Inexact Newton-like Methods Research Articles

Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems

<p style='text-indent:20px;'>An efficiently preconditioned Newton-like method for the computation of the eigenpairs of large and sparse nonsymmetric matrices is proposed. A sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system. The properties of the preconditioned matrix are investigated. Numerical results are given which reveal that the new proposed algorithms are efficient.</p>

Low-rank and sparse matrix recovery via inexact Newton-like method with non-monotone search

In this paper, a non-convex optimization model of the low-rank and sparse matrix recovery from partial samples is proposed, and an inexact Newton-like algorithm with non-monotone scheme for solving the non-convex optimization model is suggested, in which the inexact Newton-like algorithm is used for the low-rank matrix part and the non-monotone scheme is used for sparse matrix part. It is proved that the iterative sequence of the new algorithm converges to a stationary point of the model. Finally, numerical experiments show that the proposed algorithm is far superior to the missing low-rank sparse decomposition (MLSD) algorithm in Azghani et al., 2019 and Algorithm 1 in Gu et al., 2016 in CPU time, which indicates that the new algorithm is more efficient.

Weaker conditions for inexact mutitpoint Newton-like methods

In this paper we study the problem of analyzing the convergence both local and semilocal of inexact Newton-like methods for approximating the solution of an equation in which there exists nondifferentiability. We will impose conditions, to ensure that the method converges, are weaker than in the ones imposed in previous results. The theoretical results shown in this study are applied to a chemical application in order to be proven.

Local Convergence of Inexact Newton-Like Method under Weak Lipschitz Conditions

The paper develops the local convergence of Inexact Newton-Like Method (INLM) for approximating solutions of nonlinear equations in Banach space setting. We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis. The obtained results compare favorably with earlier ones such as [7, 13, 14, 18, 19]. A numerical example is also provided.

On the convergence of inexact Newton-like methods under mild differentiability conditions

In the present paper, we have introduced an inexact Newton-like algorithm and discussed its semilocal convergence analysis under average Lipschitz condition as well as γ-Lipschitz condition for solving generalized operator equations containing non differentiable operators in Banach spaces. Our results extend and improve some well established results in the context of differentiability of involved operators. As special cases of our results, we re-obtain some well established results for the Newton method and inexact Newton method. We apply our result to solve Fredholm integral equations.

An inexact Newton-like conditional gradient method for constrained nonlinear systems

In this paper, we propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate are established by using a general majorant condition. Two applications of such condition are provided: one is for functions whose derivatives satisfy a Hölder-like condition and the other is for functions that satisfy a Smale condition, which includes a substantial class of analytic functions. Some preliminary numerical experiments illustrating the applicability of the proposed method are also presented.

Identifying Lamé parameters from time-dependent elastic wave measurements

In many sectors of today’s industry it is of utmost importance to detect defects in elastic structures contained in technical devices to guarantee their failure-free operation. As currently used signal processing techniques have natural limits with respect to accuracy and significance, modern mathematical methods are crucial to improve current algorithms. We consider in this paper a parameter identification approach for isotropic and linear elastic structures described by their Lamé parameters and a material density. This approach can be employed for non-destructive defect detection, location and characterization from time-dependent measurements of one elastic wave. To this end, we show that the operator linking the static parameters with the wave measurements is Fréchet differentiable, which allows to set up Newton-like methods for the non-linear parameter identification problem. We indicate the performance of a specific inexact Newton-like regularization method by numerical examples for a testing problem of a thin plate from measurements of the normal component of the displacement field on the boundary. As an extension, we further augment this method with a total variation regularization and thereby improve reconstructed parameters that feature edges.

Convergence Analysis of Newton-Like Methods for Inverse Eigenvalue Problems with Multiple Eigenvalues

We provide in the present paper a corrected proof for the classical quadratical convergence theorem (i.e., Theorem 3.3 in Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634--667]) of the Newton-like method for solving inverse eigenvalue problems with possible multiple eigenvalues. Moreover, as a by-product, our approach developed here can be extended to establish a similar convergence result for an inexact version of the Newton-like method with possible multiple eigenvalues, which is an extension of the corresponding inexact Newton-like method for the distinct case in Chan, Chung, and Xu [BIT Numer. Math., 43 (2003), pp. 7--20].

