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Normalized explicit approximate inverse preconditioning for solving 3D boundary value problems on uniprocessor and distributed systems
Author(s) -
Gravvanis George A.,
Giannoutakis Konstantinos M.
Publication year - 2005
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1494
Subject(s) - uniprocessor system , conjugate gradient method , mathematics , linear system , preconditioner , discretization , toeplitz matrix , biconjugate gradient stabilized method , conjugate residual method , boundary value problem , rate of convergence , matrix (chemical analysis) , computational complexity theory , algorithm , computer science , mathematical analysis , parallel computing , gradient descent , computer network , channel (broadcasting) , materials science , machine learning , multiprocessing , artificial neural network , pure mathematics , composite material
Abstract 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. Normalized explicit preconditioned conjugate gradient schemes in conjunction with normalized approximate inverse matrix techniques are presented for solving sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also derived. A Parallel Normalized Explicit Preconditioned Conjugate Gradient method for distributed memory systems, using message passing interface (MPI) communication library, is also given along with theoretical estimates on speedups, efficiency and computational complexity. Application of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results are given for uniprocessor and multicomputer systems. Copyright © 2005 John Wiley & Sons, Ltd.