Yvan Notay

• Université Libre de Bruxelles

A convergence analysis of two-grid methods based on coarsening by (unsmoothed) aggregation is presented. For diagonally dominant symmetric (M-)matrices, it is shown that the analysis can be conducted locally; that is, the convergence factor can be bounded above by computing separately for each aggregate a parameter, which in some sense measures its...

We consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection-diffusion equations. The system matrix is then an M-matrix with nonnegative row sum, and, further, when the con-vective flow has zero divergence, the column sum is also nonnegative, possibly up to a small correcti...

We consider multigrid methods with V-cycle for symmetric positive definite linear systems. We compare bounds on the convergence factor that are characterized by a constant which is the maximum over all levels of an expression involving only two consecutive levels. More particularly, we consider the classical bound by Hackbusch, a bound by McCormick...

We consider the situation where a basic preconditioner is improved with a coarse grid correction. The latter can be implemented either additively (like in the standard additive Schwarz method) or multiplicatively (like in the balancing preconditioner). In [Numer. Lin. Alg. Appl., 15 (2008), pp. 355–372], Nabben and Vuik compare both variants, and s...

Two-grid methods constitute the building blocks of multigrid methods, which are among the most efficient solution techniques for solving large sparse systems of linear equations. In this paper, an analysis is developed that does not require any symmetry property. Several equivalent expressions are provided that characterize all eigenvalues of the i...

This work tackles the evaluation of a multigrid cycling strategy using inner flexible Krylov subspace iterations. It provides a valuable improvement to the Reitzinger and Sch¨oberl algebraic multigrid method for systems coming from edge element discretizations

The Jacobi-Davidson method is an eigenvalue solver which uses an inner-outer scheme. In the outer iteration one tries to approximate an eigenpair while in the inner iteration a linear system has to be solved, often iteratively, with the ultimate goal to make progress for the outer loop. In this paper we prove a relation between the residual norm of...

We consider the Fourier analysis of multigrid methods (of Galerkin type) for symmetric positive definite and semi-positive definite linear systems arising from the discretization of scalar partial differential equations (PDEs). We relate the so-called smoothing factor to the actual two-grid convergence rate and also to the convergence rate of the V...

We investigate additional condition(s) that confirm that a V-cycle multigrid method is satisfactory (say, optimal) when it is based on a two-grid cycle with satisfactory (say, level-independent) convergence properties. The main tool is McCormick's bound on the convergence factor (SIAM J. Numer. Anal. 1985; 22:634–643), which we showed in previous w...

We consider multigrid (MG) cycles based on the recursive use of a two-grid method, in which the coarse-grid system is solved by µ⩾1 steps of a Krylov subspace iterative method. The approach is further extended by allowing such inner iterations only at the levels of given multiplicity, whereas V-cycle formulation is used at all other levels. For sym...

An algebraic multigrid (AMG) method is presented to solve large systems of lin-ear equations. The coarsening is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the number of variables by a factor slightly less than...

Aggregation-based multigrid with standard piecewise constant like prolongation is investigated. Unknowns are aggregated either by pairs or by quadruplets; in the latter case the grouping may be either linewise or boxwise. A Fourier analysis is developed for a model two- dimensional anisotropic problem. Most of the results are stated for an arbitrar...

A new software code for computing selected eigenvalues and associated eigenvectors of a real symmetric matrix is described. The eigenvalues are either the smallest or those closest to some specified target, which may be in the interior of the spectrum. The underlying algorithm combines the Jacobi-Davidson method with efficient multilevel incomplete...

We consider multigrid methods for symmetric positive definite linear systems. We present a new algebraic convergence analysis of two-grid schemes with inexact solution of the coarse grid system. This analysis allows us to bound the convergence factor of such perturbed two-grid schemes, assuming only a certain bound on the convergence factor for the...

The use of preconditionings obtained by so called modified incomplete factorizations has become quite popular for the PCG solution of regular systems arising from the discretization of elliptic PDE's. Our purpose here is to review their recent extension to the singular case. Because such conditionings may themselves be singular, we first review the...

The Jacobi-Davidson method is a popular technique to compute a few eigenpairs of large sparse matrices. Its introduction, about a decade ago, was motivated by the fact that stan- dard eigensolvers often require an expensive factorization of the matrix to compute interior eigenvalues. Such a factorization may be infeasible for large matrices as aris...

We propose a preconditioning technique that is applicable in a "black box" fashion to linear systems arising from second order scalar elliptic PDEs discretized by finite differences or finite elements with nodal basis functions. This technique is based on an algebraic multilevel scheme with coarsening by aggregation. We introduce a new aggregation...

