Solving sparse linear systems with approximate inverse preconditioners on analog devices

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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Parallel Preconditioned Conjugate Gradient Square Method Based on Normalized Approximate Inverses

Scientific Programming ◽

10.1155/2005/508607 ◽

2005 ◽

Vol 13 (2) ◽

pp. 79-91 ◽

Cited By ~ 1

Author(s):

George A. Gravvanis ◽

Konstantinos M. Giannoutakis

Keyword(s):

Linear Systems ◽

Conjugate Gradient ◽

Message Passing ◽

Message Passing Interface ◽

Distributed Memory ◽

Inverse Matrix ◽

Preconditioned Conjugate Gradient ◽

Sparse Linear Systems ◽

Approximate Inverse ◽

Approximate Inverse Matrix Techniques

A new class of normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. A new parallel normalized explicit preconditioned conjugate gradient square method in conjunction with normalized approximate inverse matrix techniques for solving efficiently sparse linear systems on distributed memory systems, using Message Passing Interface (MPI) communication library, is also presented along with theoretical estimates on speedups and efficiency. The implementation and performance on a distributed memory MIMD machine, using Message Passing Interface (MPI) is also investigated. Applications on characteristic initial/boundary value problems in three dimensions are discussed and numerical results are given.

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Approximate inverse preconditionings for sparse linear systems

International Journal of Computer Mathematics ◽

10.1080/00207169208804097 ◽

1992 ◽

Vol 44 (1-4) ◽

pp. 91-110 ◽

Cited By ~ 72

Author(s):

J. D. F. Cosgrove ◽

J. C. Díaz ◽

A. Griewank

Keyword(s):

Linear Systems ◽

Sparse Linear Systems ◽

Approximate Inverse

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Parallel multilevel recursive approximate inverse techniques for solving general sparse linear systems

The Journal of Supercomputing ◽

10.1007/s11227-016-1728-5 ◽

2016 ◽

Vol 72 (6) ◽

pp. 2259-2282 ◽

Cited By ~ 1

Author(s):

Antonios T. Makaratzis ◽

Christos K. Filelis-Papadopoulos ◽

George A. Gravvanis

Keyword(s):

Linear Systems ◽

Sparse Linear Systems ◽

Approximate Inverse

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A block sparse approximate inverse with cutoff preconditioner for semi-sparse linear systems derived from Molecular Orbital calculations

JSIAM Letters ◽

10.14495/jsiaml.2.41 ◽

2010 ◽

Vol 2 (0) ◽

pp. 41-44 ◽

Cited By ~ 1

Author(s):

Ikuro Yamazaki ◽

Masayuki Okada ◽

Hiroto Tadano ◽

Tetsuya Sakurai ◽

Keita Teranishi

Keyword(s):

Linear Systems ◽

Molecular Orbital ◽

Molecular Orbital Calculations ◽

Sparse Linear Systems ◽

Approximate Inverse ◽

Sparse Approximate Inverse

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GENERIC APPROXIMATE SPARSE INVERSE MATRIX TECHNIQUES

International Journal of Computational Methods ◽

10.1142/s0219876213500849 ◽

2014 ◽

Vol 11 (06) ◽

pp. 1350084 ◽

Cited By ~ 8

Author(s):

CHRISTOS K. FILELIS-PAPADOPOULOS ◽

GEORGE A. GRAVVANIS

Keyword(s):

Linear Systems ◽

Krylov Subspace ◽

Coefficient Matrix ◽

Sparse Linear Systems ◽

Approximate Inverse ◽

Matrix Algorithm ◽

Sparsity Pattern ◽

Gradient Type ◽

Comparative Results ◽

Conjugate Gradient Type Methods

During the last decades explicit preconditioning methods have gained interest among the scientific community, due to their efficiency for solving large sparse linear systems in conjunction with Krylov subspace iterative methods. The effectiveness of explicit preconditioning schemes relies on the fact that they are close approximants to the inverse of the coefficient matrix. Herewith, we propose a Generic Approximate Sparse Inverse (GenASPI) matrix algorithm based on ILU(0) factorization. The proposed scheme applies to matrices of any structure or sparsity pattern unlike the previous dedicated implementations. The new scheme is based on the Generic Approximate Banded Inverse (GenAbI), which is a banded approximate inverse used in conjunction with Conjugate Gradient type methods for the solution of large sparse linear systems. The proposed GenASPI matrix algorithm, is based on Approximate Inverse Sparsity patterns, derived from powers of sparsified matrices and is computed with a modified procedure based on the GenAbI algorithm. Finally, applicability and implementation issues are discussed and numerical results along with comparative results are presented.

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