1,721,033 research outputs found
Advances in large scale unconstrained optimization: novel preconditioning strategies for Nonlinear Conjugate Gradient methods and new developments in Newton-Krylov methods
Full text linkIn this thesis we propose new iteratively constructed preconditioners, to be paired with Conjugate Gradient-type algorithms, in order to efficiently solve large scale unconstrained optimization problems. On this guideline, the central thread of this thesis is the use of conjugate directions given by Conjugate Gradient or SYMMBK algorithms. To this aim, in Chapter 1 we recall some results about iterative methods for solving linear systems. In particular, we introduce both the Conjugate Gradient method and the Lanczos process. Finally, the idea of preconditioning is given and the well known Preconditioned Conjugate Gradient method is provided. In Chapter 2 we deal with large scale unconstrained optimization problems, recalling the Nonlinear Conjugate Gradient (NCG) method and its preconditioned version, the quasi-Newton methods, including the damped techniques, and finally the linesearch-based Inexact Newton methods. The main contribution in this thesis is given by Chapters 3-7. In particular, Chapter 3 provides new preconditioners to be used within the NCG method. This class of preconditioners draws inspiration from quasi-Newton updates, in order to obtain a good approximation of the inverse of the Hessian matrix. On detail, at the current iteration of the NCG we consider some preconditioners based on new low-rank quasi-Newton symmetric updating formulae, obtained as by-product of the NCG method at the previous steps. In particular, we propose five classes of preconditioners: in the first three classes the major drawback is that preconditioner might not be positive definite. In the fourth class, a preconditoner that is positive definite and satisfies the secant equation at the current iteration is given. Finally, in the last class, a preconditoner that is positive definite and satisfies the modified secant equation at each iteration is provided. However, in general, the search direction pk we compute at iteration k of NCG could be not well scaled, which may introduce some ill-conditioning when applying Preconditioned Nonlinear Conjugate Gradient (PNCG). In order to reduce the scaling problem, in Chapter 4 we focus on the use of damped techniques within NCG methods. Damped techniques were introduced by Powell and recently reproposed by Al-Baali, in the framework of quasi-Newton methods. New damped techniques are proposed and are embedded within a class of preconditioners described in Chapter 3 (see Section 3.3.4). In Chapter 5, by referring to the Polak-Ribière (PR) method, we firstly give theoretical results on the global convergence for an effective PNCG algorithm. Secondly, we investigate the use of a damped vector in the definition of the scalar k, hence affecting the definition of the search direction and producing a modified NCG/PNCG method. On this purpose, we prove that some global convergence properties still hold for the modified damped Polak-Ribière (D-PR-PNCG) method, while substantially preserving numerical performance. In Chapters 6-7 we focus on linesearch-based Newton-Krylov methods for solving large scale unconstrained optimization problems. In particular, in Chapter 6 we try to improve the efficiency of Newton-Krylov methods by proposing an adaptive truncation rule, for deciding the maximum number of inner iterations allowed at each outer iteration. The adaptive truncation rule is tested both within the unpreconditioned and the preconditioned framework proposed in [48]. Finally, Chapter 7 represents a work in progress: on detail, starting from [48] we deal with a class of preconditioners specifically suited for large indefinite linear systems, and may be obtained as by-product of Krylov subspace solvers, as well as by applying L-BFGS updates. In particular, in Chapter 7, only some preliminaries and the structure of our class of preconditioners, namely AINVK, are provided. At the end, conclusions are given
How the increasing competition is reshaping Tuscan wineries: the case of Chianti Classico and Brunello di Montlcino
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Dinamiche evolutive e implicazioni di policy a livello di distretti tecnologici regionali: Distretto tecnologico regionale Scienze della Vita
No full textGoing Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Nonnegative least-squares image deblurring: improved gradient projection approaches
No full textThe least-squares approach to image deblurring leads to an ill-posed problem. The addition of the nonnegativity constraint, when appropriate, does not provide regularization, even if, as far as we know, a thorough investigation of the ill-posedness of the resulting constrained least-squares problem has still to be done. Iterative methods, converging to nonnegative least-squares solutions, have been proposed. Some of them have the "semi-convergence'' property, i.e. early stopping of the iteration provides "regularized'' solutions. In this paper we consider two of these methods: the projected Landweber (PL) method and the iterative image space reconstruction algorithm (ISRA).Even if they work well in many instances, they are not frequently used in practice because, in general, they require a large number of iterations before providing a sensible solution. Therefore the main purpose of this paper is to refresh these methods by increasing their efficiency. Starting from the remark that PL and ISRA require only the computation of the gradient of the functional, we propose the application to these algorithms of special acceleration techniques that have been recently developed in the area of the gradient methods. In particular, we propose the application of efficient step-length selection rulesand line-search strategies. Moreover, remarking that ISRA is a scaled gradient algorithm, we evaluate its behaviour in comparison with a recent scaled gradient projection (SGP) method for image deblurring. Numerical experiments demonstrate that the accelerated methods still exhibit thesemi-convergence property, with a considerable gain both in the number of iterations and in the computational time; in particular, SGP appears definitely the most efficient one
Iterative regularization algorithms for constrained image deblurring on graphics processors
No full textThe ability of the modern graphics processors to operate on large matrices in parallel can be exploited for solving constrained image deblurring problems in a short time.
In particular, in this paper we propose the parallel implementation of two iterative regularization methods: the well known expectation maximization algorithm and a recent scaled
gradient projection method. The main differences between the considered approaches and their impact on the parallel implementations are discussed. The effectiveness of the parallel
schemes and the speedups over standard CPU implementations are evaluated on test problems arising from astronomical images
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