Computational models and task scheduling for parallel sparse Cholesky factorization

Abstract In this paper, a systematic and unified treatment of computational task models for parallel sparse Cholesky factorization is presented. They are classified as fine-, medium-, and large-grained graph models. In particular, a new medium-grained model based on column-oriented tasks is introduced, and it is shown to correspond structurally to the filled graph of the given sparse matrix. The task scheduling problem for the various task graphs is also discussed. A practical algorithm to schedule the column tasks of the medium-grained model for multiple processors is described. It is based on a heuristic critical path scheduling method. This will give an overall scheme for parallel sparse Cholesky factorization, appropriate for parallel machines with shared-memory architecture like the Denelcor HEP.

[1]  Jochen A. G. Jess,et al.  A Data Structure for Parallel L/U Decomposition , 1982, IEEE Transactions on Computers.

[2]  Robert Schreiber,et al.  A New Implementation of Sparse Gaussian Elimination , 1982, TOMS.

[3]  Christopher P. Arnold,et al.  An Efficient Parallel Algorithm for the Solution of Large Sparse Linear Matrix Equations , 1983, IEEE Transactions on Computers.

[4]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[5]  Mandayam A. Srinivas Optimal Parallel Scheduling of Gaussian Elimination DAG's , 1983, IEEE Transactions on Computers.

[6]  Joseph W. Liu,et al.  A compact row storage scheme for Cholesky factors using elimination trees , 1986, TOMS.

[7]  Edward G. Coffman,et al.  Computer and job-shop scheduling theory , 1976 .

[8]  M. T. Kaufman,et al.  An Almost-Optimal Algorithm for the Assembly Line Scheduling Problem , 1974, IEEE Transactions on Computers.

[9]  T. C. Hu Parallel Sequencing and Assembly Line Problems , 1961 .

[10]  Omar Wing,et al.  A Computation Model of Parallel Solution of Linear Equations , 1980, IEEE Transactions on Computers.

[11]  Iain S. Duff,et al.  Full matrix techniques in sparse Gaussian elimination , 1982 .

[12]  Frans J. Peters,et al.  Parallel pivoting algorithms for sparse symmetric matrices , 1984, Parallel Comput..

[13]  H.F. Jordan,et al.  Experience with pipelined multiple instruction streams , 1984, Proceedings of the IEEE.

[14]  K. Mani Chandy,et al.  A comparison of list schedules for parallel processing systems , 1974, Commun. ACM.

[15]  Walter H. Kohler,et al.  A Preliminary Evaluation of the Critical Path Method for Scheduling Tasks on Multiprocessor Systems , 1975, IEEE Transactions on Computers.

[16]  Alfred V. Aho,et al.  Data Structures and Algorithms , 1983 .

[17]  J. Liu,et al.  Parallel Cholesky factorization on a multiprocessor , 1985 .

[18]  Mario J. Gonzalez Deterministic Processor Scheduling , 1977, CSUR.

[19]  F. Gustavson,et al.  Implementing Linear Algebra Algorithms for Dense Matrices on a Vector Pipeline Machine , 1984 .