scholarly journals Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

2012 ◽  
Vol 2012 ◽  
pp. 1-49 ◽  
Author(s):  
Massimiliano Ferronato
Keyword(s):  
Linear Systems ◽  
Specific Problem ◽  
General Purpose ◽  
Sparse Linear Systems ◽  
System Matrix ◽  
Preconditioning Techniques ◽  
Key Factor ◽  
Recent Developments ◽  
Approximate Inverses ◽  
Linear Systems Of Equations

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper.

On the multigrid cycle strategy with approximate inverse smoothing

2014 ◽  
Vol 31 (1) ◽  
pp. 110-122 ◽  
Author(s):  
George A. Gravvanis ◽  
Christos K. Filelis-Papadopoulos
Keyword(s):  
Finite Difference ◽  
Linear Systems ◽  
Multigrid Method ◽  
Multigrid Methods ◽  
Inverse Matrix ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Content Type ◽  
Sparsity Pattern ◽  
Approximate Inverses

Purpose – The purpose of this paper is to propose multigrid methods in conjunction with explicit approximate inverses with various cycles strategies and comparison with the other smoothers. Design/methodology/approach – The main motive for the derivation of the various multigrid schemes lies in the efficiency of the multigrid methods as well as the explicit approximate inverses. The combination of the various multigrid cycles with the explicit approximate inverses as smoothers in conjunction with the dynamic over/under relaxation (DOUR) algorithm results in efficient schemes for solving large sparse linear systems derived from the discretization of partial differential equations (PDE). Findings – Application of the proposed multigrid methods on two-dimensional boundary value problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than the V-cycle multigrid schemes presented in a recent report (Filelis-Papadopoulos and Gravvanis). Research limitations/implications – The limitations of the proposed scheme lie in the fact that the explicit finite difference approximate inverse matrix used as smoother in the multigrid method is a preconditioner for specific sparsity pattern. Further research is carried out in order to derive a generic explicit approximate inverse for any type of sparsity pattern. Originality/value – A novel smoother for the geometric multigrid method is proposed, based on optimized banded approximate inverse matrix preconditioner, the Richardson method in conjunction with the DOUR scheme, for solving large sparse linear systems derived from finite difference discretization of PDEs. Moreover, the applicability and convergence behavior of the proposed scheme is examined based on various cycles and comparative results are given against the damped Jacobi smoother.

scholarly journals Efficient Secure Outsourcing of Large-Scale Sparse Linear Systems of Equations

2018 ◽  
Vol 4 (1) ◽  
pp. 26-39 ◽  
Author(s):  
Sergio Salinas ◽  
Changqing Luo ◽  
Xuhui Chen ◽  
Weixian Liao ◽  
Pan Li
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Systems Of Equations ◽  
Sparse Linear Systems ◽  
Secure Outsourcing ◽  
Linear Systems Of Equations

scholarly journals Preconditioning Techniques for Sparse Linear Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-3
Author(s):  
Massimiliano Ferronato ◽  
Edmond Chow ◽  
Kok-Kwang Phoon
Keyword(s):  
Linear Systems ◽  
Sparse Linear Systems ◽  
Preconditioning Techniques
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