We consider large, sparse linear systems
that result from the discretization of partial
differential equations on regular and ir
regular domains, and we focus on the ap
plication of the preconditioned conjugate
gradient (PCCG) method to the solution of
such systems. More specifically, our goal
is the efficient implementation of the
PCCG method on vector supercomputers.
The contribution to the above goal is
made by the introduction of a data struc
ture that can be effectively manipulated on
vector machines, the utilization of precon
ditioning matrices obtained by incomplete
factorization with diagonal update sets,
and the introduction of new numbering
schemes for both regular and irregular
grids.
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References
1.
Axelsson, O.1984. A survey of vectorizable preconditioning method for large scale finite element matrix problems. Tech. Report CNA-190, Center for Numerical Analysis, The University of Texas at Austin .
2.
Becker, E., Carey, G., and Oden, J.T.1981. Finite Elements, An Introduction, Vol. 1. Englewood Cliffs, New Jersey: Prentice-Hall. Concus , P., Golub, G., and Meurant, G.1985. Block preconditioning for the conjugate gradient methods. SIAM J. Sci. Statist. Comput.6:220-252.
3.
Cuthill, E., and McKee, J.1969. Reducing the bandwidth of sparse symmetric matrices . Proc. of the ACM National Conf. New York: Association for Computing Machinery, pp. 157-172.
4.
Dongarra, J., Gustavson, F., and Karp, A.1984. Implementing linear algebra algorithms for dense matrices on a vector pipeline machine. SIAM Rev.26:91-112.
5.
Eisenstat, S.1981. Efficient implementation of a class of preconditioned conjugate gradient methods. SIAM J. Sci. Statist. Comput.2(1):1-4.
6.
Gustafsson, I.1978. A class of first order factorization methods. BIT.18:142-156.
7.
Hageman, A., and Young, D.1981. Applied iterative methods. New York : Academic Press.
8.
Kincaid, D., Oppe, T., and Young, D.1984. Vector computations for sparse linear systems. Tech. Report CNA-189, Center for Numerical Analysis, The University of Texas at Austin.
9.
Madsen, N., Rodrigue, G., and Karush, J.1976. Matrix multiplication by diagonals on a vector/parallel processor. Inform. Proc. Lett.5(2):41-45.
10.
Manteuffel, T.1980. An incomplete factorization technique for positive definite linear systems. Math. Comp.34(150):673-697.
11.
Meijerink, J.A., and van der Vorst, H.1977. An iterative solution method for linear systems of which the coefficient matrix is symmetric M-matrix. Math. Comp.31(137):148-162.
12.
Meijerink, J.A., and van der Vorst, H.1981. Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems. J. Comput. Phys.44:134-155.
13.
Melhem, R.1986a. Parallel solution of linear systems with striped sparse matrices. Tech. ReportICMA-86-91; Parallel Comput., in press.
14.
Melhem, R.1986b. A study of the stripe structure of finite element stiffness matrices. Tech. ReportICMA-86-92; SIAM J. Numer. Anal.. in press.
15.
Meurant, G.1984. The block preconditioned conjugate gradient method on vector computers. BIT.24:623-633.
16.
Poole, E., and Ortega, J.1986. Multicolor ICCG methods for vector computers. Applied Math. Report RM-86-06, University of Virginia.
17.
Rice, J., and Boisvert, R.1984. Solving elliptic problems with Ellpack. New York: Springer Verlag.
18.
Rice, J., Houstis, E., and Dyksen, W.1981. A population of linear second order elliptic partial differential equations on rectangular domains; parts 1 and 2. Matli. Comp.36:475-484.
19.
Saad, Y.1985. Practical use of polynomial preconditionings for the conjugate gradient method. SIAM J. Sci. Statist. Comput.6(4):865-881.
20.
Saad, Y., and Schultz, M.1985. Parallel implementations of preconditioned conjugate gradient methods. Tech. Report YALEU/DCS/ RR425, Dept. of Computer Science , Yale University.
21.
van derVorst, H.1982. A vectorizable variant of some ICCG methods. SIAM J. Sci. Statist. Camput.3(3):350-356.
22.
van derVorst, H.1986. The performance of FORTRAN implementations for preconditioned conjugate gradients on vector computers. Parallel Comput.3:49-58.
23.
Young, D., Oppe, T., Kincaid, D., and Hayes, L.1985. On the use of vector computers for solving large sparse linear systems. Tech. Report CNA-199, Center for Numerical Analysis , The University of Texas at Austin.
