An out-of-core sparse Cholesky solver

Reid, J. K.; Scott, J. A.

doi:10.1145/1499096.1499098

Cited by 33 publications

(24 citation statements)

References 22 publications

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“…This has become well established as a means of improving the factorization speed at the expense of the number of entries in the factor L and the operation counts during the factorization and subsequent solve phase. During the analyze phase, a child node in the tree is merged with its parent if both parent and child have fewer than a prescribed number nemin of variables that are eliminated, or if merging parent and child generates no additional nonzeros in L. The value of the parameter nemin determines the level of node amalgamation, with a value in the range of 8 to 32 typically recommended as providing a good balance between sparsity and efficiency in the factorize and solve phases (see [25,44]). In our experiments, we set nemin equal to 32.…”

Section: Sparse Direct Solversmentioning

confidence: 99%

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: Sparse Direct Solversmentioning

confidence: 99%

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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“…In recent years, this has been extended to super-nodal sparse Cholesky factorisation [7] and graphs [25] for parallel computation while recent articles have detailed GPU implementations [45,53]. Out-of-core methods adapted to very large problems have also been proposed [44]. The performance of sparse matrix decomposition methods are highly dependent on the fill-ins and therefore rely heavily on fill-reducing ordering algorithms.…”

Section: Introductionmentioning

confidence: 99%

Efficient Cholesky Factor Recovery for Column Reordering in Simultaneous Localisation and Mapping

Touchette

Gueaieb

Lanteigne

2016

J Intell Robot Syst

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Simultaneous Localisation And Mapping problems are inherently dynamic and the structure of the graph representing them changes significantly over time. To obtain the least square solution of such systems efficiently, it is desired to maintain a good column ordering such that fill-ins are reduced. This comes at a cost since general ordering changes require the complete re-computation of the Cholesky factor. While some methods have obtained good results with reordering at loop closing, the changes are not guaranteed to be limited to the scope of the loop, leading to suboptimal performance. In this article, it is shown that the Cholesky factorisation of an updated matrix can be efficiently recovered from the previous factorisation if the permutations are localised. This is experimentally demonstrated on 2D SLAM datasets. A method is then provided to identify when such recovery is advantageous over the complete re-computation of the Cholesky factor. Furthermore, a hybrid algorithm combining factorisation recovery and re-computation of the Cholesky factor is proposed for dynamically evolving problems and tested on SLAM datasets. Steps where reordering occurs can be executed up to 67% faster with the proposed method.

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“…We believe this is the first in depth study of the substitution phases in a parallel and out-of core environment. Our work differs and extends the work of [10,11,12,9] because firstly we consider a parallel out-of-core context, and secondly we focus on the performance of the solve phase.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the solution phase of a parallel multifrontal approach

et al. 2010

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We study the forward and backward substitution phases of a sparse multifrontal factorization. These phases are often neglected in papers on sparse direct factorization but, in many applications, they can be the bottleneck so it is crucial to implement them efficiently. In this work, we assume that the factors have been written on disk during the factorization phase, and we discuss the design of an efficient solution phase.We will look at the issues involved when we are solving the sparse systems on parallel computers and will consider in particular their solution in a limited memory environment when out-of-core working is required. Two different approaches are presented to read data from the disk, with a discussion on the advantages and the drawbacks of each.We present some experiments on realistic test problems using an out-of-core version of a sparse multifrontal code called MUMPS (MUltifrontal Massively Parallel Solver).

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An out-of-core sparse Cholesky solver

Cited by 33 publications

References 22 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

Efficient Cholesky Factor Recovery for Column Reordering in Simultaneous Localisation and Mapping

Analysis of the solution phase of a parallel multifrontal approach

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