Balanced Incomplete Factorization

In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU or LDL T factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the Sherman-Morrison formula [18]. In contrast to the RIF algorithm [11], the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. For the symmetric positive definite case, we derive the theory and present an algorithm for computing the incomplete LDL T factorization, and discuss experimental results. We call this new approximate LDL T factorization the Balanced Incomplete Factorization (BIF). Our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill-conditioned problems by preconditioned iterative methods. Moreover, the internal coupling of computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness. We also derive and present the theory for the general nonsymmetric case, but do not discuss its implementation.

[1]  Michele Benzi,et al.  A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method , 1996, SIAM J. Sci. Comput..

[2]  Alan Jennings,et al.  A Compact Storage Scheme for the Solution of Symmetric Linear Simultaneous Equations , 1966, Comput. J..

[3]  Yousef Saad,et al.  On the Relations between ILUs and Factored Approximate Inverses , 2002, SIAM J. Matrix Anal. Appl..

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

[5]  Olaf Schenk,et al.  Weighted Matchings for Preconditioning Symmetric Indefinite Linear Systems , 2006, SIAM J. Sci. Comput..

[6]  Owe Axelsson,et al.  Diagonally compensated reduction and related preconditioning methods , 1994, Numer. Linear Algebra Appl..

[7]  YereminA. Yu.,et al.  Factorized sparse approximate inverse preconditionings I , 1993 .

[8]  Robert Bridson,et al.  Ordering, Anisotropy, and Factored Sparse Approximate Inverses , 1999, SIAM J. Sci. Comput..

[9]  Juana Cerdán,et al.  Preconditioning Sparse Nonsymmetric Linear Systems with the Sherman-Morrison Formula , 2003, SIAM J. Sci. Comput..

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

[11]  N. I. Buleev.  A numerical method for solving two-dimensional diffusion equations , 1960 .

[12]  Michele Benzi,et al.  A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems , 1998, SIAM J. Sci. Comput..

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

[14]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[15]  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.

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

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

[18]  Matthias Bollhöfer,et al.  A Robust and Efficient ILU that Incorporates the Growth of the Inverse Triangular Factors , 2003, SIAM J. Sci. Comput..

[19]  Matthias Bollhöfer,et al.  A robust ILU with pivoting based on monitoring the growth of the inverse factors , 2001 .

[20]  Michele Benzi,et al.  Orderings for Factorized Sparse Approximate Inverse Preconditioners , 1999, SIAM J. Sci. Comput..

[21]  M. A. Ajiz,et al.  A robust incomplete Choleski‐conjugate gradient algorithm , 1984 .

[22]  M. Tismenetsky,et al.  A new preconditioning technique for solving large sparse linear systems , 1991 .

[23]  Ingyu Lee,et al.  Effective Preconditioning through Ordering Interleaved with Incomplete Factorization , 2005, SIAM J. Matrix Anal. Appl..

[24]  Michele Benzi,et al.  Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method , 2000, SIAM J. Sci. Comput..

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

[26]  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..

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

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

[29]  Timothy A. Davis,et al.  Algorithm 8 xx : a concise sparse Cholesky factorization package , 2004 .

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

[31]  José Marín Mateos-Aparicio,et al.  BLOCK APPROXIMATE INVERSE PRECONDITIONERS FOR SPARSE NONSYMMETRIC LINEAR SYSTEMS , 2010 .

[32]  Rudolf A. Römer,et al.  On Large-Scale Diagonalization Techniques for the Anderson Model of Localization , 2006, SIAM J. Sci. Comput..

[33]  M. Benzi,et al.  A comparative study of sparse approximate inverse preconditioners , 1999 .

[34]  Igor E. Kaporin,et al.  High quality preconditioning of a general symmetric positive definite matrix based on its U , 1998 .

[35]  Michele Benzi,et al.  A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems , 2003, SIAM J. Sci. Comput..

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

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

[38]  Anne Greenbaum,et al.  Approximating the inverse of a matrix for use in iterative algorithms on vector processors , 1979, Computing.

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

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

[41]  R. P. Kendall,et al.  An Approximate Factorization Procedure for Solving Self-Adjoint Elliptic Difference Equations , 1968 .

[42]  C. Micchelli,et al.  Polynomial Preconditioners for Conjugate Gradient Calculations , 1983 .

[43]  Igor E. Kaporin,et al.  High quality preconditioning of a general symmetric positive definite matrix based on its UTU + UTR + RTU-decomposition , 1998, Numer. Linear Algebra Appl..