RILUM : A General Framework for Robust MultilevelRecursive Incomplete LU Preconditioning Techniques

We introduce a general framework for constructing multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. This technique is based on a recursive two by two block incomplete LU factorization on the coeecient matrix. The coarse level system is constructed as an (approximate) Schur complement. A dynamic preconditioner is obtained by solving the Schur complement matrix approximately. The novelty of the proposed techniques is to solve the Schur complement matrix by a preconditioned Krylov subspace method. The very preconditioner for this secondary iteration is constructed by considering the Schur complement matrix as a general sparse matrix and by applying to it the block ILU factorization process that was applied to the original matrix. This recur-sive procedure continues for a few times and results in a multilevel preconditioner. Diierent implementation strategies are discussed. We conduct numerical experiments with two particular RILUM implementations to show the performance of the proposed techniques and to demonstrate grid independent convergence rates of RILUM for solving certain problems.

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

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

[3]  Jun Zhang,et al.  Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems , 2001 .

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

[5]  Cornelis Vuik,et al.  GMRESR: a family of nested GMRES methods , 1994, Numer. Linear Algebra Appl..

[6]  曹志浩,et al.  ON ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS , 1993 .

[7]  H. Elman.  A stability analysis of incomplete LU factorizations , 1986 .

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

[9]  G. Golub,et al.  ITERATIVE METHODS FOR CYCLICALLY REDUCED NON-SELF-ADJOINT LINEAR SYSTEMS , 1990 .

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

[11]  Lloyd N. Trefethen,et al.  How Fast are Nonsymmetric Matrix Iterations? , 1992, SIAM J. Matrix Anal. Appl..

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

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

[14]  J. G. Lewis,et al.  A fast algorithm for reordering sparse matrices for parallel factorization , 1989 .

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

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

[17]  Thomas A. Manteuffel,et al.  Algebraic Multigrid Based on Element Interpolation (AMGe) , 2000, SIAM J. Sci. Comput..

[18]  Stephen F. McCormick,et al.  Multilevel adaptive methods for partial differential equations , 1989, Frontiers in applied mathematics.

[19]  Tony F. Chan,et al.  MULTILEVEL ELLIPTIC SOLVERS ON UNSTRUCTURED GRIDS , 1998 .

[20]  Michael Griebel,et al.  Multilevel preconditioning based on discrete symmetrization for convection-diffusion equations , 1997 .

[21]  Michael Griebel,et al.  Parallel Domain-Oriented Multilevel Methods , 1995, SIAM J. Sci. Comput..

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

[23]  G. Golub,et al.  Block Preconditioning for the Conjugate Gradient Method , 1985 .

[24]  D. P. Young,et al.  Application of sparse matrix solvers as effective preconditioners , 1989 .

[25]  Saad,et al.  A Multi-Level Preconditioner with Applicationsto the Numerical Simulation of Coating ProblemsYousef , 1998 .

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

[27]  Randolph E. Bank,et al.  The Incomplete Factorization Multigraph Algorithm , 1999, SIAM J. Sci. Comput..

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

[29]  Jun Zhang,et al.  Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications , 2000 .

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

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

[32]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

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

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

[35]  Gene H. Golub,et al.  Closer to the solutions: iterative linear solvers , 1997 .

[36]  Jun Zhang,et al.  A grid-based multilevel incomplete LU factorization preconditioning technique for general sparse matrices , 2001, Appl. Math. Comput..

[37]  Howard C. Elman.  Approximate Schur complement reconditioners on serial and parallel computers , 1989 .

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