Iterative Methods for Solving Linear Systems

Greenbaum, Anne

doi:10.1137/1.9781611970937

Cited by 897 publications

(887 citation statements)

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“…The SD and OM algorithms are classical ( [21]), and, although simple, have the disadvantage of slow convergence when the condition number of A is large ( [3,23]). The BB algorithm has been much studied because of its remarkable improvement over the SD and OM algorithms ( [25,4,17] and references therein) and proofs of its convergence can be found in [20].…”

Section: Preliminariesmentioning

confidence: 99%

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Cooperative concurrent asynchronous computation of the solution of symmetric linear systems

2016

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This paper extends the idea of Brezinski's hybrid acceleration procedure, for the solution of a system of linear equations with a symmetric coefficient matrix of dimension n, to a new context called cooperative computation, involving m agents (m n), each one concurrently computing the solution of the whole system, using an iterative method. Cooperation occurs between the agents through the communication, periodic or probabilistic, of the estimate of each agent to one randomly chosen agent, thus characterizing the computation as concurrent and asynchronous. Every time one agent receives solution estimates from the others, it carries out a least squares computation, involving a small linear system of dimension m, in order to replace its current solution estimate by an affine combination of the received estimates, and the algorithm continues until a stopping criterion is met. In addition, an autocooperative algorithm, in which estimates are updated using affine combinations of current and past estimates, is also proposed. The proposed algorithms are shown to be efficient for certain matrices, specifically in relation to the popular Barzilai-Borwein algorithm, through numerical experiments.

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Section: Preliminariesmentioning

confidence: 99%

“…This method has the properties of very fast convergence, theoretically in at most n iterations, where n is the problem dimension ( [21]), superior to that of the BB algorithm ( [6]). It is also appropriate for use with large, sparse matrices, since it uses only matrix-vector products in its computations.…”

Section: Preliminariesmentioning

confidence: 99%

Cooperative concurrent asynchronous computation of the solution of symmetric linear systems

2016

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“…In the previous section we showed how Arnoldi can be used to systematically a method called the General Minimum Residual Method (GMRES), see [7 ] and [8 ]. This is an iterative method where one seeks an approximate solution x m living in the a ne space…”

Section: Gmresmentioning

confidence: 99%

A Krylov Subspace Approach to Parametric Inversion of Electromagnetic Data Based on Residual Minimization

Balidemaj¹,

Remis²

2010

PIERS Online

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“…which are the standard CG formulas (Greenbaum, 1997). From the point of view of control theory, one approach to understand this algorithm is to think of the 'parameters' a k and b k as scalar control inputs.…”

Section: Steepest Descent Plus Momentum Equals Frozen Conjugate Gradientmentioning

confidence: 99%

Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method

Bhaya

Kaszkurewicz

2004

Neural Networks

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It is pointed out that the so called momentum method, much used in the neural network literature as an acceleration of the backpropagation method, is a stationary version of the conjugate gradient method. Connections with the continuous optimization method known as heavy ball with friction are also made. In both cases, adaptive (dynamic) choices of the so called learning rate and momentum parameters are obtained using a control Liapunov function analysis of the system. q

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Iterative Methods for Solving Linear Systems

Cited by 897 publications

References 0 publications

Cooperative concurrent asynchronous computation of the solution of symmetric linear systems

Cooperative concurrent asynchronous computation of the solution of symmetric linear systems

A Krylov Subspace Approach to Parametric Inversion of Electromagnetic Data Based on Residual Minimization

Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method

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