The rate of convergence of explicit approximate inverse preconditioning

Int. J. Numer. Meth. Engng

Giannoutakis

2005

Self Cite

SUMMARYNormalized 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.

Section: Normalized Approximate Inverse Matrix Techniquesmentioning

confidence: 99%

“…, p − 1) from the main diagonal in a 'recurring wave'-like pattern, cf. References [16,18,19,21]. It is reasonable to assume that the value of the retention parameter l can be chosen as multiples of m and p.…”

Section: Class I Class Iimentioning

confidence: 99%

Normalized explicit approximate inverse preconditioning for solving 3D boundary value problems on uniprocessor and distributed systems

Int. J. Numer. Meth. Engng

Giannoutakis

2005

Self Cite

“…In order to solve efficiently large order linear systems a moving window shifted from bottom to top is used in order to reduce the memory requirements, cf. [29,30,61], yielding the Normalized Optimized Approximate Inverse Finite Element Matrix algorithmic procedure (henceforth called the NOROAIFEM algorithm) for computing the elements of a class of approximate inverses. The computational work of the NOROAIFEM algorithm is ≈ O[(r + +1)δl]n multiplicative operations, cf.…”

Section: Exact and Approximate Inverse Finite Element Matrixmentioning

confidence: 99%

High Performance Inverse Preconditioning

Arch Computat Methods Eng

2008

Self Cite

The derivation of parallel numerical algorithms for solving sparse linear systems on modern computer systems and software platforms has attracted the attention of many researchers over the years. In this paper we present an overview on the design issues of parallel approximate inverse matrix algorithms, based on an antidiagonal "wave pattern" approach and a "fish-bone" computational procedure, for computing explicitly various families of exact and approximate inverses for solving sparse linear systems. Parallel preconditioned conjugate gradienttype schemes in conjunction with parallel approximate inverses are presented for the efficient solution of sparse linear systems. Applications of the proposed parallel methods by solving characteristic sparse linear systems on symmetric multiprocessor systems and distributed systems are discussed and the parallel performance of the proposed schemes is given, using MPI, OpenMP and Java multithreading.

“…The convergence analysis of similar explicit approximate inverse preconditioning has been presented in [48,61,66,92]. The basic properties of the EPCG method, cf.…”

Section: ) Setmentioning

confidence: 99%

“…[63,66], using the Finite Element approximate inverse M~,r2, Furthermore, the total number of arithmetic operations required for the computation of the solution u. is given by 0(n3/251(51 +el + 6)-1/2(rl + r2 + el + e2)loge -t) (163) According to the convergence analysis of explicit approximate inverse preconditioning, cf. [63,66], the rate of convergence is improved when the value of the "retention" parameter 5l is chosen as multiples of the bandwidths, i.e. m and p. It is evident that when the value of the "retention" parameter 5l is chosen as multiples of the bandwidths rn and p then the required overall computational complexity of the method can be prohibitively high.…”

Section: {2 = O (~(Al + El +mentioning

confidence: 99%

Explicit approximate inverse preconditioning techniques

2002

ARCO

SummaryThe numerical treatment and the production of related software for solving large sparse linear systems of algebraic equations, derived mainly from the discretization of partial differential equation, by preconditioning techniques has attracted the attention of many researchers. In this paper we give an overview of explicit approximate inverse matrix techniques for computing explicitly various families of approximate inverses based on Choleski and LU -type approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference, finite element and the domain decomposition discretization of elliptic and parabolic partial differential equations. Composite iterative schemes, using inner-outer schemes in conjunction with Picard and Newton method, based on approximate inverse matrix techniques for solving non-linear boundary value problems, are presented. Additionally, isomorphic iterative methods are introduced for the efficient solution of non-linear systems. Explicit preconditioned conjugate gradient -type schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear system of algebraic equations. Theoretical estimates on the rate of convergence and computational complexity of the explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.