Parallel preconditioners based on partitioning sparse matrices 1

We describe a method for constructing an ecien t block diagonal preconditioner for accelerating the iterative solution of general sets of sparse linear equations. Our method uses a hypergraph partitioner on a scaled and sparsied matrix and attempts to ensure that the diagonal blocks are nonsingular and dominant. We illustrate our approach using the partitioner PaToH and the Krylov-based GMRES algorithm. We verify our approach with runs on problems from economic modelling and chemical engineering, traditionally dicult applications for iterative methods. Our approach and the block diagonal preconditioning lends itself to good exploitation of parallelism. This we also demonstrate.

[1]  R. M. Mattheyses,et al.  A Linear-Time Heuristic for Improving Network Partitions , 1982, 19th Design Automation Conference.

[2]  Charles M. Fiduccia,et al.  A linear-time heuristic for improving network partitions , 1988, 25 years of DAC.

[3]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[4]  Youcef Saad,et al.  A Basic Tool Kit for Sparse Matrix Computations , 1990 .

[5]  Ian Foster,et al.  Designing and building parallel programs , 1994 .

[6]  O. Axelsson.  Iterative solution methods , 1995 .

[7]  C. Aykanat.  Hypergraph Model for Mapping Repeated Sparse-Matrix Vector Product Computations onto Multicomputers , 1995 .

[8]  John K. Reid,et al.  The design of MA48: a code for the direct solution of sparse unsymmetric linear systems of equations , 1996, TOMS.

[9]  A. Neumaier,et al.  A NEW PIVOTING STRATEGY FOR GAUSSIAN ELIMINATION , 1996 .

[10]  Iain S. Duff,et al.  The Rutherford-Boeing sparse matrix collection , 1997 .

[11]  Vipin Kumar,et al.  Hmetis: a hypergraph partitioning package , 1998 .

[12]  Bruce Hendrickson,et al.  Graph Partitioning and Parallel Solvers: Has the Emperor No Clother? (Extended Abstract) , 1998, IRREGULAR.

[13]  Tamara G. Kolda,et al.  Partitioning Sparse Rectangular Matrices for Parallel Computations of Ax and ATv , 1998, PARA.

[14]  Luc Giraud,et al.  On the Influence of the Orthogonalization Scheme on the Parallel Performance of GMRES , 1998, Euro-Par.

[15]  T. Kolda.  Partitioning Sparse Rectangular Matrices for Parallel Processing ? , 1998 .

[16]  Iain S. Duff,et al.  The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices , 1999, SIAM J. Matrix Anal. Appl..

[17]  Ümit V. Çatalyürek,et al.  Hypergraph-Partitioning-Based Decomposition for Parallel Sparse-Matrix Vector Multiplication , 1999, IEEE Trans. Parallel Distributed Syst..

[18]  Tamara G. Kolda,et al.  Graph partitioning models for parallel computing , 2000, Parallel Comput..

[19]  Tamara G. Kolda,et al.  Partitioning Rectangular and Structurally Unsymmetric Sparse Matrices for Parallel Processing , 1999, SIAM J. Sci. Comput..

[20]  Bruce Hendrickson,et al.  Graph partitioning models for parallel computing q , 2000 .

[21]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[22]  Iain S. Duff,et al.  On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix , 2000, SIAM J. Matrix Anal. Appl..

[23]  Luc Giraud,et al.  On the round-off error analysis of the Gram-Schmidt algorithm with reorthogonalization , 2002 .

[24]  George Karypis,et al.  Multilevel Hypergraph Partitioning , 2003 .

[25]  Julien Langou,et al.  Algorithm 842: A set of GMRES routines for real and complex arithmetics on high performance computers , 2005, TOMS.

[26]  Brendan Vastenhouw,et al.  A Two-Dimensional Data Distribution Method for Parallel Sparse Matrix-Vector Multiplication , 2005, SIAM Rev..

[27]  S.,et al.  An Efficient Heuristic Procedure for Partitioning Graphs , 2022 .