We study a homogeneous variant of the JacobiñDavidson method for the generalized and polynomial eigenvalue problem. While a homogeneous form of these problems was previously considered for the subspace extraction phase, in this paper this form is also exploited for the subspace expansion phase and the projection present in the correction equation....

We consider inverse iteration-based eigensolvers, which require at each step solv-ing an "inner" linear system. We assume that this linear system is solved by some (preconditioned) Krylov subspace method. In this framework, several approaches are possible, which differ by the linear system to be solved and/or the way the pre-conditioner is used. Th...

We consider algebraic methods of the two-level type for the iterative solution of large sparse linear systems. We assume that a fine/coarse partitioning and an algebraic interpolation have been defined in one way or another, and review different schemes that may be built with these ingredients. This includes algebraic multigrid (AMG) schemes, two-l...

The Davidson method is a popular technique to compute a few of the smallest (or largest) eigenvalues of a large sparse real symmetric matrix. It is effective when the matrix is nearly diagonal, that is, when the matrix of eigenvectors is close to the identity matrix. However, its convergence properties are not yet well understood, and neither is ho...

We consider the computation of the smallest eigenvalue and associated eigenvector of a Hermitian positive definite pencil. Rayleigh quotient iteration (RQI) is known,to converge cubically, and we first analyze how this convergence is affected when the arising linear systems are solved only approximately. We introduce a special measure of the relati...

To precondition large sparse linear systems resulting from the discretization of second-order elliptic partial differential equations, many recent works focus on the so-called algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partitioned in hierarchical form. They have been shown...

To compute the smallest eigenvalues and associated eigenvectors of a real symmetric matrix, we consider the Jacobi–Davidson method with inner preconditioned conjugate gradient iterations for the arising linear systems. We show that the coefficient matrix of these systems is indeed positive definite with the smallest eigenvalue bounded away from zer...

This paper is concerned with the computation of a few of the smallest eigenvalues and associated eigenvectors of a (large sparse) real symmetric matrix A . As preconditioning techniques became increasingly popular and ecient in the context of linear systems solution, many recent works focus on their use within the context of eigenvalue computation...

Stable finite difference approximations of convection–diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M-matrix, which is highly non-symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non-symmetric case an algebraic multil...

We analyze the conjugate gradient method with preconditioning slightly variable from one iteration to the next. To maintain the optimal convergence properties, we consider a variant proposed by Axelsson that performs an explicit orthogonalization of the search directions vectors. For this method, which we refer to as exible conjugate gradient, we d...

We analyze the conjugate gradient method with preconditioning slightly variable from one iteration to the next. To maintain the optimal convergence properties, we consider a variant proposed by Axelsson that performs an explicit orthogonalization of the search directions vectors. For this method, which we refer to as flexible conjugate gradient, we...

Incomplete factorization preconditioners based on recursive red–black orderings have been shown efficient for discrete second order elliptic PDEs with isotropic coefficients. However, they suffer for some weakness in presence of anisotropy or grid stretching. Here we propose to combine these orderings with block incomplete factorization preconditio...

Approximate factorizations are probably the most powerful preconditioners at the present time in the context of iterative solution methods for FE structural analysis. In this contribution we focus on some aspects of the reduction method proposed previously, which allow the use of perturbed approximate factorizations. In particular, we show that it...

Stable finite difference approximations of convection-diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M--matrix, which is highly non symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non symmetric case an algebraic multi...

To efficiently solve second order discrete elliptic PDEs, by Krylov subspace like methods, one needs to use some robust preconditioning techniques. Relaxed incomplete factorizations (RILU) are powerful candidates. Unfortunately, their efficiency critically depends on the choice of the relaxation parameter w whose "optimal" value is not only hard to...

. A new multilevel preconditioner is proposed for the iterative solution of linear systems whose coefficient matrix is a symmetric M--matrix arising from the discretization of a second order elliptic PDE. It is based on a recursive block incomplete factorization of the matrix partitioned in a two-by-two block form, in which the submatrix related to...

This contribution describes how iterative solvers can meet specific requirements of industrial FE analyses, focusing on the case frequently met where the unknowns are subject to linear equality constraints. Standard iterative methods designed to deal with that kind of problem suffer from a significant overhead with respect to the CPU times involved...

This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the two-level type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns, we analyze the ef...

We consider algebraic multilevel preconditioning methods based on the recursive use of a 2 × 2 block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. As is well known, for discrete second-order elliptic PDEs, optimal convergence properties are proved for such basic two-level schemes under mil...

On the occasion of the third centenary of the appointment of Johann Bernoulli at the University of Groningen, a number of linear systems solvers for some Laplace-like equations have been compared during a one-day workshop. CPUtimes of several advanced solvers measured on the same computer (an HP-755 workstation) are presented, which makes it possib...

