Sym-Ildl

Greif, Chen; He, Shiwen; Liu, Paul

doi:10.1145/3054948

Cited by 9 publications

(7 citation statements)

References 25 publications

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“…More recently, a number of authors have considered incomplete LDL T factorizations of symmetric quasi-definite matrices [39], saddle-point systems [52], and general indefinite systems [22,53]. The use of a limited-memory IC factorization combined with LSMR to solve (1.1) is considered in section 5, and in section 6 we explore using incomplete LDL T factorizations to solve (1.3).…”

Section: C321mentioning

confidence: 99%

“…An alternative would be to use the symmetric solver MINRES [40]. This requires a positive definite preconditioner, and so we could employ the method presented by Gill et al [16] to modify D β and L β (note that this approach has been used recently by Greif, He, and Liu [22]). …”

Section: Regularized Augmented Systemmentioning

confidence: 99%

“…Chow and Saad [8] considered the class of incomplete LU preconditioners for solving indefinite problems, and later Li and Saad [33] integrated pivoting procedures with scaling and reordering. Building on this, Greif, He, and Liu [22] recently developed an incomplete factorization package called SYM-ILDL for general sparse symmetric indefinite matrices. Here the system matrix may be any sparse indefinite matrix; no advantage is made of the specific block structure of (1.3).…”

Section: Incomplete Factorization Of the Normal Matrix Cmentioning

confidence: 99%

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On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

Scott¹

2017

SIAM J. Sci. Comput.

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Abstract. By examining the performance of modern parallel sparse direct solvers and exploiting our knowledge of the algorithms behind them, we perform numerical experiments to study how they can be used to efficiently solve rank-deficient sparse linear least-squares problems arising from practical applications. The Cholesky factorization of the normal equations breaks down when the least-squares problem is rank-deficient, while applying a symmetric indefinite solver to the augmented system can give an unacceptable level of fill in the factors. To try to resolve these difficulties, we consider a regularization procedure that modifies the diagonal of the unregularized matrix. This leads to matrices that are easier to factorize. We consider both the regularized normal equations and the regularized augmented system. We employ the computed factors of the regularized systems as preconditioners with an iterative solver to obtain the solution of the original (unregularized) problem. Furthermore, we look at using limited-memory incomplete Cholesky-based factorizations and how these can offer the potential to solve very large problems.Key words. least-squares problems, normal equations, augmented system, sparse matrices, direct methods, iterative methods, Cholesky factorizations, preconditioning, regularization AMS subject classifications. 65F05, 65F50 DOI. 10.1137/16M10653801. Introduction. In recent years, a number of methods have been proposed for preconditioning sparse linear least-squares problems; a brief overview with a comprehensive list of references is included in the introduction to the paper of Bru et al. [4]. The recent study of Gould and Scott [20,21] reviewed many of these methods (specifically those for which software has been made available) and then tested and compared their performance using a range of examples coming from practical applications. One of the outcomes of that study was some insight into which least-squares problems in the widely used sparse matrix collections CUTEst [19] and University of Florida [10] currently pose a real challenge for direct methods and/or iterative solvers. In particular, the study found that most of the available software packages were not reliable or efficient for rank-deficient least-squares problems (at least not when run with the recommended settings for the input parameters that were employed in the study). In this paper, we look further at such problems and focus on the effectiveness of both sparse direct solvers and iterative methods with incomplete factorization preconditioners. A key theme is the use of regularization (see, for example, [15,48,49]). We propose computing a factorization (either complete or incomplete) of a regularized problem and then using this as a preconditioner for an iterative solver to recover the solution of the original (unregularized) problem.The problem we are interested in is

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Section: C321mentioning

confidence: 99%

Section: Regularized Augmented Systemmentioning

confidence: 99%

Section: Incomplete Factorization Of the Normal Matrix Cmentioning

confidence: 99%

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On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

Scott¹

2017

SIAM J. Sci. Comput.

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“…Hence, even though it does not exploit symmetry, we use it to show the robustness of the proposed scheme in Section 4.2.1. For the latter, on the other hand, we use symm-ildl which is an external package [16] that has an interface for MATLAB and is robust. Hereafter, we refer to this preconditioner as ILDL T .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…However, this preconditioner is, in general, not practical for large problems due to fill-in and large storage requirements. In [16], therefore, an incomplete LDL T factorization based preconditioner for MINRES is proposed. Another suggestion for a preconditioner of MINRES, proposed in [28], is the positive definite absolute value of A, defined as |A| := V |Λ|V T in which A = V ΛV T is the spectral decomposition of A.…”

Section: Introduction Symmetric Indefinite Linear Systemsmentioning

confidence: 99%

A Robust Iterative Scheme for Symmetric Indefinite Systems

Manguoğlu¹,

Mehrmann²

2019

SIAM J. Sci. Comput.

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We propose a two-level nested preconditioned iterative scheme for solving sparse linear systems of equations in which the coefficient matrix is symmetric and indefinite with relatively small number of negative eigenvalues. The proposed scheme consists of an outer Minimum Residual (MINRES) iteration, preconditioned by an inner Conjugate Gradient (CG) iteration in which CG can be further preconditioned. The robustness of the proposed scheme is illustrated by solving indefinite linear systems that arise in the solution of quadratic eigenvalue problems in the context of model reduction methods for finite element models of disk brakes as well as on other problems that arise in a variety of applications.Key word. symmetric indefinite systems, Krylov subspace method, sparse linear systems, deflation, preconditioned minimum residual method, preconditioned conjugate gradient method MSC 2000: 65F10, 65F15, 65F50 * Submitted DATE.Funding: The first author would like to thank Alexander von Humboldt Foundation for the support of a research stay at TU Berlin. The second author thanks Einstein Foundation Berlin for support within the project: Algorithmic Linear Algebra: High Performance, Numerical Stability, and Error Resiliency.

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Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

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When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.

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Sym-Ildl

Cited by 9 publications

References 25 publications

On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

A Robust Iterative Scheme for Symmetric Indefinite Systems

Preconditioners for Krylov subspace methods: An overview

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