Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems

Abstract This paper introduces several strategies to deal with pivot blocks in multi-level block incomplete LU factorization (BILUM) preconditioning techniques. These techniques are aimed at increasing the robustness and controlling the amount of fill-ins of BILUM for solving large sparse linear systems when large-size blocks are used to form block-independent set. Techniques proposed in this paper include double-dropping strategies, approximate singular-value decomposition, variable size blocks and use of an arrowhead block submatrix. We point out the advantages and disadvantages of these strategies and discuss their efficient implementations. Numerical experiments are conducted to show the usefulness of the new techniques in dealing with hard-to-solve problems arising from computational fluid dynamics. In addition, we discuss the relation between multi-level ILU preconditioning methods and algebraic multi-level methods.

[1]  O. Axelsson,et al.  Algebraic multilevel preconditioning methods, II , 1990 .

[2]  H.,et al.  Closer to the solution : Iterative linear solversGene , 1996 .

[3]  Owe Axelsson,et al.  Algebraic multilevel iteration method for Stieltjes matrices , 1994, Numer. Linear Algebra Appl..

[4]  Arnold Reusken,et al.  Approximate cyclic reduction preconditioning , 1998 .

[5]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

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

[7]  Murli M. Gupta,et al.  A single cell high order scheme for the convection‐diffusion equation with variable coefficients , 1984 .

[8]  Åke Björck,et al.  Numerical methods for least square problems , 1996 .

[9]  David A. H. Jacobs,et al.  The State of the Art in Numerical Analysis. , 1978 .

[10]  Masha Sosonkina,et al.  Distributed Schur Complement Techniques for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

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

[12]  Edmond Chow,et al.  Approximate Inverse Techniques for Block-Partitioned Matrices , 1997, SIAM J. Sci. Comput..

[13]  Gene H. Golub,et al.  Scientific computing: an introduction with parallel computing , 1993 .

[14]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[15]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[16]  Jun Zhang,et al.  Domain Decomposition and MultiLevel Type Techniques for General Sparse Linear Systems , 1997 .

[17]  I. Duff,et al.  The state of the art in numerical analysis , 1997 .

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

[19]  J. Mandel.  Balancing domain decomposition , 1993 .

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

[21]  P. Wesseling.  An Introduction to Multigrid Methods , 1992 .

[22]  R. Bank,et al.  The hierarchical basis multigrid method , 1988 .

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

[24]  Yvan Notay,et al.  Incomplete factorization preconditioning may lead to multigrid like speed of convergence , 1996 .

[25]  J. Zhangy.  A Sparse Approximate Inverse Technique for ParallelPreconditioning of General Sparse Matrices , 1998 .

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

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

[28]  Q. Chang,et al.  On the Algebraic Multigrid Method , 1996 .

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

[30]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

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