Considering matrices obtained by the application of a five-point stencil on a 2D rectangular grid, we analyse a preconditioning method introduced by Axelsson and Eijkhout, and by Brand and Heinemann. In this method, one performs a (modified) incomplete factorization with respect to a so-called ‘repeated’ or ‘recursive’ red–black ordering of the unk...

We investigate the ordering and fill-in strategies for approximate factorization preconditioning, focusing on automatic procedures to take care that a user friendly solver should converge equally fast whatever the original numbering of the unknowns. Considering the discrete PDE context, we pay particular attention to anisotropic problems for which...

SUMMARY The preconditioned conjugate gradient algorithm is a well-known and powerful method used to solve large sparse symmetric positive definite linear systems. Such systems are generated by the finite element discretiz- ation in structural analysis but users of finite elements in this context generally still rely on direct methods. It is our pur...

We present a parallel iterative solver for discrete second order elliptic PDEs. It is based on the conjugate gradient algorithm with incomplete factorization preconditioning, using a domain decomposed ordering to allow parallelism in the triangular solves, and resorting to some special recently developed parallelization technique to avoid communica...

We present a parallel iterative solver for discrete second order elliptic PDEs. It is based on the conjugate gradient algorithm with incomplete factorization preconditioning, using a domain decomposed ordering to allow parallelism in the triangular solves, and resorting to some special recently developed parallelization technique to avoid communica...

A new incomplete factorization method is proposed, differing from previous ones by the way in which the diagonal entries of the triangular factors are defined. A comparison is given with the dynamic modified incomplete factorization methods of Axelsson–Barker and Beauwens, and with the relaxed incomplete Cholesky method of Axelsson and Lindskog. Th...

Preconditioning by approximate factorizations is widely used in iterative methods for solving linear systems such as those arising from the finite element formulation of many engineering problems. The influence of the ordering of the unknowns on their convergence behaviour has been the subject of recent investigations because of its particular rele...

We investigate here rounding error effects on the convergence rate of the conjugate gradients. More precisely, we analyse on both theoretical and experimental basis how finite precision arithmetic affects known bounds on iteration numbers when the spectrum of the system matrix presents small or large isolated eigenvalues.

An upper triangular matrix U = (uij) is said to be ‘S/P consistently ordered’ if for all i, j, i ≠ j, the existe nce of an index k<i, j with uki≠0 and ukj≠0 entails the existence of an index l>i, j such that uil≠0 and ujl≠0. The conditioning analysis of Stieltjes matrices by (unpertubed) modified incomplete factorizations has recently been improved...

We analyse the robustness of the (perturbed) modified incomplete factorization method. In the discrete PDE context, we show that, under very general assumptions, one may derive bounds which allow to prove convergence rates similar to those observed for model problems. Thus, contrarily to presently available results, we do not only give an idea of t...

The paper is devoted to the conditioning analysis of modified block incomplete factorizations of a given Stieltjes matrix. We obtain new results, improve other theories, and compare all existing upper bounds through numerical experiments. Applied to discrete elliptic PDEs, our results show that an O(h−1) spectral bound can be achieved for a large c...

We improve the conditioning analysis of modified block incomplete factorizations of Stieltjes matrices. Letting N denote the number of diagonal blocks, our results show that the spectral condition number is bounded by N for a large class of two dimensional PDEs.

Recent works have shown that, whenA is a Stieltjes matrix, its so-called modified incomplete factorizations provide effective preconditioning matrices for solvingAx=b by polynomially accelerated iterative methods. We extend here these results to the singular case with the conclusion that the latter techniques are able to solve singular systems at t...

A subproper splitting of a matrix A is a decomposition A = B − C such that the kernel of A includes that of B while the range of B includes that of A. Our purpose in the present work is to extend the convergence analysis of polynomial acceleration to the case of iterative schemes associated with subproper splittings, in the case of Hermitian matric...

This report presents the ITSOL package for the iterative solution of symmetric positive definite systems by PCG with incomplete factorizations as preconditioner. ITSOL is a tool devised to quickly implement and test standard or new preconditioning techniques, without having to rewrite a specific program in each case. Therefore, it does not provide...

We analyze the conjugate gradient method with preconditioning slightly variable from one iteration to the next. To maintain the optimal convergence properties, we consider a variant proposed by Axelsson that performs an explicit orthogonalization of the search directions vectors. For this method, which we refer to as exible conjugate gradient, we d...

This manual gives an introduction to the use of JADAMILU, a set of Fortran 77 routines to compute selected eigenvalues and associated eigenvectors of large scale real symmetric or complex Hermitian matrices. Generalized eigenvalue problems with positive definite mass matrix can also be solved. The eigenvalues sought can either be the smallest ones...