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.     

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 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.     

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.     

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.     

Fast algorithms for hierarchically semiseparable matrices

Semiseparable matrices and many other rank‐structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low‐rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast‐structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

An Incomplete Cholesky Factorization for Dense Symmetric Positive Definite Matrices

In this paper, we study the use of an incomplete Cholesky factorization (ICF) as a preconditioner for solving dense symmetric positive definite linear systems. This method is suitable for situations where matrices cannot be explicitly stored but each column can be easily computed. Analysis and implementation of this preconditioner are discussed. We test the proposed ICF on randomly generated systems and large matrices from two practical applications: semidefinite programming and support vector machines. Numerical comparison with the diagonal preconditioner is also presented.     

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.     

A robust optimal preconditioner for the mixed finite element discretization of elliptic optimal control problems

In this paper, we consider the efficient solving of the resulting algebraic system for elliptic optimal control problems with mixed finite element discretization. We propose a block‐diagonal preconditioner for the symmetric and indefinite algebraic system solved with minimum residual method, which is proved to be robust and optimal with respect to both the mesh size and the regularization parameter. The block‐diagonal preconditioner is constructed based on an isomorphism between appropriately chosen solution space and its dual for a general control problem with both state and gradient state observations in the objective functional. Numerical experiments confirm the efficiency of our proposed preconditioner.     

Preconditioner based on the Sherman-Morrison formula for regularized least squares problems

We propose a sparse approximate inverse preconditioner based on the Sherman-Morrison formula for Tikhonov regularized least square problems. Theoretical analysis shows that, the factorization method can take the advantage of the symmetric property of the coefficient matrix and be implemented cheaply. Combined with dropping rules, the incomplete factorization leads to a preconditioner for Krylov iterative methods to solve regularized least squares problems. Numerical experiments show that our preconditioner is competitive compared to existing methods, especially for ill-conditioned and rank deficient least squares problems.     

Wilkinson's inertia‐revealing factorization and its application to sparse matrices

We propose a new inertia‐revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof‐of‐concept implementation and present experimental results, studying the method's numerical stability and performance.     

Bilus: A block version of ilus factorization

ILUS factorization has many desirable properties such as its amenability to the skyline format, the ease with which stability may be monitored, and the possibility of constructing a preconditioner with symmetric structure. In this paper we introduce a new preconditioning technique for general sparse linear systems based on the ILUS factorization strategy. The resulting preconditioner has the same properties as the ILUS preconditioner. Some theoretical properties of the new preconditioner are discussed and numerical experiments on test matrices from the Harwell-Boeing collection are tested. Our results indicate that the new preconditioner is cheaper to construct than the ILUS preconditioner.     

An Algebraic Multigrid-Based Physical Factorization Preconditioner for the Multi-Group Radiation Diffusion Equations in Three Dimensions

The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume discretization of the three-dimensional multi-group radiation diffusion equations. The key idea is to take advantage of a particular kind of block factorization of the resulting system matrix and approximate the left-hand block matrix selectively spurred by parallel processing considerations. The spectral property of the preconditioned matrix is then analyzed. The practical strategy is considered sequentially and in parallel. Finally, numerical results illustrate the numerical robustness, computational efficiency and parallel strong and weak scalabilities over the real-world structured and unstructured coupled problems, showing its competitiveness with many existing block preconditioners.     

A method for verifying the accuracy of numerical solutions of symmetric saddle point linear systems

A fast numerical verification method is proposed for evaluating the accuracy of numerical solutions for symmetric saddle point linear systems whose diagonal blocks of the coefficient matrix are semidefinite matrices. The method is based on results of an algebraic analysis of a block diagonal preconditioning. Some numerical experiments are present to illustrate the usefulness of the method.     

Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D

<正>In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems.In earlier works,the local Schur complements were computed exactly using a sparse direct solver.The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling huge problems.In this work we investigate the use of sparse approximation of the dense local Schur complements.These approximations are computed using a partial incomplete LU factorization.Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems;preliminary experiments on linear systems arising from structural mechanics are also reported.     

New fast divide‐and‐conquer algorithms for the symmetric tridiagonal eigenvalue problem

In this paper, two accelerated divide‐and‐conquer (ADC) algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N²r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices, which are Cauchy‐like matrices and are off‐diagonally low‐rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low‐rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off‐diagonal rank. Numerous experiments have been carried out to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than six times faster for some large matrices with few deflations. Copyright © 2016 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.     

A class of preconditioners based on matrix splitting for nonsymmetric linear systems

Given a general matrix splitting A=M-N where M is nonsingular, a new factorization scheme in terms of factorized and splitting matrices is given using the Sherman-Morrison formula. Theoretical analysis shows that the factorization can give an LDU decomposition of A under some special choices. We propose and implement a class of preconditioners based on this factorization combining with dropping rules. A number of numerical experiments from discrete convection diffusion equation and some practical problems show that the new preconditioner is efficient, and is comparable to existing preconditioners in term of storage requirement and computational cost.     

Fast structured LU factorization for nonsymmetric matrices

In this paper, an approximate LU factorization algorithm is developed for nonsymmetric matrices based on the hierarchically semiseparable matrix techniques. It utilizes a technique involving orthogonal transformations and approximations to avoid the explicit computation of the Schur complement in each factorization step. A modified compression method is further developed for the case when some diagonal blocks have small singular values. The complexity of the methods proposed in this paper is analyzed and shown to be $O(N^2k)$ , where $N$ is the dimension of matrix and $k$ is the maximum off-diagonal (numerical) rank. Depending on the accuracy and efficiency requirements in the approximation, this factorization can be used either as a direct solver or a preconditioner. Numerical results from applications are included to show the efficiency of our method.     

Constraint preconditioning for nonsymmetric indefinite linear systems

This paper introduces and presents theoretical analyses of constraint preconditioning via a Schilders'‐like factorization for nonsymmetric saddle‐point problems. We extend the Schilders' factorization of a constraint preconditioner to a nonsymmetric matrix by using a different factorization. The eigenvalue and eigenvector distributions of the preconditioned matrix are determined. The choices of the parameter matrices in the extended Schilders' factorization and the implementation of the preconditioning step are discussed. An upper bound on the degree of the minimum polynomial for the preconditioned matrix and the dimension of the corresponding Krylov subspace are determined, as well as the convergence behavior of a Krylov subspace method such as GMRES. Numerical experiments are presented. Copyright © 2009 John Wiley & Sons, Ltd.