This paper is the result of a study of triangular matrices with particular emphasis on those which are convergence preserving transformations.

Nowadays, many optimization problems in real applications share a separable structure in the objective and it becomes more and more common to solve these problems by decomposition methods such as fast iterative shrinkage-thresholding algorithm (FISTA), generalized alternating direction method of multipliers (GADMM), and first-order primal-dual algorithm (PD), to name just a few. In this thesis, we focus on two optimization algorithms for solving convex programs: θ-scheme and a preconditioned primal-dual algorithm. For the θ-scheme, we first present an elaborative convergence analysis in a Hilbert space and propose a general convergent inexact θ-scheme. Second, for unconstrained problems, we prove the convergence of the θ-scheme and show a sublinear convergence rate in terms of the objective function. Furthermore, a practical inexact θ-scheme is derived to solve l_2-loss based problems and its convergence is proved. Third, for constrained problems, even though the convergence of the θ-scheme is available in the literature, yet its sublinear convergence rate is unknown until we provide one via a variational reformulation of the solution set. Besides, in order to relax the condition imposed on the θ-scheme, we propose a new variant and show its convergence. Finally, some preliminary numerical experiments demonstrate the efficiency of the θ-scheme and our proposed methods. For the preconditioned primal-dual algorithm, noticing that a practical step size cannot lie in the theoretical region, we show that the range of dual step size can be enlarged by 1/3 at most and at the same time, the convergence and a sublinear convergence rate can be ensured. Therefore, this practical step size can indeed guarantee the convergence. Furthermore, if more regularity conditions are imposed on objective functions, we can obtain a linear convergence rate. Finally, some connection with other methods is revealed. In future work, we focus on the acceleration of the θ-scheme. Some preliminary numerical experiments demonstrate the potential efficiency of our proposed accelerated θ-scheme. Therefore, it would be our priority of further study.

Iteratively reweighted least squares (IRLS) algorithms provide an alternative to the more standard 1 l -minimization approach in compressive sensing. Daubechies et al. introduced a particularly stable version of an IRLS algorithm and rigorously proved its convergence in 2010. They did not, however, consider the case in which prior information on the support of the sparse domain of the solution is available. In 2009, Miosso et al. proposed an IRLS algorithm that makes use of this information to further reduce the number of measurements required to recover the solution with specified accuracy. Although Miosso et al. obtained a number of simulation results strongly confirming the utility of their approach, they did not rigorously establish the convergence properties of their algorithm. In this paper, we introduce prior information on the support of the sparse domain of the solution into the algorithm of Daubechies et al. We then provide a rigorous proof of the convergence of the resulting algorithm.

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This thesis is missing page 147 in all printed copies. - Digitization Centre / Techniques for accelerating the convergence of multi-variable iterative calculations were studied. The study is mainly concerned with iterations which result from the calculation of recycle processes of complex chemical plants. The conditions for convergence and the existence of the solution of linear and non-linear iterations were obtained. Special attention was given to the asymptotic behaviour of the iterations since, in most cases, they approach geometric progression. The convergence of an Alkylation unit simulation was studied by implementation of the "full matrix" technique for accelerating the rate of convergence. The geometric extrapolation technique was described and applied successfully for accelerating the convergence of the calculation of multi-component distillation columns. / Thesis / Doctor of Philosophy (PhD)

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Yes / This chapter reviews the substantive issue of monetary union through evaluating countries readiness for entry utilising the experience of the European Union’s process of Economic and Monetary Union (EMU). The European single currency system came under unprecedented strain following the Global Financial Crisis induced Great Recession and there is little reason to assume that this will diminish, in any significant way, in the near future. Crucially it is important to reflect that each economy is unique in its blend of sectoral strengths and weaknesses and comparative advantage, therefore the national interest will be distinctively different for each potential participant. In particular, there is no set rule in which to weigh the relative merits of the arguments associated with membership of a monetary union. In terms of the eurozone the chapter critically evaluates the convergence criteria stipulated in the Treaty on European Union (TEU) and empirically reviews the compliance of EU member states. It questions whether the TEU criteria satisfactorily perform this role, such that the convergence criteria present a series of financial tests, of which some are theoretically spurious, while the remainder are inadequate to indicate the range of consequences of participation. Finally, the chapter undertakes an analysis of the macroeconomic performance of Greece. Specifically, it focuses on the main features, economic events and key economic indicators (GDP per capita, inflation, unemployment, twin deficit of current account and net lending/borrowing, output gap and gross debt) during the crucial 2000-09 period, between eurozone membership and the crisis.

An extensometer has been developed to continuously monitor roof extension in underground mines. The extensometer is designed to be installed in the MSHA-mandated test holes in the roof and measures the displacement between an anchorage point at the top of the borehole and the hole in the roof of the excavation. Once installed, the extensometer will report displacement through semi-wireless communications network. The extensometer is hard-wired into the permissible MIDAS datalogger, where results can be obtained wirelessly via the MIDAS user interface. Lab tests have indicated that the device produces displacement data. The device was installed in one underground coal mine to review its effectiveness in the field. / Master of Science

The purpose of this paper is to examine certain questions concerning infinite series. The first chapter introduces several basic definitions and theorems from calculus. In particular, this chapter contains the proofs for various convergence tests for series of real numbers. The second chapter deals primarily with the equivalence of absolute convergence, unconditional convergence, bounded multiplier convergence, and c0 multiplier convergence for series of real numbers. Also included in this chapter is a proof that an unconditionally convergent series may be rearranged so that it converges to any real number desired. The third chapter contains a proof of the Silverman-Toeplitz Theorem together with several applications.