Preconditioning sparse matrices for computing eigenvalues and solving linear systems of equations

Preconditioning Sparse Matrices for Computing Eigenvalues and Solving Linear Systems of Equations

[1]  J. A. ScottyJanuary.  An Evaluation of Software for Computing Eigenvalues of Sparse Nonsymmetric Matrices , 1996 .

[2]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

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

[4]  Marcus J. Grote,et al.  Parallel Preconditioning with Sparse Approximate Inverses , 1997, SIAM J. Sci. Comput..

[5]  D. Hartfiel.  Concerning diagonal similarity of irreducible matrices , 1971 .

[6]  Takumi Washio,et al.  Ordering strategies and related techniques to overcome the trade-off between parallelism and convergence in incomplete factorizations , 1999, Parallel Comput..

[7]  Kincho H. Law,et al.  A robust incomplete factorization based on value and space constraints , 1995 .

[8]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[9]  E. Ng,et al.  Predicting structure in nonsymmetric sparse matrix factorizations , 1993 .

[10]  C. Reinsch,et al.  Balancing a matrix for calculation of eigenvalues and eigenvectors , 1969 .

[11]  James Hardy Wilkinson,et al.  Error Analysis of Direct Methods of Matrix Inversion , 1961, JACM.

[12]  Rudnei Dias Da Cunha,et al.  PIM 2.2 The Parallel Iterative Methods package for Systems of Linear Equations User's Guide (Fortran , 1996 .

[13]  Jack Dongarra,et al.  A Test Matrix Collection for Non-Hermitian Eigenvalue Problems , 1997 .

[14]  A. Pothen,et al.  Efficient Parallel Computation of ILU(k) Preconditioners , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[15]  William Gropp,et al.  PETSc 2.0 Users Manual: Revision 2.0.16 , 1997 .

[16]  Bruce Hendrickson,et al.  Support Theory for Preconditioning , 2003, SIAM J. Matrix Anal. Appl..

[17]  Y. Saad.  BILUM : Block versions of multi-elimination ILU preconditioner for general sparse linear systems , 1999 .

[18]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[19]  W. James,et al.  A Conjugate Gradient-Truncated Direct Method for the Iterative Solution of the Reservoir Simulation Pressure Equation , 1981 .

[20]  P. Sadayappan,et al.  On improving the performance of sparse matrix-vector multiplication , 1997, Proceedings Fourth International Conference on High-Performance Computing.

[22]  William G. Poole,et al.  An algorithm for reducing the bandwidth and profile of a sparse matrix , 1976 .

[23]  V. Kumar,et al.  Parallel Threshold-based ILU Factorization , 1997, ACM/IEEE SC 1997 Conference (SC'97).

[24]  Gary L. Miller,et al.  Performance evaluation of a new parallel preconditioner , 1995, Proceedings of 9th International Parallel Processing Symposium.

[25]  Sivan Toledo,et al.  Improving the memory-system performance of sparse-matrix vector multiplication , 1997, IBM J. Res. Dev..

[26]  Yousef Saad,et al.  Distributed ILU(0) and SOR Preconditioners for Unstructured Sparse Linear Systems , 1998 .

[27]  Walter L. Smith.  Probability and Statistics , 1959, Nature.

[28]  Mark T. Jones,et al.  An improved incomplete Cholesky factorization , 1995, TOMS.

[29]  Michele Benzi,et al.  Preconditioning Highly Indefinite and Nonsymmetric Matrices , 2000, SIAM J. Sci. Comput..

[30]  Wei-Pai Tang,et al.  Ordering Methods for Preconditioned Conjugate Gradient Methods Applied to Unstructured Grid Problems , 1992, SIAM J. Matrix Anal. Appl..

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

[32]  E. L. Poole,et al.  Multicolor ICCG methods for vector computers , 1987 .

[33]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[34]  Joseph W. H. Liu,et al.  Robust Ordering of Sparse Matrices using Multisection , 1998 .

[35]  Iain S. Duff,et al.  Preconditioning and parallel preconditioning , 1998 .

