Student Number : 9804484F MSc dissertation - School of Statistics and Acturial Science - Faculty of Science / This dissertation conducts a comprehensive and up to date review of measures of component and module importance within monotone systems, where it is assumed that components work independent of each other. The dissertation traces the development of these important measures from the initial definition of Birnbaum importance right through to the definition of Meng's criticality importance. Furthermore, the dissertation draws a distinction between time independent measures and time dependent measures (such as the Barlow-Proschan measures). The dissertation demonstrates how such measures may be implemented in analysing the importance of components within the monotone systems by evaluating these measures for a well known bridge structure example. This evaluation also reveals how each defined measure can be compared to each other. In conclusion, the dissertation describes how these measures can be extended to non-monotone systems or systems with dependent components.

This thesis is concerned with the formulation of a suitable notion of monotonicity of discrete and continuous-time dynamical systems on Lie groups and homogeneous spaces. In a linear space, monotonicity refers to the property of a system that preserves an ordering of the elements of the space. Monotone systems have been studied in detail and are of great interest for their numerous applications, as well as their close connections to many physical and biological systems. In a linear space, a powerful local characterisation of monotonicity is provided by differential positivity with respect to a constant cone field, which combines positivity theory with a local analysis of nonlinear systems. Since many dynamical systems are naturally defined on nonlinear spaces, it is important to seek a suitable adaptation of monotonicity on such spaces. However, the question of how one can develop a suitable notion of monotonicity on a nonlinear manifold is complicated by the general absence of a clear and well-defined notion of order on such a space. Fortunately, for Lie groups and important examples of homogeneous spaces that are ubiquitous in many problems of engineering and applied mathematics, symmetry provides a way forward. Specifically, the existence of a notion of geometric invariance on such spaces allows for the generation of invariant cone fields, which in turn induce notions of conal orders. We propose differential positivity with respect to invariant cone fields as a natural and powerful generalisation of monotonicity to nonlinear spaces and develop the theory in this thesis. We illustrate the ideas with numerous examples and apply the theory to a number of areas, including the theory of consensus on Lie groups and order theory on the set of positive definite matrices.

<p>U disertaciji su posmatrani projektivni postupci tipa konjugovanih gradijenata za re&scaron;avanje nelinearnih monotonih sistema velikih dimenzija. Ovi postupci kombinuju projektivnu metodu sa pravcima pretraživanja tipa konjugovanih gradijenata. Zbog osobine monotonosti sistema, projektivna metoda omogućava jednostavnu globalizaciju, a pravci pretraživanja tipa konjugovanih gradijenata zahtevaju malo<br />računarske memorije pa su pogodni za re&scaron;avanje sistema velikih dimenzija. Projektivni postupci tipa konjugovanih gradijenata ne koriste izvode niti funkciju cilja i zasnovani su samo na izračunavanju vrednosti funkcije sistema, pa su pogodni i za re&scaron;avanje neglatkih monotonih sistema. Po&scaron;to se globalna konvergencija dokazuje bez pretpostavki o regularnosti, ovi postupci se mogu koristiti i za re&scaron;avanje sistema sa singularnim re&scaron;enjima. U disertaciji su definisana tri nova tročlana pravca pretraživanja<br />tipa Flečer-Rivs i dva nova hibridna pravca tipa Hu-Stori. Formulisani su projektivni postupci sa novim pravcima pretraživanja i dokazana je njihova globalna konvergencija. Numeričke performanse postupaka testirane su na relevantnim primerima i poređene sa poznatim postupcima iz literature. Numerički rezultati potvrđuju da su novi postupci robusni, efikasni i uporedivi sa postojećim postupcima.</p> / <p>Projection based CG methods for solving large-scale nonlinear monotone systems are considered in this thesis. These methods combine hyperplane projection technique with conjugate gradient (CG) search directions. Hyperplane projection method is suitable for monotone systems, because it enables simply globalization, while CG directions are efficient for large-scale nonlinear systems, due to low memory. Projection based CG methods are funcion-value based, they don&rsquo;t use merit function and derivatives, and because of that they are also suitable for solving nonsmooth monotone systems. The global convergence of these methods are ensured without additional regularity assumptions, so they can be used for solving singular systems.Three new three-term search directions of Fletcher-Reeves type and two new hybrid search directions of Hu-Storey type are defined. PCG algorithm with five new CG type directions is proposed and its global convergence is established. Numerical performances of methods are tested on relevant examples from literature. These results point out that new projection based CG methods have good computational performances. They are efficient, robust and competitive with other methods.</p>