Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

The choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for large‐size sparse eigenproblems. Although incomplete factorizations with partial fill‐in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large‐size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi–Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid‐dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. Copyright © 2011 John Wiley & Sons, Ltd.     

Schur complement‐based domain decomposition preconditioners with low‐rank corrections

This paper introduces a robust preconditioner for general sparse matrices based on low‐rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low‐rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low‐rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Preconditioner updates for solving sequences of linear systems in matrix‐free environment

We present two new ways of preconditioning sequences of nonsymmetric linear systems in the special case where the implementation is matrix free. Both approaches are fully algebraic, they are based on the general updates of incomplete LU decompositions recently introduced in (SIAM J. Sci. Comput. 2007; 29 (5):1918–1941), and they may be directly embedded into nonlinear algebraic solvers. The first of the approaches uses a new model of partial matrix estimation to compute the updates. The second approach exploits separability of function components to apply the updated factorized preconditioner via function evaluations with the discretized operator. Experiments with matrix‐free implementations of test problems show that both new techniques offer useful, robust and black‐box solution strategies. In addition, they show that the new techniques are often more efficient in matrix‐free environment than either recomputing the preconditioner from scratch for every linear system of the sequence or than freezing the preconditioner throughout the whole sequence. Copyright © 2010 John Wiley & Sons, Ltd.     

Multicolor low‐rank preconditioner for general sparse linear systems

This article 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 adjacency graph of the subdomains, resulting in a two‐level block diagonal structure. A full binary tree structure $\mathrm{??}$ is then built to facilitate the construction of the preconditioner. A key property that is exploited is the observation that the difference between the inverse of the original matrix and that of its block diagonal approximation is often well approximated by a low‐rank matrix. This property and the block diagonal structure of the reordered matrix lead to a multicolor low‐rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom‐up traversal of the tree $\mathrm{??}$. 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 nonleaf 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.     

An algebraic multifrontal preconditioner that exploits the low‐rank property

We present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factorization is used instead. This latter exploits the presence of low numerical rank in some off‐diagonal blocks of the frontal matrices. An algebraic procedure is presented that allows to identify the hierarchy of the off‐diagonal blocks with low numerical rank based on the sparsity of the system matrix. This procedure is motivated by a model problem analysis, yet numerical experiments show that it is successful beyond the model problem scope. Further aspects relevant for the algebraic structured preconditioner are discussed and illustrated with numerical experiments. The preconditioner is also compared with other solvers, including the corresponding direct solver. Copyright © 2015 John Wiley & Sons, Ltd.     

Factorized-Sparse-Approximate-Inverse Preconditionings of Linear Systems with Unsymmetric Matrices

The paper considers the application of factorized-sparse-approximate-inverse (FSAI) preconditionings to linear algebraic systems with nonsingular unsymmetric coefficient matrices. Special attention is paid to methods for optimizing the sparsity pattern of such preconditioners. It is numerically demonstrated that the efficiency of FSAI preconditionings can be substantially improved by applying the so-called postfiltering and prefiltering in order to sparsify preconditioning matrices. Bibliography: 20 titles.     

A Comparative Study on Dynamic and Static Sparsity Patterns in Parallel Sparse Approximate Inverse Preconditioning

Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing an SAI preconditioner. Both dynamic and static sparsity pattern selection approaches have been proposed by researchers. Through a few numerical experiments, we conduct a comparable study on the properties and performance of the SAI preconditioners using the different sparsity patterns for solving some sparse linear systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Banded target matrices and recursive FSAI for parallel preconditioning

In this paper we propose a parallel preconditioner for the CG solver based on successive applications of the FSAI preconditioner. We first compute an FSAI factor G _out for coefficient matrix A, and then another FSAI preconditioner is computed for either the preconditioned matrix $S = G_{\rm out} A G_{\rm out}^T$ or a sparse approximation of S. This process can be iterated to obtain a sequence of triangular factors whose product forms the final preconditioner. Numerical results onto large SPD matrices arising from geomechanical models account for the efficiency of the proposed preconditioner which provides a reduction of the iteration number and of the CPU time of the iterative phase with respect to the original FSAI preconditioner. The proposed preconditioner reveals particularly efficient for accelerating an iterative procedure to find the smallest eigenvalues of SPD matrices, where the increased setup cost of the RFSAI preconditioner does not affect the overall performance, being a small percentage of the total CPU time.     

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.     

A supernodal block factorized sparse approximate inverse for non-symmetric linear systems

The concept of supernodes, originally developed to accelerate direct solution methods for linear systems, is generalized to block factorized sparse approximate inverse (Block FSAI) preconditioning of non-symmetric linear systems. It is shown that aggregating the unknowns in clusters that are processed together is particularly useful both to reduce the cost for the preconditioner setup and accelerate the convergence of the iterative solver. A set of numerical experiments performed on matrices arising from the meshfree discretization of 2D and 3D potential problems, where a very large number of nodal contacts is usually found, shows that the supernodal Block FSAI preconditioner outperforms the native algorithm and exhibits a much more stable behavior with respect to the variation of the user-specified parameters.     