[36]  Cleve Ashcraft,et al.  Compressed Graphs and the Minimum Degree Algorithm , 1995, SIAM J. Sci. Comput..

[37]  N. Munksgaard,et al.  Solving Sparse Symmetric Sets of Linear Equations by Preconditioned Conjugate Gradients , 1980, TOMS.

[38]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[39]  T. Manteuffel.  An incomplete factorization technique for positive definite linear systems , 1980 .

[40]  Marcus J. Grote,et al.  Effective Parallel Preconditioning with Sparse Approximate Inverses , 1995, PPSC.

[41]  H. Elman,et al.  Ordering techniques for the preconditioned conjugate gradient method on parallel computers , 1989 .

[42]  Alan George,et al.  A Fast Implementation of the Minimum Degree Algorithm Using Quotient Graphs , 1980, TOMS.

[43]  G. Pagallo,et al.  A bipartite quotient graph model for unsymmetric matrices , 1983 .

[44]  Stavros A. Zenios,et al.  A Comparative Study of Algorithms for Matrix Balancing , 1990, Oper. Res..

[45]  Yousef Saad,et al.  ILUs And Factorized Approximate Inverses are stronglyrelated . Part II : Applications to stabilization , 2000 .

[46]  Edmond Chow,et al.  Parallel Approximate Inverse Preconditioners , 1997, PPSC.

[47]  H. Wilf,et al.  Direct Solutions of Sparse Network Equations by Optimally Ordered Triangular Factorization , 1967 .

[48]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[49]  N. S. Mendelsohn,et al.  Coverings of Bipartite Graphs , 1958, Canadian Journal of Mathematics.

[50]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

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

[52]  Sivan Toledo,et al.  Toward an Efficient Column Minimum Degree Code for Symmetric Multiprocessors , 1999, PPSC.

[53]  D. Kershaw.  The incomplete Cholesky—conjugate gradient method for the iterative solution of systems of linear equations , 1978 .

[54]  D. Chen.  Analysis , Implementation , and Evaluation of Vaidya ’ s Preconditioners , 2001 .

[55]  S. Eisenstat,et al.  Node Selection Strategies for Bottom-Up Sparse Matrix Ordering , 1998, SIAM J. Matrix Anal. Appl..

[56]  Yousef Saad,et al.  ARMS: an algebraic recursive multilevel solver for general sparse linear systems , 2002, Numer. Linear Algebra Appl..

[57]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[58]  A. Pothen.  Sparse null bases and marriage theorems , 1984 .

[59]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[60]  Jack Dongarra,et al.  Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.

[61]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[62]  R. Betancourt,et al.  An efficient heuristic ordering algorithm for partial matrix refactorization , 1988 .

[63]  Fred G. Gustavson,et al.  LAWRA: Linear Algebra with Recursive Algorithms , 2000, PARA.

[64]  E. F. F. Botta,et al.  Matrix Renumbering ILU: An Effective Algebraic Multilevel ILU Preconditioner for Sparse Matrices , 1999, SIAM J. Matrix Anal. Appl..

[65]  Edmond Chow,et al.  A Scalable Parallel Computation of Sparse Approximate Inverse Preconditioners , 1999, PPSC.

[66]  Alex Pothen,et al.  Computing the block triangular form of a sparse matrix , 1990, TOMS.

[67]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[68]  Jun Zhang.  Sparse approximate inverse and multilevel block ILU preconditioning techniques for general sparse matrices , 2000 .

[69]  A. George,et al.  On the application of the minimum degree algorithm to finite element systems , 1978 .

[70]  E. D'Azevedo,et al.  Towards a cost-effective ILU preconditioner with high level fill , 1992 .

[71]  Sergio Pissanetzky,et al.  Sparse Matrix Technology , 1984 .

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

[73]  Michele Benzi,et al.  Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems , 1999, SIAM J. Sci. Comput..

[74]  Jason Wu,et al.  The Reference Manual for SPOOLES, Release 2.2: An Object Oriented Software Library for Solving Sparse Linear Systems of Equations , 1999 .

