Vicente F. Candela

Recurrence relations for rational cubic methods II: the Chebyshev method

We continue the analysis of rational cubic methods, initiated in [7]. In this paper, we obtain a system of a priori error bounds for the Chebyshev method in Banach spaces through a local convergence theorem that provides sufficient conditions on the initial point in order to ensure the convergence of Chebyshev iterates. The error estimates are exact for second degree polynomials. We also discuss some applications.ZusammenfassungWir betrachten ein System von a priori Fehlerabschätzungen für die Konvergenz des Chebyshev-Verfahrens in, Banachräumen. Unsere Sätze geben hinreichende Bedingungen an, den Startwert, welche die Konvergenz der Chebyshev-Iteration sichern. Sie bestehen aus einem System rekursiver Beziehungen, ähnlich den Bedingungen von Kantorvich für das Newton-Verfahren.

Recurrence relations for rational cubic methods, part I. The Halley method

In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of “recurrence relations”, analogous to those given for the Newton method by Kantorovich, improving previous results by Döring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.ZusammenfassungWir betrachten ein System von a priori Fehlerabschätzungen für die Konvergenz des Halley-Verfahrens in Banachräumen. Unsere Sätze geben hinreichende Bedingungen an den Startwert, welche die Konvergenz der Halley-Iteration sichern. Sie bestehen aus einem System rekursiver Beziehungen, ähnlich den Bedingungen von Kantorovich für das Newton-Verfahren. Weitere rationale kubische Verfahren werden in einer künftigen Arbeit untersucht.

A class of quasi-Newton generalized Steffensen methods on Banach spaces

We consider a class of generalized Steffensen iterations procedure for solving nonlinear equations on Banach spaces without any derivative. We establish the convergence under the Kantarovich-Ostrowskis conditions. The majorizing sequence will be a Newtons type sequence, thus the convergence can have better properties. Finally, a numerical comparation with the classical methods is presented.

A local spectral inversion of a linearized TV model for denoising and deblurring

In this paper, we propose a model for denoising and deblurring consisting of a system of linear partial differential equations with locally constant coefficients, obtained as a local linearization of the total variation models. The keypoint of our model is to get the local inversion of the Laplacian operator, which will be done via the Fast Fourier Transform (FFT). Two local schemes will be developed: a pointwise and a piecewise one. We will analyze both, their advantages and their limitations.

A Polynomial Approach to the Piecewise Hyperbolic Method

In this paper, a local (third-order accurate) shock capturing method for hyperbolic conservation laws is presented. The method has been made with the same idea as the PHM method, but with a simpler reconstruction. A comparison with the classic high order methods is discussed.

Blind Deconvolution Models Regularized by Fractional Powers of the Laplacian

We present a method for deconvolution of images by means of an inversion of fractional powers of the Gaussian. The main feature of our model is the introduction of a regularizing term which is also a fractional power of the Laplacian. This term allows us to recover higher frequencies. The model is particularly useful to devise an algorithm for blind deconvolution. We will show, analyze and illustrate through examples the performance of this algorithm.

The convergence of the perturbed Newton method and its application for ill-conditioned problems

Abstract Iterative methods, such as Newton’s, behave poorly when solving ill-conditioned problems: they become slow (first order), and decrease their accuracy. In this paper we analyze deeply and widely the convergence of a modified Newton method, which we call perturbed Newton, in order to overcome the usual disadvantages Newton’s one presents. The basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization. Choices for that parameter are proposed. The theoretical analysis will be illustrated through examples.

Third-order iterative methods without using any Fréchet derivative

A modification of classical third-order methods is proposed. The main advantage of these methods is they do not need to evaluate any Frechet derivative. A convergence theorem in Banach spaces, just assuming the second divided difference is bounded and a punctual condition, is analyzed. Finally, some numerical results are presented.

Mathematical models for restoration of baroque paintings

In this paper we adapt different techniques for image deconvolution, to the actual restoration of works of arts (mainly paintings and sculptures) from the baroque period. We use the special characteristics of these works in order to both restrict the strategies and benefit from those properties. We propose an algorithm which presents good results in the pieces we have worked. Due to the diversity of the period and the amount of artists who made it possible, the algorithms are too general even in this context. This is a first approach to the problem, in which we have assumed very common and shared features for the works of art. The flexibility of the algorithm, and the freedom to choose some parameters make it possible to adapt the problem to the knowledge that restorators in charge may have about a particular work.

On the Numerical Approximation of the Length of (Implicit) Level Curves

Abstract The evaluation of the length of a curve, represented in an Eulerian way as the zero level set of an implicit function, depends mainly on the representation of the curve. In this paper, we propose a parameter to measure the complexity of the curve, and therefore the accuracy of the evaluation, based on the evolution of the representation in different scales. We will analyze this parameter, its properties and its relations with the regularity of the curve.