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Schur complement preconditioned conjugate gradient methods for spline collocation equations

Author:

Christina C. ChristaraAuthors Info & Claims

ICS '90: Proceedings of the 4th international conference on Supercomputing

June 1990

Pages 108 - 120

https://doi.org/10.1145/77726.255146

Published: 01 June 1990 Publication History

Abstract

We are interested in the efficient solution of linear second order Partial Differential Equation (PDE) problems on rectangular domains. The PDE discretisation scheme used is of Finite Element type and is based on quadratic splines and the collocation methodology. We integrate the Quadratic Spline Collocation (QSC) discretisation scheme with a Domain Decomposition (DD) technique. We develop DD motivated orderings of the QSC equations and unknowns and apply the Preconditioned Conjugate Gradient (PCG) method for the solution of the Schur Complement (SC) system. Our experiments show that the SC-PCG-QSC method in its sequential mode is very efficient compared to standard direct band solvers for the QSC equations. We have implemented the SC-PCG-QSC method on the iPSC/2 hypercube and present performance evaluation results for up to 32 processors configurations. We discuss a type of nearest neighbour communication scheme, in which the amount of data transfer per processor does not grow with the number of processors. The estimated efficiencies are at the order of 90%.

References

[1]

Ahlberg, J. H. and T. Ito, A collocation method for two-point boundary value problems, Math. Comp., 29 (1975), pp. 129-131,761-776.

[2]

Archer, D. A., Some collocation methods for dif. ferential equations, Rice University, Houston, "IX, Ph.D. thesis, 1973.

[3]

Axelason, O. and V. A. Barker, Finite element solution of boundary value problems, Academic Press, 1984.

Digital Library

[4]

Bjorstand, P. E. and O. B. Widlund, lterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), pp. 1097-1120.

Digital Library

[5]

Bramble, J. H., J. E. Pasciak and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring i, Math. Comp., 47 (1986), pp. 103-134.

Digital Library

[6]

Bramley, R. and A. Sameh, A robust parallel solver for block tridiagonal systems, Proc~cdings of the 1988 International Conference on Supercomputing (ICS88), July 1988, St. Malo, France.

Digital Library

[7]

Cavendish, J. C., A collocation method for elliptic and parabolic boundary value problems, using cubic splines, Univ. of Pittsburgh, PA, Ph.D. thesis, 1972.

[8]

Chan, T. F., Analysis of preconditioners for domain decomposition, SIAM J. Numer. Anal., (1987), pp. 382-390.

Digital Library

[9]

Christara, C. C., E. N. Houstis and J. R. Rice, A Parallel Spline Collocation - Capacitance Method for Elliptic PDEs, Proceedings of the 1988 International Conference on Supercomputing (ICS88), July I988, St. Malo, France.

Digital Library

[10]

Christara, C. C., @line collocation methods, software and architectures for linear elh'ptic boundary value problems, Ph.D. thesis, Purdue University, IN, U.S.A., 1988.

Digital Library

[11]

Christara, C. C. and E. N. Houstis, A domain decomposition spline collocation method for elliptic partial differential equations, Proceedings of the fourth Conference on Hypereubes, Concurrent Computers and Applications (HCCA4), March 1989, Monterey, CA, U.S.A.

[12]

Christara, C. C., Conjugate Gradient Methods for Spline Collocation Equations, to appear in Proceedings of the fifth Distributed Momory Computing Conference (DMCC5), April 1990, Charl~ston, SC, U.S.A.

[13]

Christara, C. C., Quadraffc Spline Collocation Methods for Elliptic Partial Differential Equations, Univ. of Toronto, DCS Tech. Rcp. (1990), submitted for publication.

[14]

Christara, C. C., {terative Solution of Spline CoUocation Equations, Univ. of Toronto, DCS Teeh. Rep., in preparation.

[15]

Concus, P., G. H. Golub and D. O'Leary, A generalised Conjugate Gradient method for the numerical solution of elliptic Partial Differential Equations, Sparse Matrix Computations, Bunch I. and D. Rose (eds.), Aoademic Press, New York 1976, pp. 309-322.

[16]

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

Digital Library

[17]

Daniel, J. W. and B. K. Swartz, Extrapolated collocation for two point boundary value problems using cubic splines, J. Inst. Maths Applies, 16 (1975), pp. 161-174.