[75]  I. Duff,et al.  The effect of ordering on preconditioned conjugate gradients , 1989 .

[76]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[77]  J. Reif.  Efficient approximate solution of sparse linear systems , 1998 .

[78]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[79]  L. Grosz.  Preconditioning by incomplete block elimination , 2000, Numer. Linear Algebra Appl..

[80]  Barry W. Peyton,et al.  A Blocked Incomplete Cholesky Preconditioner For Hierarchical-Memory Computers , 1999 .

[81]  John N. Shadid,et al.  Official Aztec user''s guide: version 2.1 , 1999 .

[82]  E. Cuthill,et al.  Reducing the bandwidth of sparse symmetric matrices , 1969, ACM '69.

[83]  Chih-Jen Lin,et al.  Incomplete Cholesky Factorizations with Limited Memory , 1999, SIAM J. Sci. Comput..

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

[85]  Xiaoye S. Li,et al.  Computing Row and Column Counts for Sparse QR and LU Factorization , 2001 .

[86]  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..

[87]  Fred G. Gustavson,et al.  Recursion leads to automatic variable blocking for dense linear-algebra algorithms , 1997, IBM J. Res. Dev..

[88]  Christian Wagner,et al.  Multilevel ILU decomposition , 1999, Numerische Mathematik.

[89]  E. E. Osborne.  On pre-conditioning matrices , 1959, ACM '59.

[90]  Giovanni Manzini,et al.  On the Ordering of Sparse Linear Systems , 1996, Theor. Comput. Sci..

[91]  J. Demmel,et al.  Balancing sparse matrices for computing eigenvalues , 2000 .

[92]  Yousef Saad,et al.  Diagonal Threshold Techniques in Robust Multi-level ILU Preconditioners for General Sparse Linear Systems , 1999 .

[93]  Yousef Saad,et al.  ILUs and Factorized Approximate Inverses are Strongly Related. Part I: Overview of Results , 2000 .

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

[95]  Yousef Saad,et al.  ILUM: A Multi-Elimination ILU Preconditioner for General Sparse Matrices , 1996, SIAM J. Sci. Comput..

[96]  Y. Saad,et al.  Iterative solution of linear systems in the 20th century , 2000 .

[97]  Jun Zhang,et al.  BILUM: Block Versions of Multielimination and Multilevel ILU Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

[98]  John R. Gilbert,et al.  Sparse Matrices in MATLAB: Design and Implementation , 1992, SIAM J. Matrix Anal. Appl..

[99]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[100]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[101]  A. H. Sherman,et al.  Comparative Analysis of the Cuthill–McKee and the Reverse Cuthill–McKee Ordering Algorithms for Sparse Matrices , 1976 .

[102]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[103]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[104]  David E. Culler,et al.  REXEC: A Decentralized, Secure Remote Execution Environment for Clusters , 2000, CANPC.

[105]  A. George,et al.  An Implementation of Gaussian Elimination with Partial Pivoting for Sparse Systems , 1985 .

[106]  James Demmel,et al.  Making Sparse Gaussian Elimination Scalable by Static Pivoting , 1998, Proceedings of the IEEE/ACM SC98 Conference.

[107]  I. Gustafsson.  A class of first order factorization methods , 1978 .

[108]  R. Betancourt.  Efficient parallel processing technique for inverting matrices with random sparsity , 1986 .

[109]  Iain S. Duff,et al.  Users' guide for the Harwell-Boeing sparse matrix collection (Release 1) , 1992 .

[110]  J. A. George.  Computer implementation of the finite element method , 1971 .

[111]  Bonnie S. Heck-Ferri,et al.  Matrix scaling for large-scale system decomposition , 1996, Autom..

[112]  L. Khachiyan,et al.  On the Complexity of Matrix Balancing , 1997 .

[113]  Bruce Hendrickson,et al.  Effective Sparse Matrix Ordering: Just Around the BEND , 1997, PPSC.

[114]  Yousef Saad,et al.  ILUT: A dual threshold incomplete LU factorization , 1994, Numer. Linear Algebra Appl..

