Jacobian-free diagonal Newton's method for solving nonlinear systems with singular Jacobian

The basic requirement of Newton’s method in solving systems of nonlinear equations is, the Jacobian must be non-singular. This condition restricts to some extent the application of Newton method. In this paper we present a modification of Newton’s method for systems of nonlinear equations where the Jacobian is singular. This is made possible by approximating the Jacobian inverse into a diagonal matrix by means of variational techniques. The anticipation of our approach is to bypass the point in which the Jacobian is singular. The local convergence of the proposed method has been proven under suitable assumptions. Numerical experiments are carried out which show that, the proposed method is very encouraging

A Simulated Annealing Based Optimization Algorithm

In modern engineering finding an optimal design is formulated as an optimization problem which involves evaluating a computationally expensive black-box function. To alleviate these difficulties, such problems are often solved by using a metamodel, which approximates the computer simulation and provides predicted values at a much lower computational cost. While metamodels can significantly improve the efficiency of the design process, they also introduce several challenges, such as a high evaluation cost, the need to effectively search the metamodel landscape and to locate good solutions, and the selection of which metamodel is most suitable to the problem being solved. To address these challenges, this chapter proposes an algorithm that uses a hybrid simulated annealing and SQP search to effectively search the metamodel. It also uses ensembles that combine prediction of several metamodels to improve the overall prediction accuracy. To further improve the ensemble accuracy, it adapts the ensemble topology during the search. Finally, to ensure convergence to a valid optimum in the presence of metamodel inaccuracies, the proposed algorithm operates within a trust-region framework. An extensive performance analysis based on both mathematical test functions and an engineering application shows the effectiveness of the proposed algorithm

Reliable preliminary space mission design: Optimisation under uncertainties in the frame of evidence theory

In the early phase of the design of a space mission it is generally desirable to investigate as many feasible alternative solutions as possible. Traditionally a system margin approach is used in order to estimate the correct value of subsystem budgets. While this is a consolidated and robust approach, it does not give a measure of the reliability of any of the investigated solutions. In addition the mass budget is typically overdimensioned, where a more accurate design could lead to improvements in payload mass. This study will address two principal issues typically associated with the design of a space mission: (i) the effective and efficient generation of preliminary solutions by properly treating their inherent multi-disciplinary elements and (ii) the minimisation of the impact of uncertainties on the overall design, which in turn will lead to an increase in the reliability of the produced results. The representation and treatment of the uncertainties are key aspects of reliable design. An insufficient consideration of uncertainty or an unadapted mathematical representation leads to misunderstanding of the real issues of a design, to delay in the future development of the project or even potentially to its failure. The most common way to deal with uncertainty is the probabilistic approach. However, this theory is not suitable to represent epistemic uncertainties, arising from lack of knowledge. Alternative theories have been recently developed, amongst which we find Evidence Theory which is implemented in this work. Developed by Shafer from Dempster's original work, it is regarded by many as a suitable paradigm to accurately represent uncertainties. Evidence Theory is presented and discussed from an engineering point of view and special attention given to the implementation of this approach. Once mathematically represented, the uncertainties can be taken into account in the design optimisation problem. However, the computational complexity of Evidence Theory can be overwhelming and therefore more efficient ways to solve the reliable design problem are required. Existing methods are considered and improvements developed by the author, to increase the efficiency of the algorithm by making the most of the available data, are proposed and tested. Additionally, a new sample-based approximation technique to tackle large scale problems, is introduced in this thesis. Assuming that the uncertainties are modelled by means of intervals, the cluster approximation method, and especially implemented as a Binary Space Partition, appears to be very well-suited to the task. The performance of the various considered methods to solve the reliable design optimisation problem in the frame of Evidence Theory is tested and analysed. The dependency on the problem characteristics, such as dimensionality, complexity, or multitude of local solutions are carefully scrutinised. The conclusions of these tests enables the author to propose guidelines on how to tackle the problem depending on its specificity. Finally, two examples of preliminary space mission design are used to illustrate how the proposed methodology can be applied. Using realistic and current mission designs, the results show the benefits that could be achieved during the preliminary analyses and feasibility studies of space exploration

Η Εξ αποστάσεως εκπαίδευση μέσω των Τεχνολογιών της Πληροφορίας και των Επικοινωνιών: Διερεύνηση των απόψεων και αντιλήψεων των διευθυντών και προϊσταμένων των νηπιαγωγείων της Λάρισας ως προς την εφαρμογή της

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