The objective of this work is to develop both theory and formal optimization-based numerical techniques for the optimal design of process and control systems under uncertainty. While previous work mainly concentrated on steady-state considerations and time-invariant uncertainty, the emphasis in this thesis is on the use of dynamic mathematical models to describe the process system and time-varying uncertainty. Flexibility considerations, robust stability criteria and explicit control structure selection and controller design aspects are considered as an integral part of the process synthesis/design task. For the incorporation of flexibility and control system design in process design and optimization, a mixed-integer stochastic optimal control formulation was proposed, the solution of which results in process design and control systems which are economically optimal while being able to cope with parametric uncertainty and process disturbances. Regarding robust stability criteria, a combined flexibility-stability analysis method was developed which provides a quantitative measure of the size of the parameter space over which feasible and stable operation can be attained by proper adjustment of the control variables. Such an analysis step can then be included in the simultaneous process and control design formulation. Algorithms and numerical techniques for the solution of the resulting mathematical formulations have also been developed. In particular, an iterative decomposition algorithm was proposed, which alternates between two subproblems: A multiperiod design subproblem, which determines the process and control structure and design to satisfy a set of critical uncertain parameters over time, and the combined flexibility-stability analysis step, which identifies a new set of critical parameters for a fixed design and control. Since both steps of the algorithm involve the solution of mixed-integer optimal control formulations, a novel technique for the solution of this class of problems was also proposed, featuring an implicit Runge-Kutta method for time discretization, an efficient integration step size selection procedure and adjoint equations for obtaining the reduced gradients. Numerical examples together with detailed process design examples, such as a ternary distillation system and a binary double effect distillation system, are also presented to demonstrate the potential of the proposed methodology.

Partial differential equations occur in a variety of forms in many different branches of Mathematical Physics. These equations can be classified according to various criteria, which may include such formal aspects as the number of dependent or independent variables, the order of the derivatives involved and the degree of non-linearity of the equations. These equations can also be categorised according to the methods which are employed for solving the partial differential equations, or according to particular properties which their solutions may possess. In pure analysis, to solve partial differential equations we may use methods such as transformation or separation of variables. However, these methods can only be applied to very special classes of problems [WEINBERGER, 1965]. In practice, we employ numerical methods for the solution of such systems and although the types of methods used in numerical analysis of differential equations do not generally correspond with those used in mathematical analysis, both depend upon particular properties of the solution. This thesis is concerned with the numerical solution of certain types of partial differential equations and therefore, with practical problems which can be treated by certain numerical methods. In practice, the numerical methods for solving the differential problems depend upon the nature of other auxiliary conditions, such as boundary or initial conditions. Certain types of auxiliary conditions are suitable only for certain corresponding types of differential equations, and in general, physical problems suggest auxiliary conditions which 3 are suitable for the differential equations involved in the problem. If, the auxiliary conditions are specified in such a way that there exists one and only one solution (uniqueness) for the differential problem, and in addition, a small change in these given auxiliary conditions result in a small change in the solution (stability), then the problem is said to be well-posed. Since numerical methods are by nature approximate processes, however, these methods rarely produce exact solutions for a given problem. However, it can be shown [see STEPHENSON, 1970], that if the differential problem is well-posed then the solution of this problem is expected to be accurate.

We consider the convective Cahn-Hilliard equation that is used as a model of coarsening dynamics in driven systems and that in two spatial dimensions (x; y) has the form ut + Duuₓ + ∇²(u - u³ + ∇²u) = 0. Here t denotes time, u = u(x; y; t) is the order parameter and D is the parameter measuring the strength of driving. We primarily consider the case of one spatial dimension, when there is no y-dependence. For the case of no driving, when D = 0, the standard Cahn-Hilliard equation is recovered, and it is known that solutions to this equation are characterised by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters or droplets/holes or islands are obtained) followed by the coarsening process, where the average size of the clusters grows in time and the number of the clusters decreases. Moreover, two main coarsening mechanisms have been identified in the literature, namely, coarsening due to volume and translational modes. On the other hand, for the case of strong driving, when D ! 1, the well-known Kuramoto-Sivashinsky equation is recovered, solutions of which are characterised by complicated chaotic oscillations in both space and time. The primary aim of the present thesis is to perform a detailed and systematic investigation of the transitions in the solutions of the convective Cahn-Hilliard equation for a wide range of parameter values as the driving-force parameter is increased, and, in particular, to understand in detail how the coarsening dynamics is affected by driving. We find that one of the coarsening modes is stabilised at relatively small values of D, and the type of the unstable coarsening mode may change as D increases. In addition, we find that there may be intervals in the driving-force parameter D where coarsening is completely stabilised. On the other hand, there may be intervals where twomode solutions are unstable and the solutions can evolve, for example, into one-droplet/hole solutions, symmetry-broken two-droplet/hole solutions or time-periodic solutions. We present detailed stability diagrams for 2-mode solutions in the parameter planes and corroborate our findings by time-dependent simulations. Finally, we present preliminary results for the case of the (convective) Cahn-Hilliard equation in two spatial dimensions.

