A Max-Plus Approach to Incomplete Cholesky Factorization Preconditioners

Hook, James; Scott, J. A.; Tisseur, Françoise; Hogg, JD

doi:10.1137/16m1107735

Cited by 6 publications

(5 citation statements)

References 32 publications

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“…Finally, we observe that there is a lack of iterative methods and preconditioners that can be used to extend the size of LSE problems that can be solved. We have shown that using an incomplete factorization within a block factorization of an augmented system can be effective, but most current incomplete factorizations that result in efficient preconditioners are serial in nature and not able to tackle extremely large problems (but see [1,25] for novel approaches that are designed to exploit parallelism). Addressing the lack of iterative approaches is a challenging subject for future work.…”

Section: Discussionmentioning

confidence: 99%

Solving large linear least squares problems with linear equality constraints

Scott

Tůma

2022

Bit Numer Math

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We consider the problem of solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. While some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. We propose modifications with an emphasis on requiring that the constraints be satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null-space basis matrix for a “wide” matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive for solving a sequence of related problems. Numerical experiments on problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches.

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Section: Discussionmentioning

confidence: 99%

Solving large linear least squares problems with linear equality constraints

Scott

Tůma

2022

Bit Numer Math

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“…Finally, we observe that there is a lack of iterative methods and preconditioners that can be used to extend the size of LSE problems that can be solved. We have shown that using an incomplete factorization within a block factorization of an augmented system can be effective but most current incomplete factorizations that result in efficient preconditioners are serial in nature and not able to tackle extremely large problems (but see [20] for a novel approach that is straightforward to parallelise). Addressing the lack of iterative approaches is a challenging subject for future work.…”

Section: Discussionmentioning

confidence: 99%

Solving large linear least squares problems with linear equality constraints

Scott¹,

Tůma²

2021

Preprint

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We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. To address this, we propose modifications and new ideas, with an emphasis on requiring the constraints are satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null space basis matrix for a "wide" matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive when solving a sequence of related problems. Numerical experiments using problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches.

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“…We observe that the counts are relatively insensitive to p but, as p increases, it The first is the Matlab variant ichol with the global diagonal shift set to 0.1 and default values for other parameters and the second is the Matlab interface to the incomplete Cholesky (IC) factorization preconditioner HSL_MI28 [39] from the HSL library [20] using the default parameter settings. IC preconditioners are widely used but their construction is often serial, potentially limiting their suitability for very large problems (see [19] for an IC preconditioner that can be parallelised). In terms of iteration counts, the Nyström-Schur and the HSL_MI28 preconditioners are clearly superior to the simple ichol preconditioner, with neither consistently offering the best performance.…”

Section: 3mentioning

confidence: 99%

Two-Level Nyström--Schur Preconditioner for Sparse Symmetric Positive Definite Matrices

Daas¹,

Rees²,

Scott³

2021

SIAM J. Sci. Comput.

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Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners.In particular, we propose a new and efficient approach that employs Nyström's method for computing low rank approximations to develop robust algebraic two-level preconditioners. Construction of the new preconditioners involves iteratively solving a smaller but denser symmetric positive definite Schur complement system with multiple right-hand sides. Numerical experiments on problems coming from a range of application areas demonstrate that this inner system can be solved cheaply using block conjugate gradients and that using a large convergence tolerance to limit the cost does not adversely affect the quality of the resulting Nyström-Schur two-level preconditioner.

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A Max-Plus Approach to Incomplete Cholesky Factorization Preconditioners

Cited by 6 publications

References 32 publications

Solving large linear least squares problems with linear equality constraints

Solving large linear least squares problems with linear equality constraints

Solving large linear least squares problems with linear equality constraints

Two-Level Nyström--Schur Preconditioner for Sparse Symmetric Positive Definite Matrices

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