Multiple Cooling Multi Objective Simulated Annealing algorithm has been combined with a conceptual design code written by the author to carry out a multi objective design optimization of an Agricultural Aerial Robot. Both the single and the multi objective optimization problems are solved. The performance figures of merits for different aircraft configurations are compared. In this thesis the potential of optimization as a powerful design tool to the aerospace problems is demonstrated.

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Optimal control problems (OCPs) governed by convection dominated diffusion-convection-reaction equations arise in many science and engineering applications such as shape optimization of the technological devices, identification of parameters in environmental processes and flow control problems. A characteristic feature of convection dominated optimization problems is the presence of sharp layers. In this case, the Galerkin finite element method performs poorly and leads to oscillatory solutions. Hence, these problems require stabilization techniques to resolve boundary and interior layers accurately. The Streamline Upwind Petrov-Galerkin (SUPG) method is one of the most popular stabilization technique for solving convection dominated OCPs. The focus of this thesis is the application and analysis of the SUPG method for distributed and boundary OCPs governed by evolutionary diffusion-convection-reaction equations. There are two approaches for solving these problems: optimize-then-discretize and discretize-then-optimize. For the optimize-then-discretize method, the time-dependent OCPs is transformed to a biharmonic equation, where space and time are treated equally. The resulting optimality system is solved by the finite element package COMSOL. For the discretize-then-optimize approach, we have used the so called allv at-once method, where the fully discrete optimality system is solved as a saddle point problem at once for all time steps. A priori error bounds are derived for the state, adjoint, and controls by applying linear finite element discretization with SUPG method in space and using backward Euler, Crank- Nicolson and semi-implicit methods in time. The stabilization parameter is chosen for the convection dominated problem so that the error bounds are balanced to obtain L2 error estimates. Numerical examples with and without control constraints for distributed and boundary control problems confirm the effectiveness of both approaches and confirm a priori error estimates for the discretize-then-optimize approach.

Multivariate adaptive regression spline (MARS) denotes a modern methodology from statistical learning which is very important in both classification and regression, with an increasing number of applications in many areas of science, economy and technology. MARS is very useful for high dimensional problems and shows a great promise for fitting nonlinear multivariate functions. MARS technique does not impose any particular class of relationship between the predictor variables and outcome variable of interest. In other words, a special advantage of MARS lies in its ability to estimate the contribution of the basis functions so that both the additive and interaction effects of the predictors are allowed to determine the response variable. The function fitted by MARS is continuous, whereas the one fitted by classical classification methods (CART) is not. Herewith, MARS becomes an alternative to CART. The MARS algorithm for estimating the model function consists of two complementary algorithms: the forward and backward stepwise algorithms. In the first step, the model is built by adding basis functions until a maximum level of complexity is reached. On the other hand, the backward stepwise algorithm is began by removing the least significant basis functions from the model. In this study, we propose not to use the backward stepwise algorithm. Instead, we construct a penalized residual sum of squares (PRSS) for MARS as a Tikhonov regularization problem, which is also known as ridge regression. We treat this problem using continuous optimization techniques which we consider to become an important complementary technology and alternative to the concept of the backward stepwise algorithm. In particular, we apply the elegant framework of conic quadratic programming which is an area of convex optimization that is very well-structured, herewith, resembling linear programming and, hence, permitting the use of interior point methods. The boundaries of this optimization problem are determined by the multiobjective optimization approach which provides us many alternative solutions. Based on these theoretical and algorithmical studies, this MSc thesis work also contains applications on the data investigated in a T&Uuml / BiTAK project on quality control. By these applications, MARS and our new method are compared.

Interval uncertainty affects our decision making activities on a daily basis making the data structure of intervals of real numbers more and more popular in theoretical models and related software applications. Natural questions for people or businesses that face interval uncertainty in their data when dealing with cooperation are how to form the coalitions and how to distribute the collective gains or costs. The theory of cooperative interval games is a suitable tool for answering these questions. In this thesis, the classical theory of cooperative games is extended to cooperative interval games. First, basic notions and facts from classical cooperative game theory and interval calculus are given. Then, the model of cooperative interval games is introduced and basic definitions are given. Solution concepts of selection-type and interval-type for cooperative interval games are intensively studied. Further, special classes of cooperative interval games like convex interval games and big boss interval games are introduced and various characterizations are given. Some economic and Operations Research situations such as airport, bankruptcy and sequencing with interval data and related interval games have been also studied. Finally, some algorithmic aspects related with the interval Shapley value and the interval core are considered.