In this thesis several problems in Partial Differential Equations in unbounded domains are studied using the techniques of uniformly local spaces and weighted energy theory. First Coupled Burger's equations are studied on the whole space R and existence of solutions in uniformly local spaces is proven in the case where the non-linearity is gradient. Moreover the uniqueness of these solutions and some additional regularity is proven. Second the Cahn-Hilliard, and closely related Cahn-Hilliard-Oono, equations are studied on the whole space R3 with both polynomial and singular potentials and existence of solutions in uniformly local spaces is proven. Moreover uniqueness and additional regularity of these equations is also proven. Third the Navier-Stokes equations are studied on the whole space R2 and, building on the work of Zelik who showed the existence of solutions in uniformly local spaces, the existence of a finite dimensional globally compact attractor is proven in the case where the forcing term has arbitrarily slow decay at infinity.

The main aim of this thesis is to study certain extensions of Fourier transforms and Bessel potentials. The transforms can be written as composites (or compositions) of some elementary transformations, Fourier operators and Bessel potentials. The Bessel potentials have first been introduced by Aronszajn and Smith. Further, the extended Bessel potentials which are weighted forms of the operators previously considered, can be expressed as composites of the extended Fourier transforms.

The k nearest neighbor rule or the kNN rule is a nonparametric algorithm that search for the k nearest neighbors of a query set in another set of points. In this thesis, application of the kNN rule in predictability analysis of stock and share returns is proposed. The first experiment tests the possibility of prediction for ‘success’ (or ‘winner’) components of four stock and shares market indices in a selected time period [1]. We have developed a method of labeling the component with either ‘winner’ or ‘loser’. We analyze the existence of information on the winner–loser separation in the initial fragments of the daily closing prices log–returns time series. The Leave–One–Out Cross–Validation with the kNN algorithm is applied on the daily log–returns of components. Two distance measurements are used in our experiment, a correlation distance, and its proximity. By analyzing the error, for the HANGSENG and the DAX index, there are clear signs of possibility to evaluate the probability of long–term success. The correlation distance matrix histograms and 2–D/3–D elastic maps generated from the ViDaExpert show that the ‘winner’ components are closer to each other and ‘winner’/‘loser’ components are separable on elastic maps for the HANGSENG and the DAX index while for the negative possibility indices, there is no sign of separation. In the second experiment, for a selected time interval, daily log–return time series is split into “history”, “present” and “future” parts. The kNN rule is used to search for nearest neighbors of “present” from a set. This set is created by using the sliding window strategy. The nearest neighbors are considered as the predicted “future” part. We then use ideas from dynamical systems and to regenerate “future” part closing prices from nearest neighbors log–returns. Different sub–experiments are created in terms of the difference in generation of “history” part, different market indices, and different distance measurements. This approach of modeling or forecasting works for both the ergodic dynamic systems and the random processes. The Lorenz attractor with noise is used to generate data and the data are used in the kNN experiment with the Euclidean distance. The sliding window strategy is applied in both test and training set. The kNN rule is used to find the k nearest neighbors and the next ‘window’ is used as the prediction. The error analysis of the relative mean squared error RMSE shows that k = 1 can give the best prediction and when k → 100, the average RMSE values converge. The average standard deviation values converge when k → 100. The solution Z(t) is predicted quite accurate using the kNN experiment.

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In many science and engineering problems there is uncertainty in the input data. The ability to suitably model and handle this uncertainty is crucial for obtaining meaningful information about solutions. In this thesis, we consider the numerical approximation of statistics of solutions to partial differential equations (PDEs) with uncertain inputs. We focus on PDEs with random coefficients and random domains.We consider a general set of numerical methods known collectively as stochastic finite element methods. We can distinguish stochastic Galerkin methods and stochastic sampling methods. For the latter, samples of the random inputs are generated and the deterministic PDE is solved for each one. Averages of the quantities of interest are calculated using solutions obtained from the samples. We focus in particular on a specific type of sampling method, the stochastic collocation method.The main computational cost associated with solving PDEs with random data using stochastic finite element methods is the solution of the resulting linear system(s) obtained from the fully discrete problem. The main aim in this thesis is to identify efficient and robust techniques for solving the sequence of linear systems obtained from stochastic collocation methods and to reduce the computational costs by recycling as much information as possible. New iterative solution strategies are presented for stochastic collocation discretisations of PDEs with random coefficients and for stochastic collocation discretisations of PDEs on random domains. Substantial savings on computing costs have been demonstrated.

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