choleski fill-in strategies and we have chosen the simplest, ICCGCO), in which the non-zero elements in the incomplete Choleski factors are in the same diagonals as the original matrix; that is, no fill is allowed outside these diagonals. We will briefly discuss, however, how other fill strategies can be implemented. We also con sider an m-step ICCG(O) method in the sense of Adams and Ortega [1982]; m=l is then ICCGCO)." @default.
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- W2288591330 date "1985-01-01" @default.
- W2288591330 modified "2023-09-27" @default.
- W2288591330 title "INCOMPLETE CHOLESKI CONJUGATE GR.ADIENT ON THE CYBER. 2031205" @default.
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