On the convergence of inexact two-point Newton-like methods on Banach spaces

We present a unified convergence analysis of Inexact Newton-like methods in order to approximate a locally unique solution of a nonlinear operator equation containing a nondifferentiable term in a Banach space setting. The convergence conditions are more general and the error analysis more precise than in earlier studies such as (Argyros, 2007; Cătinaş, 2005; Cătinaş, 1994; Chen and Yamamoto, 1989; Dennis, 1968; Hernández and Romero, 2005; Potra and Pták, 1984; Rheinboldt, 1977). Special cases of our results can be used to find zeros of derivatives. Numerical examples are also provided in this study.

THE CONVERGENCE BALL OF INEXACT NEWTON-LIKE METHOD IN BANACH SPACE UNDER WEAK LIPSHITZ CONDITION

We present a local convergence analysis for inexact Newton-like method in a Banach space under weaker Lipschitz condition. The convergence ball is enlarged and the estimates on the error distances are more precise under the same computational cost as in earlier studies such as [6, 7, 11, 18]. Some special cases are considered and applications for solving nonlinear systems using the Newton-arithmetic mean method are improved with the new convergence technique.

Improved local convergence analysis of inexact Newton-like methods under the majorant condition

Improved local convergence analysis of inexact Newton-like methods under the majorant condition

A line search filter algorithm with inexact step computations for equality constrained optimization

In this paper, a new line search filter algorithm for equality constrained optimization is presented. The approach belongs to the class of inexact Newton-like methods. It can also be regarded as an inexact version of generic sequential quadratic programming (SQP) methods. The trial step is obtained by truncatedly solving the primal–dual system based on any robust and efficient linear system solver. Practical termination tests for the linear system solver are established to ensure global convergence. Preliminary numerical results demonstrate the approach is potentially useful.

Computational experience with numerical methods for nonnegative least-squares problems

We discuss the solution of large-scale box-constrained linear least-squares problems by two recent affine-scaling methods: a cyclic Barzilai–Borwein strategy and an Inexact Newton-like method where a preconditioning technique allows for an efficient computation of the steps. A robust globally and fast locally convergent method based on the combination of the two procedures is presented along with extensive numerical results. Copyright © 2010 John Wiley & Sons, Ltd.

A Ulm-like method for inverse eigenvalue problems

We propose a Ulm-like method for solving inverse eigenvalue problems, which avoids solving approximate Jacobian equations comparing with other known methods. A convergence analysis of this method is provided and the R-quadratic convergence property is proved under the assumption of the distinction of given eigenvalues. Numerical experiments as well as the comparison with the inexact Newton-like method are given in the last section.

Local convergence analysis of inexact Newton-like methods under majorant condition

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.

Local convergence of inexact methods under the Hölder condition

We study the convergence properties for some inexact Newton-like methods including the inexact Newton methods for solving nonlinear operator equations on Banach spaces. A new type of residual control is presented. Under the assumption that the derivative of the operator satisfies the Hölder condition, the radius of convergence ball of the inexact Newton-like methods with the new type of residual control is estimated, and a linear and/or superlinear convergence property is proved, which extends the corresponding result of [B. Morini, Convergence behaviour of inexact Newton methods, Math. Comput. 68 (1999) 1605–1613]. As an application, we show that the inexact Newton-like method presented in [R.H. Chan, H.L. Chung, S.F. Xu, The inexact Newton-like method for inverse eigenvalue problem, BIT Numer. Math. 43 (2003) 7–20] for solving inverse eigenvalue problems can be regarded equivalently as one of the inexact Newton-like methods considered in this paper. A numerical example is provided to illustrate the convergence performance of the algorithm.

The Inexact Newton-Like Method for Inverse Eigenvalue Problem

In this paper, we consider using the inexact Newton-like method for solving inverse eigenvalue problem. This method can minimize the oversolving problem of Newton-like methods and hence improve the efficiency. We give the convergence analysis of the method, and provide numerical tests to illustrate the improvement over Newton-like methods.

Local convergence theorems for Newton methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Frechet-derivative whereas the second theorem employs hypotheses on themth (m ≥ 2 an integer). Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the mth Frechet-derivative our radius of convergence can sometimes be larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].

Inexact Methods: Forcing Terms and Conditioning

In this paper, we consider inexact Newton and Newton-like methods andprovide new convergence conditions relating the forcing terms to theconditioning of the iteration matrices. These results can be exploited wheninexact methods with iterative linear solvers are used. In this framework,preconditioning techniques can be used to improve the performance ofiterative linear solvers and to avoid the need of excessively small forcingterms. Numerical experiments validating the theoretical results arediscussed.