PaStiX: a high-performance parallel direct solver for sparse symmetric positive definite systems

Solving large sparse symmetric positive definite systems of linear equations is a crucial and time-consuming step, arising in many scientific and engineering applications. The block partitioning and scheduling problem for sparse parallel factorization without pivoting is considered. There are two major aims to this study: the scalability of the parallel solver, and the compromise between memory overhead and efficiency. Parallel experiments on a large collection of irregular industrial problems validate our approach.

[1]  Robert Schreiber,et al.  Improved load distribution in parallel sparse Cholesky factorization , 1994, Proceedings of Supercomputing '94.

[2]  Jean-Yves L'Excellent,et al.  Analysis, tuning and comparison of two general sparse solvers for distributed memory computers , 2000 .

[3]  Jean Roman,et al.  Sparse Matrix Ordering with SCOTCH , 1997, HPCN Europe.

[4]  James Demmel,et al.  A Scalable Sparse Direct Solver Using Static Pivoting , 1999, PPSC.

[5]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[6]  Patrick R. Amestoy,et al.  Multifrontal parallel distributed symmetric and unsymmetric solvers , 2000 .

[7]  Vipin Kumar,et al.  Highly Scalable Parallel Algorithms for Sparse Matrix Factorization , 1997, IEEE Trans. Parallel Distributed Syst..

[8]  Pascal Hénon,et al.  A Mapping and Scheduling Algorithm for Parallel Sparse Fan-In Numerical Factorization , 1999, Euro-Par.

[9]  I. Duff,et al.  Sparse numerical linear algebra: Direct methods and preconditioning , 1996 .

[10]  George Karypis,et al.  PSPASES: Building a High Performance Scalable Parallel Direct Solver for Sparse Linear Systems , 1999 .

[11]  Anoop Gupta,et al.  An Evaluation of Left-Looking, Right-Looking and Multifrontal Approaches to Sparse Cholesky Factorization on Hierarchical-Memory Machines , 1991, Int. J. High Speed Comput..

[12]  Iain S. Duff,et al.  The Multifrontal Solution of Unsymmetric Sets of Linear Equations , 1984 .

[13]  Pascal Hénon,et al.  PaStiX: A Parallel Sparse Direct Solver Based on a Static Scheduling for Mixed 1D/2D Block Distributions , 2000, IPDPS Workshops.

[14]  J. Bunch,et al.  Some stable methods for calculating inertia and solving symmetric linear systems , 1977 .

[15]  Vipin Kumar,et al.  A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering , 1998, J. Parallel Distributed Comput..

[16]  Patrick R. Amestoy,et al.  Hybridizing nested dissection and halo approximate minimum degree for efficient sparse matrix ordering , 2000 .

[17]  Alex Pothen,et al.  A Mapping Algorithm for Parallel Sparse Cholesky Factorization , 1993, SIAM J. Sci. Comput..

[18]  Anoop Gupta,et al.  An Efficient Block-Oriented Approach to Parallel Sparse Cholesky Factorization , 1994, SIAM J. Sci. Comput..

[19]  Joseph W. H. Liu,et al.  Comparisons of Three Distributed Algorithms for Sparse Cholesky Factorization , 1989, PPSC.

[20]  Cleve Ashcraft,et al.  A Fan-In Algorithm for Distributed Sparse Numerical Factorization , 1990, SIAM J. Sci. Comput..

[21]  Joseph W. H. Liu,et al.  Modification of the minimum-degree algorithm by multiple elimination , 1985, TOMS.

[22]  P. Charrier,et al.  Algorithmique et calculs de complexité pour un solveur de type dissections emboîtées , 1989 .

[23]  Cleve Ashcraft,et al.  The Fan-Both Family of Column-Based Distributed Cholesky Factorization Algorithms , 1993 .

[24]  Edward Rothberg.  Performance of Panel and Block Approaches to Sparse Cholesky Factorization on the iPSC/860 and Paragon Multicomputers , 1996, SIAM J. Sci. Comput..

[25]  Patrick R. Amestoy,et al.  An Approximate Minimum Degree Ordering Algorithm , 1996, SIAM J. Matrix Anal. Appl..

[26]  Xiaoye Sherry Li,et al.  Sparse Gaussian Elimination on High Performance Computers , 1996 .

[27]  Alan George,et al.  Computer Solution of Large Sparse Positive Definite , 1981 .

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

[29]  A. George,et al.  Graph theory and sparse matrix computation , 1993 .

[30]  Patrick Amestoy,et al.  Memory Management Issues in Sparse Multifrontal Methods On Multiprocessors , 1993, Int. J. High Perform. Comput. Appl..

[31]  John M. Conroy,et al.  Multifrontal Sparse Solvers in Message Passing and Data Parallel Environments - A Comparative Study , 1993, PARCO.

[32]  Michael T. Heath,et al.  Parallel Algorithms for Sparse Linear Systems , 1991, SIAM Rev..

[33]  Wolfgang Fichtner,et al.  Efficient Sparse LU Factorization with Left-Right Looking Strategy on Shared Memory Multiprocessors , 2000 .