A multilevel block incomplete cholesky preconditioner for solving normal equations in linear least squares problems

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.     

The importance of structure in incomplete factorization preconditioners

In this paper, we consider level-based preconditioning, which is one of the basic approaches to incomplete factorization preconditioning of iterative methods. It is well-known that while structure-based preconditioners can be very useful, excessive memory demands can limit their usefulness. Here we present an improved strategy that considers the individual entries of the system matrix and restricts small entries to contributing to fewer levels of fill than the largest entries. Using symmetric positive-definite problems arising from a wide range of practical applications, we show that the use of variable levels of fill can yield incomplete Cholesky factorization preconditioners that are more efficient than those resulting from the standard level-based approach. The concept of level-based preconditioning, which is based on the structural properties of the system matrix, is then transferred to the numerical incomplete decomposition. In particular, the structure of the incomplete factorization determined in the symbolic factorization phase is explicitly used in the numerical factorization phase. Further numerical results demonstrate that our level-based approach can lead to much sparser but efficient incomplete factorization preconditioners.     

New breakdown-free variant of AINV method for nonsymmetric positive definite matrices

This paper proposes a new breakdown-free preconditioning technique, called SAINV-NS, of the AINV method of Benzi and Tuma for nonsymmetric positive definite matrices. The resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for QMR, BiCGSTAB and GMRES(m) methods. The preconditoner is reliable (pivot breakdown can not occur) and effective at reducing the number of iterations. Some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the AINV-A algorithm.     

A class of multilevel recursive incomplete LU preconditioning techniques

We introduce a class of 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 coefficient 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. Such a reduction process is repeated to yield a multilevel recursive preconditioner.     

On the eigenvalue distribution of a class of preconditioning methods

Summary A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationAx=b, whereA is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrixA and include both pointwise and blockwise factorization. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofC ^–1 A, whereC is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.     

ILUT: A dual threshold incomplete LU factorization

In this paper we describe an Incomplete LU factorization technique based on a strategy which combines two heuristics. This ILUT factorization extends the usual ILU(O) factorization without using the concept of level of fill-in. There are two traditional ways of developing incomplete factorization preconditioners. The first uses a symbolic factorization approach in which a level of fill is attributed to each fill-in element using only the graph of the matrix. Then each fill-in that is introduced is dropped whenever its level of fill exceeds a certain threshold. The second class of methods consists of techniques derived from modifications of a given direct solver by including a dropoff rule, based on the numerical size of the fill-ins introduced, traditionally referred to as threshold preconditioners. The first type of approach may not be reliable for indefinite problems, since it does not consider numerical values. The second is often far more expensive than the standard ILU(O). The strategy we propose is a compromise between these two extremes.     

On the semi-conjugate direction methods with dynamic preconditioning

The semi-conjugate residual methods and semi-conjugate gradient methods with dynamic preconditioners in Krylov subspaces are considered for solving the systems of linear algebraic equations whose matrices are not symmetric. Their orthogonal and variational properties are under study. New algorithms are proposed for choosing the inner iteration parameters in the preconditioning matrices corresponding to incomplete factorization methods. The efficiency of the resulting iterative processes is demonstrated by a set of numerical experiments for finite difference diffusion-convection equations.     

Weighted graph based ordering techniques for preconditioned conjugate gradient methods

We describe the basis of a matrix ordering heuristic for improving the incomplete factorization used in preconditioned conjugate gradient techniques applied to anisotropic PDE's. Several new matrix ordering techniques, derived from well-known algorithms in combinatorial graph theory, which attempt to implement this heuristic, are described. These ordering techniques are tested against a number of matrices arising from linear anisotropic PDE's, and compared with other matrix ordering techniques. A variation of RCM is shown to generally improve the quality of incomplete factorization preconditioners.This work was supported by by the Natural Sciences and Engineering Research Council of Canada, and by the Information Technology Research Center, which is funded by the Province of Ontario.     

Combination of numerical and structured approaches to the construction of a second-order incomplete triangular factorization in parallel preconditioning methods

Parallel versions of the stabilized second-order incomplete triangular factorization conjugate gradient method in which the reordering of the coefficient matrix corresponding to the ordering based on splitting into subdomains with separators are considered. The incomplete triangular factorization is organized using the truncation of fill-in “by value” at internal nodes of subdomains, and “by value” and ‘by positions” on the separators. This approach is generalized for the case of constructing a parallel version of preconditioning the second-order incomplete LU factorization for nonsymmetric diagonally dominant matrices with. The reliability and convergence rate of the proposed parallel methods is analyzed. The proposed algorithms are implemented using MPI, results of solving benchmark problems with matrices from the collection of the University of Florida are presented.     

A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problems

We present a parallel preconditioned iterative solver for large sparse symmetric positive definite linear systems. The preconditioner is constructed as a proper combination of advanced preconditioning strategies. It can be formally seen as being of domain decomposition type with algebraically constructed overlap. Similar to the classical domain decomposition technique, inexact subdomain solvers are used, based on incomplete Cholesky factorization. The proper preconditioner is shown to be near optimal in minimizing the so‐called K‐condition number of the preconditioned matrix. The efficiency of both serial and parallel versions of the solution method is illustrated on a set of benchmark problems in linear elasticity. Copyright © 2002 John Wiley & Sons, Ltd.     

