Inexact Newton Methods

Dembo, Ron S.; Eisenstat, Stanley C.; Steihaug, Trond

doi:10.1137/0719025

Cited by 1,370 publications

(1,035 citation statements)

References 7 publications

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“…Estimate (2.9) holds as a strict inequality. Therefore, the new error bounds are more precise than the old ones using only L 1 [7][8][9] or L 0 and L 1 [4,5].…”

Section: Applications and Examplesmentioning

confidence: 99%

“…In this paper we are concerned with the problem of approximating a solution x * of the nonlinear least squares problem: min G(x) := 1 2 F (x) T F (x), (1.1) where F is Fréchet-differentiable defined on R n with values in R m , m ≥ n. The solution of these problems are very important in computational sciences and other disciplines [2,9,10,12,15] and the solution of these problems can be found in closed form only in special cases. In computational sciences the practice of numerical analysis is essentially connected to variants of Newton's method [1]- [15].…”

Section: Introductionmentioning

confidence: 99%

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Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

Argyros

George²

2016

Annals of West University of Timisoara - Mathematics and Computer Science

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Abstract. We present a local convergence analysis of inexact Gauss-Newton-like method (IGNLM) for solving nonlinear leastsquares problems in a Euclidean space setting. The convergence analysis is based on our new idea of restricted convergence domains. Using this idea, we obtain a more precise information on the location of the iterates than in earlier studies leading to smaller majorizing functions. This way, our approach has the following advantages and under the same computational cost as in earlier studies: A large radius of convergence and more precise estimates on the distances involved to obtain a desired error tolerance. That is, we have a larger choice of initial points and fewer iterations are also needed to achieve the error tolerance. Special cases and numerical examples are also presented to show these advantages.

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“…Estimate (2.9) holds as a strict inequality. Therefore, the new error bounds are more precise than the old ones using only L 1 [7][8][9] or L 0 and L 1 [4,5].…”

Section: Applications and Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

Argyros

George²

2016

Annals of West University of Timisoara - Mathematics and Computer Science

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“…We also tested GMRESDR (30,5) at the resolution of the linear system generated at the first outer iteration of Algorithm 1. It was slightly better than GM-RESH(30) in solving Bratu problem but much better when solving convectiondiffusion problem which has a very ill conditioned Jacobian matrix.…”

Section: Boundary Value Problemsmentioning

confidence: 99%

“…In the inexact Newton methods, [5], the sequence x k (called sequence of outer iterations) is generated by …”

Section: Numerical Experimentsmentioning

confidence: 99%

A safeguard approach to detect stagnation of GMRES(m) with applications in Newton-Krylov methods

Gomes-Ruggiero¹,

Lopes²,

Toledo-Benavides³

2008

Comput.appl.math.

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Abstract.Restarting GMRES, a linear solver frequently used in numerical schemes, is known to suffer from stagnation. In this paper, a simple strategy is proposed to detect and avoid stagnation, without modifying the standard GMRES code. Numerical tests with the proposed modified GMRES(m) procedure for solving linear systems and also as part of an inexact Newton procedure, demonstrate the efficiency of this strategy.Mathematical subject classification: 65H10, 65F10.

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“…In these cases the most important drawback of the Newton method is the necessity to solve the Newton system. A possible way of overcoming this difficulty is to apply the inexact Newton method, [4,8], and to solve the Newton system only approximately by means of some iterative method.…”

Section: Introductionmentioning

confidence: 99%

Newton-like method with modification of the right-hand-side vector

Krejić

Lužanin

2001

Math. Comp.

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Abstract. This paper proposes a new Newton-like method which defines new iterates using a linear system with the same coefficient matrix in each iterate, while the correction is performed on the right-hand-side vector of the Newton system. In this way a method is obtained which is less costly than the Newton method and faster than the fixed Newton method. Local convergence is proved for nonsingular systems. The influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented. Relevant numerical examples are used to demonstrate the advantages of the proposed method.

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Inexact Newton Methods

Cited by 1,370 publications

References 7 publications

Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

A safeguard approach to detect stagnation of GMRES(m) with applications in Newton-Krylov methods

Newton-like method with modification of the right-hand-side vector

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