On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems

Bellavia, Stefania; Riccietti, Elisa

doi:10.1007/s10957-018-1318-1

Cited by 7 publications

(41 citation statements)

References 26 publications

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“…Avoiding this computation we introduce several sources of inexactness, that will be described in details in the following section. Due to this, from a theoretical point of view, we cannot expect the same regularizing properties of its exact counterpart in [2], but the proposed approach reduces to the method in [2] when the dimension of the Lancsoz space ℓ reaches n. However, we are still able to show that it is possible to control how the distance of the current solution approximation from an exact solution of the problem changes from an iteration to the other, and that this decreases if the dimension of the Lanczos space is large enough.…”

Section: Introductionmentioning

confidence: 95%

“…Many approaches have been proposed in the literature to deal with this problem, such as nonstationary iterated Tikhonov regularization or regularized versions of trust region methods [1,2,14,28,29,15,6,30].…”

Section: Introductionmentioning

confidence: 99%

“…This method represents an inexact version of the approach proposed in [2], which is tailored for smallmedium scale problems. The method indeed employs the singular value decomposition (SVD) of the Jacobian matrix, which can be too costly to compute for large-scale problems.…”

Section: Introductionmentioning

confidence: 99%

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An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Bellavia¹,

Donatelli²,

Riccietti³

2019

Preprint

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In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Bellavia¹,

Donatelli²,

Riccietti³

2019

Preprint

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“…Differently from previous local analyses in the literature [1][2][3][4], we neither assume zero residue at a solution nor full rank of the Jacobian at such a point. In applied contexts, such as data fitting, parameter estimation, experimental design, and imaging problems, to name a few, admitting nonzero residue is essential for achieving meaningful solutions (see, e.g., [5][6][7][8][9][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition

Behling

Gonçalves

Santos

2019

J Optim Theory Appl

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The Levenberg-Marquardt method is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares problems. In this paper, we consider local convergence properties of the method, when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full rank of the Jacobian in a neighborhood of a stationary point. Differently from the zero-residue case, the choice of the Levenberg-Marquardt parameter is shown to be dictated by (i) the behavior of the rank of the Jacobian and (ii) a combined measure of nonlinearity and residue size in a neighborhood of the set of (possibly non-isolated) stationary points of the sum of squares function.

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“…We are not aware of Levenberg-Marquardt methods for both zero and non-zero residual nonlinear least-squares problems with approximated function and gradient, for which both local and global convergence is proved. Contributions on this topic are given by [5] where the inexactness is present only in the gradient and in the Jacobian and local convergence is not proved and by [3,4] where the Jacobian is exact and only local convergence is considered.…”

Section: Introductionmentioning

confidence: 99%

A Levenberg–Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients

2018

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This is an author's version published in: http://oatao.univ-toulouse.fr/24842To cite this version: Bellavia, Stefania and Gratton, Serge and Riccietti, Elisa A Levenberg-Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients. (2018) Numerische Mathematik, 140 (3). 791-825. ISSN 0029-599X Official URL: https://doi. AbstractIn this paper we consider large scale nonlinear least-squares problems for which function and gradient are evaluated with dynamic accuracy and propose a Levenberg-Marquardt method for solving such problems. More precisely, we consider the case in which the exact function to optimize is not available or its evaluation is computationally demanding, but approximations of it are available at any prescribed accuracy level. The proposed method relies on a control of the accuracy level, and imposes an improvement of function approximations when the accuracy is detected to be too low to proceed with the optimization process. We prove global and local convergence and complexity of our procedure and show encouraging numerical results on test problems arising in data assimilation and machine learning. Mathematics Subject ClassificationAchieving the Cauchy decrease is a sufficient condition to get global convergence of the method, so one can rely on approximated solutions of problem (2.1), see [11]. A solution of (2.1) can alternatively be found solving the optimality conditions(2.3)Then, we approximately solve (2.3), i.e. we compute a step p such that J δ k (x k ) T J δ k (x k ) + λ k I p = −g δ k (x k ) + r k ,

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On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems

Cited by 7 publications

References 26 publications

An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition

A Levenberg–Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients

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