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Article
Publication date: 1 April 2006

Konstantinos M. Giannoutakis and George A. Gravvanis

To propose novel parallel/distributed normalized explicit finite element (FE) approximate inverse preconditioning for solving sparse FE linear systems.

218

Abstract

Purpose

To propose novel parallel/distributed normalized explicit finite element (FE) approximate inverse preconditioning for solving sparse FE linear systems.

Design/methodology/approach

The design of suitable methods was the main objective for which several families of the normalized approximate inverse, based on sparse normalized approximate factorization, are produced. The main motive for the derivation of the new normalized approximate inverse FE matrix algorithmic techniques is that they can be efficiently used in conjunction with normalized explicit preconditioned conjugate gradient (NEPCG) – type schemes on parallel and distributed systems. Theoretical estimates on the rate of convergence and computational complexity of the NEPCG method are also derived.

Findings

Application of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results for uniprocessor systems along with speed‐ups and efficiency for multicomputer systems are given. These results tend to become optimum, which are in qualitative agreement with the theoretical results presented for uniprocessor and distributed memory systems, using message passing interface (MPI) communication library.

Research limitations/implications

Further parallel algorithmic techniques will be investigated in order to improve the speed‐ups and the computational complexity of the parallel normalized explicit approximate inverse preconditioning.

Originality/value

The proposed parallel/distributed normalized explicit approximate inverse preconditioning, using approximate factorization and approximate inverse algorithms, is an efficient computational method that is valuable for computer scientists and for scientists and engineers in engineering computations.

Details

Engineering Computations, vol. 23 no. 3
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 7 March 2016

Christos K. Filelis-Papadopoulos and George A. Gravvanis

– The purpose of this paper is to propose novel factored approximate sparse inverse schemes and multi-level methods for the solution of large sparse linear systems.

146

Abstract

Purpose

The purpose of this paper is to propose novel factored approximate sparse inverse schemes and multi-level methods for the solution of large sparse linear systems.

Design/methodology/approach

The main motive for the derivation of the various generic preconditioning schemes lies to the efficiency and effectiveness of factored preconditioning schemes in conjunction with Krylov subspace iterative methods as well as multi-level techniques for solving various model problems. Factored approximate inverses, namely, Generic Factored Approximate Sparse Inverse, require less fill-in and are computed faster due to the reduced number of nonzero elements. A modified column wise approach, namely, Modified Generic Factored Approximate Sparse Inverse, is also proposed to further enhance performance. The multi-level approximate inverse scheme, namely, Multi-level Algebraic Recursive Generic Approximate Inverse Solver, utilizes a multi-level hierarchy formed using Block Independent Set reordering scheme and an approximation of the Schur complement that results in the solution of reduced order linear systems thus enhancing performance and convergence behavior. Moreover, a theoretical estimate for the quality of the multi-level approximate inverse is also provided.

Findings

Application of the proposed schemes to various model problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than results by other researchers for some of the model problems.

Research limitations/implications

Further enhancements are investigated for the proposed factored approximate inverse schemes as well as the multi-level techniques to improve quality of the schemes. Furthermore, the proposed schemes rely on the definition of multiple parameters that for some problems require thorough testing, thus adaptive techniques to define the values of the various parameters are currently under research. Moreover, parallel schemes will be investigated.

Originality/value

The proposed approximate inverse preconditioning schemes as well as multi-level schemes are efficient computational methods that are valuable for computer scientists and for scientists and engineers in engineering computations.

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Article
Publication date: 1 March 2000

George A. Gravvanis

A new class of explicit preconditioning methods based on the concept of sparse approximate factorization procedures and inverse matrix techniques is introduced for solving…

261

Abstract

A new class of explicit preconditioning methods based on the concept of sparse approximate factorization procedures and inverse matrix techniques is introduced for solving biharmonic equations. Isomorphic methods in conunction with explicit preconditioned schemes based on approximate inverse matrix techniques are presented for the efficient solution of biharmonic equations. Application of the proposed method on linear systems is discussed and numerical results are given.

Details

Engineering Computations, vol. 17 no. 2
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 25 February 2014

George A. Gravvanis and Christos K. Filelis-Papadopoulos

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…

118

Abstract

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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Article
Publication date: 1 May 1999

George A. Gravvanis

A new class of approximate inverse banded matrix techniques based on the concept of LU‐type factorization procedures is introduced for computing explicitly approximate inverses…

852

Abstract

A new class of approximate inverse banded matrix techniques based on the concept of LU‐type factorization procedures is introduced for computing explicitly approximate inverses without inverting the decomposition factors. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of banded linear systems. Applications of the method on a linear system are discussed and numerical results are given.

Details

Engineering Computations, vol. 16 no. 3
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 6 November 2017

Christos K. Filelis-Papadopoulos and George A. Gravvanis

Large sparse least-squares problems arise in different scientific disciplines such as optimization, data analysis, machine learning and simulation. This paper aims to propose a

69

Abstract

Purpose

Large sparse least-squares problems arise in different scientific disciplines such as optimization, data analysis, machine learning and simulation. This paper aims to propose a two-level hybrid direct-iterative scheme, based on novel block independent column reordering, for efficiently solving large sparse least-squares linear systems.

Design/methodology/approach

Herewith, a novel block column independent set reordering scheme is used to separate the columns in two groups: columns that are block independent and columns that are coupled. The permutation scheme leads to a two-level hierarchy. Using this two-level hierarchy, the solution of the least-squares linear system results in the solution of a reduced size Schur complement-type square linear system, using the preconditioned conjugate gradient (PCG) method as well as backward substitution using the upper triangular factor, computed through sparse Q-less QR factorization of the columns that are block independent. To improve the convergence behavior of the PCG method, the upper triangular factor, computed through sparse Q-less QR factorization of the coupled columns, is used as a preconditioner. Moreover, to further reduce the fill-in, then the column approximate minimum degree (COLAMD) algorithm is used to permute the block consisting of the coupled columns.

Findings

The memory requirements for solving large sparse least-squares linear systems are significantly reduced compared to Q-less QR decomposition of the original as well as the permuted problem with COLAMD. The memory requirements are reduced further by choosing to form larger blocks of independent columns. The convergence behavior of the iterative scheme is improved due to the chosen preconditioning scheme. The proposed scheme is inherently parallel due to the introduction of block independent column reordering.

Originality/value

The proposed scheme is a hybrid direct-iterative approach for solving sparse least squares linear systems based on the implicit computation of a two-level approximate pseudo-inverse matrix. Numerical results indicating the applicability and effectiveness of the proposed scheme are given.

Details

Engineering Computations, vol. 34 no. 8
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 1 February 1996

Jaroslav Mackerle

Presents a review on implementing finite element methods on supercomputers, workstations and PCs and gives main trends in hardware and software developments. An appendix included…

677

Abstract

Presents a review on implementing finite element methods on supercomputers, workstations and PCs and gives main trends in hardware and software developments. An appendix included at the end of the paper presents a bibliography on the subjects retrospectively to 1985 and approximately 1,100 references are listed.

Details

Engineering Computations, vol. 13 no. 1
Type: Research Article
ISSN: 0264-4401

Keywords

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