Robust multigrid preconditioners for the high‐contrast biharmonic plate equation

We study the high‐contrast biharmonic plate equation with Hsieh–Clough–Tocher discretization. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 2008; 11 :319–331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretization, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high‐contrast elliptic PDEs of order 2k, k>2. Moreover, we prove a fundamental qualitative property of the solution to the high‐contrast biharmonic plate equation. Namely, the solution over the highly bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction. Copyright © 2010 John Wiley & Sons, Ltd.     

A robust algebraic multilevel preconditioner for non‐symmetric M‐matrices

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 multilevel preconditioning method formerly proposed for pure diffusion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis that applies to constant coefficient problems with periodic boundary conditions whenever using an ‘idealized’ version of the two‐level preconditioner. Within this setting, it is proved that any eigenvalue λ of the preconditioned system satisfies for some real constant c such that . This result holds independently of the grid size and uniformly with respect to the ratio between convection and diffusion. Extensive numerical experiments are conducted to assess the convergence of practical two‐ and multi‐level schemes. These experiments, which include problems with highly variable and rotating convective flow, indicate that the convergence is grid independent. It deteriorates moderately as the convection becomes increasingly dominating, but the convergence factor remains uniformly bounded. This conclusion is supported for both uniform and some non‐uniform (stretched) grids. Copyright © 2000 John Wiley & Sons, Ltd.     

Inexact constraint preconditioners for linear systems arising in interior point methods

Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint preconditioners have shown much promise in this context. However, there are situations in which an unfavorable sparsity pattern of Jacobian matrix may adversely affect the preconditioner and make its inverse representation unacceptably dense hence too expensive to be used in practice. A remedy to such situations is proposed in this paper. An approximate constraint preconditioner is considered in which sparse approximation of the Jacobian is used instead of the complete matrix. Spectral analysis of the preconditioned matrix is performed and bounds on its non-unit eigenvalues are provided. Preliminary computational results are encouraging.     

Extended Kaczmarz‐like methods with oblique projections

In this paper we describe two “sparse preconditioning” techniques for accelerating the convergence of Kaczmarzlike algorithms. The first method, uses projections with respect to the “energy scalar product” generated by an appropriate symmetric and positive definite matrix. The second one starts from some recent results of Y. Censor and T. Elfving on “sparsity pattern oriented” (SPO) oblique projections and uses an “algebraic multigrid interpolationlike” construction of the (SPO) family. Numerical experiments are described on a system comming from a bioelectric field simulation problem.     

Block filtering decomposition

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so‐called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low‐frequency modes on convergence and so decrease or eliminate the plateau that is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process. We show the efficiency of our preconditioner through a set of numerical experiments on symmetric and nonsymmetric matrices. Copyright © 2014 John Wiley & Sons, Ltd.     

A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems

We consider the numerical solution of linear systems arising from the discretization of the electric field integral equation (EFIE). For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. The electromagnetic scattering problem is here solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM). The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix. In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations in the solution of the resulting system of equations.     

Using groups in the splitting preconditioner computation for interior point methods

Interior point methods usually rely on iterative methods to solve the linear systems of large scale problems. The paper proposes a hybrid strategy using groups for the preconditioning of these iterative methods. The objective is to solve large scale linear programming problems more efficiently by a faster and robust computation of the preconditioner. In these problems, the coefficient matrix of the linear system becomes ill conditioned during the interior point iterations, causing numerical difficulties to find a solution, mainly with iterative methods. Therefore, the use of preconditioners is a mandatory requirement to achieve successful results. The paper proposes the use of a new columns ordering for the splitting preconditioner computation, exploring the sparsity of the original matrix and the concepts of groups. This new preconditioner is designed specially for the final interior point iterations; a hybrid approach with the controlled Cholesky factorization preconditioner is adopted. Case studies show that the proposed methodology reduces the computational times with the same quality of solutions when compared to previous reference approaches. Furthermore, the benefits are obtained while preserving the sparse structure of the systems. These results highlight the suitability of the proposed approach for large scale problems.     

Preconditioning for accurate solutions of ill‐conditioned linear systems

This article develops the preconditioning technique as a method to address the accuracy issue caused by ill‐conditioning. Given a preconditioner M for an ill‐conditioned linear system Ax=b, we show that, if the inverse of the preconditioner M^?1 can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse‐equivalent accuracy is introduced to describe the high accuracy that is achieved and an error analysis will be presented. Numerical examples are presented to illustrate the error analysis and the performance of the methods.     

Preconditioning of elliptic problems by approximation in the transform domain

Preconditioned conjugate gradient method is applied for solving linear systemsAx=b where the matrixA is the discretization matrix of second-order elliptic operators. In this paper, we consider the construction of the trnasform based preconditioner from the viewpoint of image compression. Given a smooth image, a major portion of the energy is concentrated in the low frequency regions after image transformation. We can view the matrixA as an image and construct the transform based preconditioner by using the low frequency components of the transformed matrix. It is our hope that the smooth coefficients of the given elliptic operator can be approximated well by the low-rank matrix. Numerical results are reported to show the effectiveness of the preconditioning strategy. Some theoretical results about the properties of our proposed preconditioners and the condition number of the preconditioned matrices are discussed.     

BCCB preconditioners for systems of BVM‐based numerical integrators

Boundary value methods (BVMs) for ordinary differential equations require the solution of non‐symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is A_k1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.