[18]

Dryja, M., A finite element - capacitance method for elliptic problems on regions partitioned into aubregions, Numer. Math., 44, (1984), pp. 153-168.

Digital Library

[19]

Dryja, M. and W. Proskurowski, lterat~'ve methods in subspaces for solving elliptic problems using domain decomposition, University of Southern California, Tech. Rep. CRI-86-10 (1986).

[20]

Fyfe, D. J, The use of cubic splines in the solution of two-point boundary value problems, Comput. I. 17 (1968), pp. 188-192.

[21]

Golub, G. H. and C. F. van Loan, Matrix computations, John Hopkins University Press, 1985 (476 pgs).

[22]

Gustafson, J. L., G. R. Montry and R. E. Benner, Development of Parallel Methods for a 1024- Processor Hypercube, SIAM J. Sci. Statist. Comput., 9, 4 (1988), pp. 609-638.

Digital Library

[23]

Hageman, L. A. and D. M. Young, Applied Iterative Methods, Academic Press 1981.

[24]

Hestenes, M. and E. Stiefel, Methods of Conjugate Gradients for solving linear systems, J. Res. Natl. Bur. Stand. Sect B,19 (1952), pp. 409--436.

[25]

Houstis, E. N., E. A. Vavalis and J. R. Rice, Convergence of an O(h4) cubic spline collocation method for elliptic partial differential equations, SIAM J. Numer. Anal., 25 (1988), pp. 54--74.

Digital Library

[26]

Houstis, E. N., J. R. Rice, C. C. Christara and E. A. Vavalis, Performance of scientific software, Mathematical Aspects of Scientific Software, (J. R. Rice, ed.), Springer Verlag, 1988, pp. 123-156.

Digital Library

[27]

Housl~ E. N., J. R. Rt~ and E. A. Vavalis, A Schwartz Splitting variant of Cubic Spline Collocation Methods for Elliptic PDE.~, PrOngs of the third Conference on Hypereub~, Concta-tent Computers and Applioations, January 1988, Pasad~na, CA, U.S.A.

Digital Library

[28]

Intel Scientific Computers, iP$CI2 Performance report, Intel Scientific Computers, January 1988, Beav~rton, OR, U.S.A.

[29]

Irodotou-Ellina, M., Spline collocation methods for high order elliptic boundary value problems, Aristotle University of Thessaloniki, Caeece, Ph.D. thesis, 1987.

[30]

Irodotou-Ellina, M. and E. N. Houstis, An O (/16) quintic xpline collocation method for fourth-ord~ two-point boundary value problems BIT, 28 (1988), pp. 288-301.

Digital Library

[31]

Kammener, W. J., G. W. Redden and R. S. Varga, Quadratic interpolatory splines, Numer. Math., 22 (1974), pp. 241-259.

Digital Library

[32]

Keyes, D. E. and W. D. Gropp, A co~ison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM J. Sci. Statist. Comput., 8 (1987), pp. s 166--s2ff2.

Digital Library

[33]

Khalifa, A. K. and j. C. Eilbeek, Collocation with quadratic and cubic splines, IM L Num~r. Anal., 2 (1982), pp. 111-121.

[34]

O'Leary, D., Parallel Implementation of Block Conjugate Gradient Algorithm. Parallel Comput. 5 (1987). pp. 127-140.

[35]

Ortega, J. M, Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, 1988 (305 pgs).

Digital Library

[36]

Rodrigue, G., Some ideas for decomposing the domain of elliptic Partial Differential Equations in the Schwarz process, Cornmun. Appl. Numer. Method, 2 (1986), pp. 245---.'249.

[37]

Russell, R. D. and L. F. Shampine, A collocation method for boundary value problems, Numer. Math., 19 (1972), pp. 1-28.

Digital Library

[38]

Sakai, M. and R. Usmani, Quadratic ~pline solutions and ~o-point boundary value problems, Publ. RIMS, Kyoto University, 19 (1983), pp. 7-13.

[39]

Tang, Wei-Pai, Schwartz Splitling and Template Operators, Stanford University, Ph.D thesis, 1987.

Digital Library

[40]

Varga, R. S., Matrix iterative analysis, Prentice Hall, 1962.

[41]

Young, D. M., lterative Solution of Large Linear Systems, Academic Press 197 l.