Incomplete block factorization methods for complex-structure matrices

Two classes of SSOR-type incomplete block factorization methods are proposed for preconditioning of linear algebraic systems of equations with block banded matrices of complex structure. Correctness conditions are derived for these methods in application to M-matrices and their efficiency is demonstrated by numerical experiments with linear algebraic systems obtained by discretization of the three-dimensional Poisson equation using quadratic and cubic serendipity finite elements. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 5–22, 1987.     

Algebraic multilevel preconditioning of finite element matrices using local Schur complements

We consider an algebraic multilevel preconditioning technique for SPD matrices arising from finite element discretization of elliptic PDEs. In particular, we address the case of non‐M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The left upper block of the considered multiplicative two‐level preconditioner is approximated using incomplete factorization techniques. The coarse‐grid element matrices are simply Schur complements computed from local neighbourhood matrices, i.e. small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation that can be analysed by regarding (local) macro elements. These components, when combined in the framework of an algebraic multilevel iteration, yield a robust and efficient linear solver. The presented numerical experiments include also the Lamé differential equation for the displacements in the two‐dimensional plane‐stress elasticity problem. Copyright © 2005 John Wiley & Sons, Ltd.     

On some versions of incomplete block-matrix factorization iterative methods

Various forms of preconditioning matrices for iterative acceleration methods are discussed. The preconditioning is based on two versions of incomplete block-matrix factorization.     

Diagonally compensated reduction and related preconditioning methods

When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.     

Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization

In this work, we provide new analysis for a preconditioning technique called structured incomplete factorization (SIF) for symmetric positive definite matrices. In this technique, a scaling and compression strategy is applied to construct SIF preconditioners, where off‐diagonal blocks of the original matrix are first scaled and then approximated by low‐rank forms. Some spectral behaviors after applying the preconditioner are shown. The effectiveness is confirmed with the aid of a type of two‐dimensional and three‐dimensional discretized model problems. We further show that previous studies on the robustness are too conservative. In fact, the practical multilevel version of the preconditioner has a robustness enhancement effect, and is unconditionally robust (or breakdown free) for the model problems regardless of the compression accuracy for the scaled off‐diagonal blocks. The studies give new insights into the SIF preconditioning technique and confirm that it is an effective and reliable way for designing structured preconditioners. The studies also provide useful tools for analyzing other structured preconditioners. Various spectral analysis results can be used to characterize other structured algorithms and study more general problems.     

Spectral analysis of parallel incomplete factorizations with implicit pseudo‐overlap

Two general parallel incomplete factorization strategies are investigated. The techniques may be interpreted as generalized domain decomposition methods. In contrast to classical domain decomposition methods, adjacent subdomains exchange data during the construction of the incomplete factorization matrix, as well as during each local forward elimination and each local backward elimination involved in the application of the preconditioner. Local renumberings of nodes are combined with suitable global fill‐in strategy in an (successful) attempt to overcome the well‐known trade‐off between high parallelism (locality) and fast convergence (globality). From an algebraic viewpoint, our techniques may be implemented as global renumbering strategies. Theoretical spectral analysis is provided, which displays that the convergence rate weakly depends on the number of subdomains. Numerical results obtained on a 16‐processor SGI Origin 2000 are reported, showing the efficiency of our parallel preconditionings. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical comparison of preconditionings for large sparse finite element problems

A numerical study of the efficiency of the modified conjugate gradients (MCG) is performed using different preconditioning schemes. The MCG behavior is evaluated in connection with the solution of large linear sets of symmetric positive definite (p.d.) equations, arising from the finite element (f.e.) integration of partial differential equations of parabolic and elliptic type and the analysis of the leftmost eingenspectrum of the corresponding matrices. A simple incomplete Cholesky factorization ICCG(O) having the same sparsity pattern as the original problem is compared with a more complex technique ICAJ (Ψ) where the triangular factor is allowed to progressively fill in depending on a rejection parameter Ψ. The performance of the preconditioning algorithms is explored on finite element equations whose size N ranges between 150 and 2300. The results show that an optimal Ψ_opt may be found which minimizes the overall CPU time for the solution of both the linear system and the eigenproblem. The comparison indicates that ICAJ (Ψ_opt) is not significantly more efficient than ICCG(O), which therefore appears to be a simple, robust, and reliable method for the preconditioning of large sparse finite element models.     

A chordal preconditioner for large-scale optimization

We propose an automatic preconditioning scheme for large sparse numerical optimization. The strategy is based on an examination of the sparsity pattern of the Hessian matrix: using a graph-theoretic heuristic, a block-diagonal approximation to the Hessian matrix is induced. The blocks are submatrices of the Hessian matrix; furthermore, each block is chordal. That is, under a positive definiteness assumption, the Cholesky factorization can be applied to each block without creating any new nonzeros (fill). Therefore the preconditioner is space efficient. We conduct a number of numerical experiments to determine the effectiveness of the preconditioner in the context of a linear conjugate-gradient algorithm for optimization.     

Eigenvalue bounds for block preconditioned matrices

In the symmetric positive definite case, two-sided eigenvalue bounds for block Jacobi scaled matrices and upper eigenvalue bounds for matrices preconditioned with an incomplete block factorization are derived. A quantitative characterization of block matrix partitionings is also suggested, which can be used when analyzing various block preconditioning methods. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 5–41.