Multidata inverse problems and Bayesian solution methods in astronomy

Statistical analyses of measurements that can be described by statistical models are of essence in astronomy and in scientific inquiry in general. The sensitivity of such analyses, modelling approaches, and the consequent predictions, is sometimes highly dependent on the exact techniques applied, and improvements therein can result in significantly better understanding of the observed system of interest. Particularly, optimising the sensitivity of statistical techniques in detecting the faint signatures of low-mass planets orbiting the nearby stars is, together with improvements in instrumentation, essential in estimating the properties of the population of such planets, and in the race to detect Earth-analogs, i.e. planets that could support liquid water and, perhaps, life on their surfaces. We review the developments in Bayesian statistical techniques applicable to detections planets orbiting nearby stars and astronomical data analysis problems in general. We also discuss these techniques and demonstrate their usefulness by using various examples and detailed descriptions of the respective mathematics involved. We demonstrate the practical aspects of Bayesian statistical techniques by describing several algorithms and numerical techniques, as well as theoretical constructions, in the estimation of model parameters and in hypothesis testing. We also apply these algorithms to Doppler measurements of nearby stars to show how they can be used in practice to obtain as much information from the noisy data as possible. Bayesian statistical techniques are powerful tools in analysing and interpreting noisy data and should be preferred in practice whenever computational limitations are not too restrictive.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Modal testing and numerical modeling of the dynamic properties of layered sheet-steel structure

Recently, due to the increasing total construction and transportation cost and difficulties associated with handling massive structural components or assemblies, there has been increasing financial pressure to reduce structural weight. Furthermore, advances in material technology coupled with continuing advances in design tools and techniques have encouraged engineers to vary and combine materials, offering new opportunities to reduce the weight of mechanical structures. These new lower mass systems, however, are more susceptible to inherent imbalances, a weakness that can result in higher shock and harmonic resonances which leads to poor structural dynamic performances. The objective of this thesis is the modeling of layered sheet steel elements, to accurately predict dynamic performance. During the development of the layered sheet steel model, the numerical modeling approach, the Finite Element Analysis and the Experimental Modal Analysis are applied in building a modal model of the layered sheet steel elements. Furthermore, in view of getting a better understanding of the dynamic behavior of layered sheet steel, several binding methods have been studied to understand and demonstrate how a binding method affects the dynamic behavior of layered sheet steel elements when compared to single homogeneous steel plate. Based on the developed layered sheet steel model, the dynamic behavior of a lightweight wheel structure to be used as the structure for the stator of an outer rotor Direct-Drive Permanent Magnet Synchronous Generator designed for high-power wind turbines is studied.

Advanced analysis and design methods for preparative chromatographic separation processes

Preparative liquid chromatography is one of the most selective separation techniques in the fine chemical, pharmaceutical, and food industries. Several process concepts have been developed and applied for improving the performance of classical batch chromatography. The most powerful approaches include various single-column recycling schemes, counter-current and cross-current multi-column setups, and hybrid processes where chromatography is coupled with other unit operations such as crystallization, chemical reactor, and/or solvent removal unit. To fully utilize the potential of stand-alone and integrated chromatographic processes, efficient methods for selecting the best process alternative as well as optimal operating conditions are needed. In this thesis, a unified method is developed for analysis and design of the following singlecolumn fixed bed processes and corresponding cross-current schemes: (1) batch chromatography, (2) batch chromatography with an integrated solvent removal unit, (3) mixed-recycle steady state recycling chromatography (SSR), and (4) mixed-recycle steady state recycling chromatography with solvent removal from fresh feed, recycle fraction, or column feed (SSR–SR). The method is based on the equilibrium theory of chromatography with an assumption of negligible mass transfer resistance and axial dispersion. The design criteria are given in general, dimensionless form that is formally analogous to that applied widely in the so called triangle theory of counter-current multi-column chromatography. Analytical design equations are derived for binary systems that follow competitive Langmuir adsorption isotherm model. For this purpose, the existing analytic solution of the ideal model of chromatography for binary Langmuir mixtures is completed by deriving missing explicit equations for the height and location of the pure first component shock in the case of a small feed pulse. It is thus shown that the entire chromatographic cycle at the column outlet can be expressed in closed-form. The developed design method allows predicting the feasible range of operating parameters that lead to desired product purities. It can be applied for the calculation of first estimates of optimal operating conditions, the analysis of process robustness, and the early-stage evaluation of different process alternatives. The design method is utilized to analyse the possibility to enhance the performance of conventional SSR chromatography by integrating it with a solvent removal unit. It is shown that the amount of fresh feed processed during a chromatographic cycle and thus the productivity of SSR process can be improved by removing solvent. The maximum solvent removal capacity depends on the location of the solvent removal unit and the physical solvent removal constraints, such as solubility, viscosity, and/or osmotic pressure limits. Usually, the most flexible option is to remove solvent from the column feed. Applicability of the equilibrium design for real, non-ideal separation problems is evaluated by means of numerical simulations. Due to assumption of infinite column efficiency, the developed design method is most applicable for high performance systems where thermodynamic effects are predominant, while significant deviations are observed under highly non-ideal conditions. The findings based on the equilibrium theory are applied to develop a shortcut approach for the design of chromatographic separation processes under strongly non-ideal conditions with significant dispersive effects. The method is based on a simple procedure applied to a single conventional chromatogram. Applicability of the approach for the design of batch and counter-current simulated moving bed processes is evaluated with case studies. It is shown that the shortcut approach works the better the higher the column efficiency and the lower the purity constraints are.