scholarly journals A Class of Generalized Approximate Inverse Solvers for Unsymmetric Linear Systems of Irregular Structure Based on Adaptive Algorithmic Modelling for Solving Complex Computational Problems in Three Space Dimensions

2016 ◽  
Vol 07 (11) ◽  
pp. 1225-1240 ◽  
Author(s):  
Anastasia-Dimitra Lipitakis
Keyword(s):  
Linear Systems ◽  
Approximate Inverse ◽  
Irregular Structure ◽  
Three Space ◽  
Unsymmetric Linear Systems

Axes of Points of Certain Linear Systems of Polarities

1967 ◽  
Vol 19 ◽  
pp. 1072-1077
Author(s):  
Seymour Schuster
Keyword(s):  
Fixed Point ◽  
Linear System ◽  
Linear Systems ◽  
Complex Projective ◽  
Three Space

In this issue honouring Professor Coxeter, I am pleased to present some results of an investigation that was prompted by questions which he himself raised over a decade ago.With respect to a linear system of polarities in complex projective three-space, the polars of a fixed point Q form an axial pencil of planes. The axis of the pencil is called the axis of point Q with respect to the linear system of polarities. Since there are ∞3 axes and ∞4 lines in the space, not every line is an axis. The following discussion answers the questions of how many and which lines are axes with respect to the linear systems of polarities that have a fixed self-polar tetrahedron.

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.

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