An algebraic approach for $${\mathcal{H}}$$ -matrix preconditioners

Simulation with models based on partial differential equations require very often the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present some ideas for updating approximate inverse preconditioners in factorized form. Computational costs, reorderings and implementation issues are considered.

Numerical Linear Algebra with Applications

This paper presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the quotient graph of the adjacency graph of the subdomains, resulting in a two-level block diagonal structure. A full binary tree structure T is then built to facilitate the construction of the preconditioner. We show that the difference between the inverse of a general block 2-by-2 SPD matrix and that of its block diagonal part can be well approximated by a low-rank matrix. This property and the block diagonal structure of the reordered matrix are exploited to develop a Multi-Color Low-Rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom-up traversal of the tree T. All irregular matrix computations, such as ILU factorizations and related triangular solves, are restricted to leaf nodes where these operations can be performed independently. Computations in non-leaf nodes only involve easy-to-optimize dense matrix operations. In order to further reduce the number of iteration of the Preconditioned Krylov subspace procedure, we combine MCLR with a few classical block-relaxation techniques. Numerical experiments on various test problems are proposed to illustrate the robustness and efficiency of the proposed approach for solving large sparse symmetric and nonsymmetric linear systems.

Recent Advances in Iterative Methods, 1994

The degree of parallelism in the preconditioned Krylov subspace method using standard preconditioners is limited and can lead to poor performance on massively parallel computers. In this paper we examine this problem and consider a number of alternatives based both on multi-coloring ideas and polynomial preconditioning. The emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured grids. It is argued that multi-coloring can be combined with multiple-step relaxation preconditioners to achieve a good level of parallelism while keeping the rates of convergence to good levels. We also exploit the idea of multi-coloring and independent set orderings to introduce a multielimination incomplete LU factorization named ILUM, which is related to multifrontal elimination. The main goal of the paper is to discuss some of the prevailing ideas and to compare them on a few test problems.

SIAM Journal on Matrix Analysis and Applications, 2021

A parallel preconditioner is proposed for general large sparse linear systems that combines a power series expansion method with low-rank correction techniques. To enhance convergence, a power series expansion is added to a basic Schur complement iterative scheme by exploiting a standard matrix splitting of the Schur complement. One of the goals of the power series approach is to improve the eigenvalue separation of the preconditioner thus allowing an effective application of a low-rank correction technique. Experiments indicate that this combination can be quite robust when solving highly indefinite linear systems. The preconditioner exploits a domain-decomposition approach and its construction starts with the use of a graph partitioner to reorder the original coefficient matrix. In this framework, unknowns corresponding to interface variables are obtained by solving a linear system whose coefficient matrix is the Schur complement. Unknowns associated with the interior variables are obtained by solving a block diagonal linear system where parallelism can be easily exploited. Numerical examples are provided to illustrate the effectiveness of the proposed preconditioner, with an emphasis on highlighting its robustness properties in the indefinite case.

The main topic of this thesis is updating preconditioners for solving large sparse linear systems Ax = b by using Krylov iterative methods. Two interesting types of problems are considered. In the first one is studied the iterative solution of nonsingular, non-symmetric linear systems where the coefficient matrix A has a skewsymmetric part of low-rank or can be well approximated with a skew-symmetric low-rank matrix. Systems like this arise from the discretization of PDEs with certain Neumann boundary conditions, the discretization of integral equations as well as path following methods, for example, the Bratu problem and the Love's integral equation. The second type of linear systems considered are least squares (LS) problems that are solved by considering the solution of the equivalent normal equations system. More precisely, we consider the solution of modified and rank deficient LS problems. By modified LS problem, it is understood that the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Rank deficient LS problems are characterized by a coefficient matrix that has not full rank, which makes difficult the computation of an incomplete factorization of the normal equations. LS problems arise in many large-scale applications of the science and engineering as for instance neural networks, linear programming, exploration seismology or image processing. Usually, incomplete LU or incomplete Cholesky factorization are used as preconditioners for iterative methods. The main contribution of this thesis is the development of a technique for updating preconditioners by bordering. It consists in the computation of an approximate decomposition for an equivalent augmented linear system, that is used as preconditioner for the original problem. The theoretical study and the results of the numerical experiments presented in this thesis show the performance of the preconditioner technique proposed and its competitiveness compared with other methods available in the literature for computing preconditioners for the problems studied. To my family. A special feeling of gratitude to my wife Elisa Savoia and my daughter Lucía Guerrero. I also dedicate this thesis to my friends who have supported me throughout the process. To each professor I had during my education, in particular, my project coordinators José Marín, José Mas and Juana Cerdán who have been more than generous with their expertise and precious time spent with me for preparing this thesis. To the

2010

This work analyzes the influenc of matrices reordering algorithms on solving linear systems using non-stationary iterative methods GMRES and Conjugate Gradient, both with and without preconditioning. The algorithms referenced most often in the literature for the reordering of matrices are Reverse Cuthill-McKee (RCM), Gibbs-Poole-Stockmeyer (GPS), Nested Dissection (ND) and Spectral (ES). We analyze these algorithms and propose some modification comparing their solution qualities (minimizing bandwidth and minimizing envelope) and CPU times. Moreover, the linear systems associated with sparse matrices are solved via preconditioned Krylov-type iterative methods considering the incomplete LU factorization preconditioners. For the computational tests, we consider a set of structurally symmetric matrices that can come from various field of knowledge. We conclude that the reordering of matrices, in most cases, reduces the number of iterations in the iterative methods, but that reducing the...

SIAM Journal on Matrix Analysis and Applications, 2017

For general symmetric positive definite (SPD) matrices, we present a framework for designing effective and robust black-box preconditioners via structured incomplete factorization. In a scaling-and-compression strategy, off-diagonal blocks are first scaled on both sides (by the inverses of the factors of the corresponding diagonal blocks) and then compressed into low-rank approximations. ULV-type factorizations are then computed. A resulting prototype preconditioner is always positive definite. Generalizations to practical hierarchical multilevel preconditioners are given. Systematic analysis of the approximation error, robustness, and effectiveness is shown for both the prototype preconditioner and the multilevel generalization. In particular, we show how local scaling and compression control the approximation accuracy and robustness, and how aggressive compression leads to efficient preconditioners that can significantly reduce the condition number and improve the eigenvalue clustering. A result is also given to show when the multilevel preconditioner preserves positive definiteness. The costs to apply the multilevel preconditioners are about O(N), where N is the matrix size. Numerical tests on several ill-conditioned problems show the effectiveness and robustness even if the compression uses very small numerical ranks. In addition, significant robustness and effectiveness benefits can be observed as compared with a standard rank structured preconditioner based on direct off-diagonal compression.

Central European Journal of Mathematics, 2013

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the ef...

Electronic Transactions on Numerical Analysis

In this paper block approximate inverse preconditioners to solve sparse nonsymmetric linear systems with iterative Krylov subspace methods are studied. The computation of the preconditioners involves consecutive updates of variable rank of an initial and nonsingular matrix A0 and the application of the Sherman-Morrison-Woodbury formula to compute an approximate inverse decomposition of the updated matrices. Therefore, they are generalizations of the preconditioner presented in Bru et al. [SIAM J. Sci. Comput., 25 (2003), pp. 701–715]. The stability of the preconditioners is studied and it is shown that their computation is breakdown-free for H-matrices. To test the performance the results of numerical experiments obtained for a representative set of matrices are presented.

Numerische Mathematik, 2009

Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an H-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based H-matrices, this new approach yields H-LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an H-matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based H-LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.