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Stefania Bellavia

Publications and source records attributed to Stefania Bellavia.

25 records · Page 2Linked to original sources
arXiv2020

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

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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arXiv2019

Adaptive Cubic Regularization Methods with Dynamic Inexact Hessian Information and Applications to Finite-Sum Minimization

We consider the Adaptive Regularization with Cubics approach for solving nonconvex optimization problems and propose a new variant based on inexact Hessian information chosen dynamically. The theoretical analysis of the proposed procedure is given. The key property of ARC framework, constituted by optimal worst-case function/derivative evaluation bounds for first- and second-order critical point, is guaranteed. Application to large-scale finite-sum minimization based on subsampled Hessian is discussed and analyzed in both a deterministic and probabilistic manner and equipped with numerical experiments on synthetic and real datasets.

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arXiv2019

Subsampled Nonmonotone Spectral Gradient Methods

This paper deals with subsampled spectral gradient methods for minimizing finite sum. Subsample function and gradient approximations are employed in order to reduce the overall computational cost of the classical spectral gradient methods. The global convergence is enforced by a nonmonotone line search procedure. Global convergence is proved when functions and gradients are approximated with increasing accuracy. R-linear convergence and worst-case iteration complexity is investigated in case of strongly convex objective function. Numerical results on well known binary classification problems are given to show the effectiveness of this framework and analyze the effect of different spectral coefficient approximations arising from variable sample nature of this procedure.

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arXiv2018

Subsampled Inexact Newton methods for minimizing large sums of convex functions

This paper deals with the minimization of large sum of convex functions by Inexact Newton (IN) methods employing subsampled functions, gradients and Hessian approximations. The Conjugate Gradient method is used to compute the inexact Newton step and global convergence is enforced by a nonmonotone line search procedure. The aim is to obtain methods with affordable costs and fast convergence. Assuming strongly convex functions, R-linear convergence and worst-case iteration complexity of the procedure are investigated when functions and gradients are approximated with increasing accuracy. A set of rules for the forcing parameters and subsample Hessian sizes are derived that ensure local q-linear/superlinear convergence of the proposed method. The random choice of the Hessian subsample is also considered and convergence in the mean square, both for finite and infinite sums of functions, is proved. Finally, global convergence with asymptotic R-linear rate of IN methods is extended to the case of sum of convex function and strongly convex objective function. Numerical results on well known binary classification problems are also given. Adaptive strategies for selecting forcing terms and Hessian subsample size, streaming out of the theoretical analysis, are employed and the numerical results showed that they yield effective IN methods.

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arXiv2018

Stable interior-point method for convex quadratic programming with strict error bounds

We present a short step interior point method for solving a class of nonlinear programming problems with quadratic objective function. Convex quadratic programming problems can be reformulated as problems in this class. The method is shown to have weak polynomial time complexity. A complete proof of the numerical stability of the method is provided. No requirements on feasibility, row-rank of the constraint Jacobian, strict complementarity, or conditioning of the problem are made. Infeasible problems are solved to an optimal interior least-squares solution.

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arXiv2015

Improved regularizing iterative methods for ill-posed nonlinear systems

In this paper we address the numerical solution of nonlinear ill-posed systems by iterative regularization methods in the classes of Levenberg-Marquardt, trust-region and adaptive quadratic regularization procedures. Both with exact and noisy data, our focus is on the potential to approach a solution of the unperturbed systems without assumptions on its vicinity to the initial guess. Regularizing properties of the methods proposed are shown theoretically and validated numerically along with enhanced convergence.

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