[115]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[116]  Daniel B. Szyld,et al.  A Block Ordering Method for Sparse Matrices , 1990, SIAM J. Sci. Comput..

[117]  D. Sorensen.  IMPLICITLY RESTARTED ARNOLDI/LANCZOS METHODS FOR LARGE SCALE EIGENVALUE CALCULATIONS , 1996 .

[118]  Eun Im,et al.  Optimizing the Performance of Sparse Matrix-Vector Multiplication , 2000 .

[119]  Yousef Saad,et al.  PSPARSLIB Users Manual: A Portable Library of parallel Sparse Iterative Solvers , 1998 .

[120]  James Demmel,et al.  An Asynchronous Parallel Supernodal Algorithm for Sparse Gaussian Elimination , 1997, SIAM J. Matrix Anal. Appl..

[121]  Jun Zhang,et al.  BILUTM: A Domain-Based Multilevel Block ILUT Preconditioner for General Sparse Matrices , 1999, SIAM J. Matrix Anal. Appl..

[122]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[123]  Jun Zhang,et al.  Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems , 1999, Numer. Linear Algebra Appl..

[124]  J. Grad,et al.  Matrix Balancing , 1971, Comput. J..

[125]  Jun Zhang,et al.  A class of multilevel recursive incomplete LU preconditioning techniques , 2001 .

[126]  Victor Eijkhout.  Overview of Iterative Linear System Solver Packages , 1998 .

[127]  George Karypis,et al.  Parmetis parallel graph partitioning and sparse matrix ordering library , 1997 .

[128]  Y. Saad,et al.  Experimental study of ILU preconditioners for indefinite matrices , 1997 .

[129]  Stephen Guattery.  Graph Embedding Techniques for Bounding Condition Numbers of Incomplete Factor Preconditioners , 1997 .

[130]  Mark T. Jones,et al.  BlockSolve95 users manual: Scalable library software for the parallel solution of sparse linear systems , 1995 .

[131]  A. Pinar,et al.  Improving Performance of Sparse Matrix-Vector Multiplication , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[132]  Sivan Toledo,et al.  An Assessment of Incomplete-LU Preconditioners for Nonsymmetric Linear Systems , 2000, Informatica.

[133]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[134]  Katherine A. Yelick,et al.  Optimizing Sparse Matrix Vector Multiplication on SMP , 1999, SIAM Conference on Parallel Processing for Scientific Computing.

[135]  W. Joubert,et al.  Numerical experiments with parallel orderings for ILU preconditioners. , 1999 .

[136]  John K. Reid,et al.  An Implementation of Tarjan's Algorithm for the Block Triangularization of a Matrix , 1978, TOMS.

[137]  W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations , 1993 .

[138]  Leopoldo García Franquelo,et al.  An efficient ordering algorithm to improve sparse vector methods , 1988 .

[139]  Z. Zlatev.  Use of Iterative Refinement in the Solution of Sparse Linear Systems , 1982 .

[140]  Alan George,et al.  The Evolution of the Minimum Degree Ordering Algorithm , 1989, SIAM Rev..

[141]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[142]  U. Rothblum,et al.  Line-sum-symmetric scalings of square nonnegative matrices , 1985 .

[143]  J. Meijerink,et al.  Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems , 1981 .

[144]  Yousef Saad,et al.  High-order ILU preconditioners for CFD problems , 2000 .

[145]  Hans Schneider,et al.  Max-Balancing Weighted Directed Graphs and Matrix Scaling , 1991, Math. Oper. Res..

[146]  Gene H. Golub,et al.  Matrix computations , 1983 .

[147]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[148]  David S. Kershaw.  On the problem of unstable pivots in the incomplete LU-conjugate gradient method , 1980 .

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

[150]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[151]  Erik Elmroth,et al.  Applying recursion to serial and parallel QR factorization leads to better performance , 2000, IBM J. Res. Dev..

[152]  D. Rose,et al.  Generalized nested dissection , 1977 .

[153]  Bruce Hendrickson,et al.  Improving the Run Time and Quality of Nested Dissection Ordering , 1998, SIAM J. Sci. Comput..