New Parallel Sparse Direct Solvers for Multicore Architectures

At the heart of many computations in science and engineering lies the need to efficiently and accurately solve large sparse linear systems of equations. Direct methods are frequently the method of choice because of their robustness, accuracy and potential for use as black-box solvers. In the last few years, there have been many new developments, and a number of new modern parallel general-purpose sparse solvers have been written for inclusion within the HSL mathematical software library. In this paper, we introduce and briefly review these solvers for symmetric sparse systems. We describe the algorithms used, highlight key features (including bit-compatibility and out-of-core working) and then, using problems arising from a range of practical applications, we illustrate and compare their performances. We demonstrate that modern direct solvers are able to accurately solve systems of order 106 in less than 3 minutes on a 16-core machine.

[1]  Nicholas I. M. Gould,et al.  A numerical evaluation of HSL packages for the direct solution of large sparse, symmetric linear systems of equations , 2004, TOMS.

[2]  Jennifer A. Scott,et al.  The effects of scalings on the performance of a sparse symmetric indefinite solver , 2008 .

[3]  Jennifer A. Scott,et al.  An out-of-core sparse Cholesky solver , 2009, TOMS.

[4]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[5]  Jennifer A. Scott,et al.  A fast and robust mixed-precision solver for the solution of sparse symmetric linear systems , 2010, TOMS.

[6]  Joseph W. H. Liu,et al.  The Multifrontal Method for Sparse Matrix Solution: Theory and Practice , 1992, SIAM Rev..

[7]  Jennifer A. Scott,et al.  Algorithm 891: A Fortran virtual memory system , 2009, TOMS.

[8]  Jennifer A. Scott,et al.  A modern analyse phase for sparse tree-based direct methods , 2010 .

[9]  A. George.  Nested Dissection of a Regular Finite Element Mesh , 1973 .

[10]  Jennifer A. Scott,et al.  An indefinite sparse direct solver for large problems on multicore machines , 2010 .

[11]  Jennifer A. Scott,et al.  HSL_MA97 : a bit-compatible multifrontal code for sparse symmetric systems , 2011 .

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

[13]  Timothy A. Davis,et al.  Algorithm 837: AMD, an approximate minimum degree ordering algorithm , 2004, TOMS.

[14]  Jennifer A. Scott,et al.  Pivoting strategies for tough sparse indefinite systems , 2013, TOMS.

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

[16]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[17]  John G. Lewis,et al.  Accurate Symmetric Indefinite Linear Equation Solvers , 1999, SIAM J. Matrix Anal. Appl..

[18]  I. Duff,et al.  Towards an automatic ordering for a symmetric sparse direct solver , 2005 .

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

[20]  Joseph W. H. Liu.  The role of elimination trees in sparse factorization , 1990 .

[21]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[22]  Jennifer A. Scott,et al.  Optimal Weighted Matchings for Rank-Deficient Sparse Matrices , 2013, SIAM J. Matrix Anal. Appl..

[23]  Jennifer A. Scott,et al.  Design of a Multicore Sparse Cholesky Factorization Using DAGs , 2010, SIAM J. Sci. Comput..

[24]  I. Duff,et al.  The factorization of sparse symmetric indefinite matrices , 1991 .

[25]  Pascal Hénon,et al.  PaStiX: a high-performance parallel direct solver for sparse symmetric positive definite systems , 2002, Parallel Comput..

[26]  Bora Uçar,et al.  A Symmetry Preserving Algorithm for Matrix Scaling , 2014, SIAM J. Matrix Anal. Appl..

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

[28]  Roger Grimes,et al.  The influence of relaxed supernode partitions on the multifrontal method , 1989, TOMS.

[29]  Jennifer A. Scott,et al.  Achieving bit compatibility in sparse direct solvers , 2012 .

[30]  Timothy A. Davis,et al.  The university of Florida sparse matrix collection , 2011, TOMS.

[31]  J. W. Walker,et al.  Direct solutions of sparse network equations by optimally ordered triangular factorization , 1967 .

[32]  Julien Langou,et al.  The Impact of Multicore on Math Software , 2006, PARA.

[33]  Iain S. Duff,et al.  MA57---a code for the solution of sparse symmetric definite and indefinite systems , 2004, TOMS.

[34]  O. Schenk,et al.  ON FAST FACTORIZATION PIVOTING METHODS FOR SPARSE SYMMETRI C INDEFINITE SYSTEMS , 2006 .

[35]  Julien Langou,et al.  A Class of Parallel Tiled Linear Algebra Algorithms for Multicore Architectures , 2007, Parallel Comput..

[36]  Pascal Hénon,et al.  PaStiX: A High-Performance Parallel Direct Solver for Sparse Symmetric Definite Systems , 2000 .

[37]  Jennifer A. Scott,et al.  Partial factorization of a dense symmetric indefinite matrix , 2012, ACM Trans. Math. Softw..

[38]  Olaf Schenk,et al.  Solving unsymmetric sparse systems of linear equations with PARDISO , 2002, Future Gener. Comput. Syst..

[39]  P. Gács,et al.  Algorithms , 1992 .

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

[41]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[42]  Nicholas I. M. Gould,et al.  A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations , 2007, TOMS.

[43]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[44]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[45]  Jennifer A. Scott,et al.  Compressed Threshold Pivoting for Sparse Symmetric Indefinite Systems , 2013, SIAM J. Matrix Anal. Appl..