A survey of direct methods for sparse linear systems

Davis, Timothy A.; Rajamanickam, Sivasankaran; Sid-Lakhdar, Wissam M.

doi:10.1017/s0962492916000076

Cited by 174 publications

(84 citation statements)

References 588 publications

(1,082 reference statements)

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“…Examples, besides the AINV algorithms already mentioned, include FSAI [23] and more recent variants [11]. Methods making use of a recursive partitioning of the matrix is used in direct methods for factorization such as multifrontal methods [10]. In this case the matrix is seen as the adjacency matrix of a graph and the matrix is partitioned using a three-by-three block partition corresponding to a vertex separator of the graph.…”

Section: Alkane Chains and Water Clustersmentioning

confidence: 99%

Localized inverse factorization

Rubensson

Artemov

Kruchinina

et al. 2020

IMA Journal of Numerical Analysis

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We propose a localized divide and conquer algorithm for inverse factorization S −1 = ZZ * of Hermitian positive definite matrices S with localized structure, e.g. exponential decay with respect to some given distance function on the index set of S. The algorithm is a reformulation of recursive inverse factorization [J. Chem. Phys., 128 (2008), 104105] but makes use of localized operations only. At each level of recursion, the problem is cut into two subproblems and their solutions are combined using iterative refinement [Phys. Rev. B, 70 (2004), 193102] to give a solution to the original problem. The two subproblems can be solved in parallel without any communication and, using the localized formulation, the cost of combining their results is proportional to the cut size, defined by the binary partition of the index set. This means that for cut sizes increasing as o(n) with system size n the cost of combining the two subproblems is negligible compared to the overall cost for sufficiently large systems.We also present an alternative derivation of iterative refinement based on a sign matrix formulation, analyze the stability, and propose a parameterless stopping criterion. We present bounds for the initial factorization error and the number of iterations in terms of the condition number of S when the starting guess is given by the solution of the two subproblems in the binary recursion. These bounds are used in theoretical results for the decay properties of the involved matrices.The localization properties of our algorithms are demonstrated for matrices corresponding to nearest neighbor overlap on one-, two-, and three-dimensional lattices as well as basis set overlap matrices generated using the Hartree-Fock and Kohn-Sham density functional theory electronic structure program Ergo [SoftwareX, 7 (2018), 107].

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Section: Alkane Chains and Water Clustersmentioning

confidence: 99%

Localized inverse factorization

Rubensson

Artemov

Kruchinina

et al. 2020

IMA Journal of Numerical Analysis

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“…Apart from this block‐upper triangular decomposition, the algorithm shifts most nonzero elements to the upper triangular part of the Jacobian, leaving the lower triangular part extremely sparse except for the entries of the simultaneous blocks. Via block backward substitution, the system can be solved starting with the first entry in the bottom right corner of the Jacobian .…”

Section: Properties and Solution Of Nonlinear Systemsmentioning

confidence: 99%

Analysis and Decomposition for Improved Convergence of Nonlinear Process Models in Chemical Engineering

Bublitz

Esche

Tolksdorf

et al. 2017

Chemie Ingenieur Technik

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Solving nonlinear equation systems as they occur in process simulation simultaneously, often fails due to ill-conditioned models or bad initialization. To counteract these issues two methods have been implemented in the web-based platform MOSAICmodeling. The Dulmage-Mendelsohn algorithm forms a block diagonal Jacobian matrix. In addition, the bordered block transformation identifies variables that need to be carefully initialized due to their great influence on the system. Both methods are applied on two process models and their convergence and sensitivity towards initialization is analyzed. Motivation and IntroductionIn process simulation tools for chemical engineering applications, such as Aspen Plus, Aspen Hysis, ChemCAD, gPROMS, and Pro/II, the initialization and solution procedure is still mostly governed by the so-called sequential modular approach. This implies the selection of so-called tear streams, which act as entry points to flowsheets and the reconciliation of recycle streams. Starting with these tear streams, unit operations are solved sequentially and the whole flowsheet is solved repeatedly until the properties of the tear stream remain constant between iterations. Within each unit operation, similar approaches are combined with heuristic rules and algorithms on how to solve each independently, e.g., a sequential tray by tray calculation of a distillation column. These methods typically feature an internal modularization regarding property calculations, thermodynamics, etc., all of which are yet again solved with dedicated algorithms. Some of these process simulation tools have been under development for more than 35 years and major advancements have been achieved regarding the size and complexity of flow sheets, the incorporation of thermodynamic models, numerical simulation, and graphical user interfaces [1].Despite this progress, process simulation of flow sheets with many recycle streams still requires some form of manual intervention, implying a certain level of user interaction to steadily improve starting values for tear streams [2]. Such interaction typically requires detailed engineering knowledge. Mere guessing of starting values of temperatures, pressures, compositions, or flows is of course highly undesirable and should be minimized as far as possible. This becomes especially relevant in view of the increasing size of flowsheets with up to several millions of variables.If one chooses an equation-oriented approach instead of the sequential modular approach, this issue becomes even more imminent. The equation-oriented mode attempts a simultaneous solution of the whole process system, usually with some quasi-Newton-type solver. To ensure convergence, a good initial point close to the desired solution is required. In practice, equation-oriented tools, e.g., gPROMS, use sequential-modular techniques as initialization methods before switching over to the equation-oriented mode, Pantelides et al. [3].In general, the nonlinear models underlying any process simulation in chemical engineering ...

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“…Application and analysis of the preconditioner for an ice sheet modeling problem, including numerical comparisons with ILU. 1 The remainder of this paper is organized as follows. Section 2 introduces the deferred-compression technique and provides an error analysis.…”

Section: Introductionmentioning

confidence: 99%

A robust hierarchical solver for ill-conditioned systems with applications to ice sheet modeling

Chen

Cambier

Boman

et al. 2019

Journal of Computational Physics

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A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strong-coupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using a thousand processors.

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A survey of direct methods for sparse linear systems

Cited by 174 publications

References 588 publications

Localized inverse factorization

Localized inverse factorization

Analysis and Decomposition for Improved Convergence of Nonlinear Process Models in Chemical Engineering

A robust hierarchical solver for ill-conditioned systems with applications to ice sheet modeling

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