Cited By

Christara CSmith B(1997)Multigrid and multilevel methods for quadratic spline collocationBIT Numerical Mathematics10.1007/BF0251035237:4(781-803)Online publication date: Dec-1997
https://doi.org/10.1007/BF02510352
Christara C(1994)Quadratic spline collocation methods for elliptic partial differential equationsBIT10.1007/BF0193501534:1(33-61)Online publication date: Mar-1994
https://doi.org/10.1007/BF01935015
Christara C(1991)Domain Decomposition and Incomplete Factorisation Methods for Partial Differential EquationsThe Sixth Distributed Memory Computing Conference, 1991. Proceedings10.1109/DMCC.1991.633166(369-377)Online publication date: 1991
https://doi.org/10.1109/DMCC.1991.633166
Show More Cited By

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Schur complement preconditioned conjugate gradient methods for spline collocation equations

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Published In

cover image ACM Conferences

ICS '90: Proceedings of the 4th international conference on Supercomputing

June 1990

492 pages

ISBN:0897913698

DOI:10.1145/77726

Chairmen:
Ahmed Sameh
Univ. of Illinois
,
Henk van der Vorst
Delft Univ. of Technology and CWI, The Netherlands

ACM SIGARCH Computer Architecture News Volume 18, Issue 3b
Special Issue: Proceedings of the 4th international conference on Supercomputing
Sept. 1990
489 pages
ISSN:0163-5964
DOI:10.1145/255129
Chairmen:
Ahmed Sameh
Univ. of Illinois at Urbana-Champaign, Urbana
,
Henk van der Vorst
Delft Univ. of Technology and CWI, The Netherlands
Issue’s Table of Contents

Copyright © 1990 ACM.

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Association for Computing Machinery

New York, NY, United States

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Published: 01 June 1990

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IC'90: ACM SIGARCH International Conference on Supercomputing

June 11 - 15, 1990

Amsterdam, The Netherlands

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Overall Acceptance Rate 629 of 2,180 submissions, 29%

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Cited By

Christara CSmith B(1997)Multigrid and multilevel methods for quadratic spline collocationBIT Numerical Mathematics10.1007/BF0251035237:4(781-803)Online publication date: Dec-1997
https://doi.org/10.1007/BF02510352
Christara C(1994)Quadratic spline collocation methods for elliptic partial differential equationsBIT10.1007/BF0193501534:1(33-61)Online publication date: Mar-1994
https://doi.org/10.1007/BF01935015
Christara C(1991)Domain Decomposition and Incomplete Factorisation Methods for Partial Differential EquationsThe Sixth Distributed Memory Computing Conference, 1991. Proceedings10.1109/DMCC.1991.633166(369-377)Online publication date: 1991
https://doi.org/10.1109/DMCC.1991.633166
Christara C(1990)Schur complement preconditioned conjugate gradient methods for spline collocation equationsACM SIGARCH Computer Architecture News10.1145/255129.25514618:3b(108-120)Online publication date: Jun-1990
https://doi.org/10.1145/255129.255146
Christara C(1990)Conjugate Gradient Methods for Spline Collocation EquationsProceedings of the Fifth Distributed Memory Computing Conference, 1990.10.1109/DMCC.1990.555433(550-558)Online publication date: 1990
https://doi.org/10.1109/DMCC.1990.555433
Bialecki BFairweather GKarageorghis A(2002)Matrix Decomposition Algorithms for Modified Spline Collocation for Helmholtz ProblemsSIAM Journal on Scientific Computing10.1137/S106482750139964X24:5(1733-1753)Online publication date: 1-May-2002
https://dl.acm.org/doi/10.1137/S106482750139964X
Bialecki BFairweather G(2001)Orthogonal spline collocation methods for partial differential equationsJournal of Computational and Applied Mathematics10.1016/S0377-0427(00)00509-4128:1-2(55-82)Online publication date: 1-Mar-2001
https://dl.acm.org/doi/10.1016/S0377-0427%2800%2900509-4
Bialecki BFairweather G(2001)Orthogonal spline collocation methods for partial differential equationsPartial Differential Equations10.1016/B978-0-444-50616-0.50005-1(55-82)Online publication date: 2001
https://doi.org/10.1016/B978-0-444-50616-0.50005-1

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