In this thesis, a general pseudospectral formulation for a class of Sturm-Liouville eigenvalue problems is consructed. It is shown that almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schr&ouml / dinger form may be transformed into a more tractable form. This tractable form will be called here a weighted equation of hypergeometric type with a perturbation (WEHTP) since the non-weighted and unperturbed part of it is known as the equation of hypergeometric type (EHT). It is well known that the EHT has polynomial solutions which form a basis for the Hilbert space of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of WEHTP, and hence the energy eigenvalues of the Schr&ouml / dinger equation. Exemplary computations are performed to support the convergence numerically.

In this thesis, a fairly detailed survey on the q-classical orthogonal polynomials of the Hahn class is presented. Such polynomials appear to be the bounded solutions of the so called qhypergeometric difference equation having polynomial coefficients of degree at most two. The central idea behind our study is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference equation by means of a qualitative analysis of the relevant q-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every posssible rational form of the polynomial coefficients, together with various relative positions of their zeros, in the q-Pearson equation to describe a desired q-weight function on a suitable orthogonality interval. Therefore, our method differs from the standard ones which are based on the Favard theorem and the three-term recurrence relation.

Numerical study of transition in the Rayleigh-B&#039 / enard problem of thermal convection between rigid plates heated from below under the influence of gravity with and without rotation is presented. The first numerical approach uses spectral element method with Fourier expansion for horizontal extent and Legendre polynomal for vertical extent for the purpose of generating a database for the subsequent analysis by using Karhunen-Lo&#039 / eve (KL) decomposition. KL decompositions is a statistical tool to decompose the dynamics underlying a database representing a physical phenomena to its basic components in the form of an orthogonal KL basis. The KL basis satisfies all the spatial constraints such as the boundary conditions and the solenoidal (divergence-free) character of the underlying flow field as much as carried by the flow database. The optimally representative character of the orthogonal basis is used to investigate the convective flow for different parameters, such as Rayleigh and Prandtl numbers. The second numerical approach uses divergence free basis functions that by construction satisfy the continuity equation and the boundary conditions in an expansion of the velocity flow field. The expansion bases for the thermal field are constructed to satisfy the boundary conditions. Both bases are based on the Legendre polynomials in the vertical direction in order to simplify the Galerkin projection procedure, while Fourier representation is used in the horizontal directions due to the horizontal extent of the computational domain taken as periodic. Dual bases are employed to reduce the governing Boussinesq equations to a dynamical system for the time dependent expansion coefficients. The dual bases are selected so that the pressure term is eliminated in the projection procedure. The resulting dynamical system is used to study the transitional regimes numerically. The main difference between the two approaches is the accuracy with which the solenoidal character of the flow is satisfied. The first approach needs a numerically or experimentally generated database for the generation of the divergence-free KL basis. The degree of the accuracy for the KL basis in satisfying the solenoidal character of the flow is limited to that of the database and in turn to the numerical technique used. This is a major challenge in most numerical simulation techniques for incompressible flow in literature. It is also dependent on the parameter values at which the underlying flow field is generated. However the second approach is parameter independent and it is based on analytically solenoidal basis that produces an almost exactly divergence-free flow field. This level of accuracy is especially important for the transition studies that explores the regions sensitive to parameter and flow perturbations.

In this thesis, a survey on pseudospectral methods for differential equations is presented. Properties of the classical orthogonal polynomials required in this context are reviewed. Differentiation matrices corresponding to Jacobi, Laguerre,and Hermite cases are constructed. A fairly detailed investigation is made for the Hermite spectral methods, which is applied to the Schr&ouml / dinger eigenvalue equation with several potentials. A discussion of the numerical results and comparison with other methods are then introduced to deduce the effciency of the method.

Recent developments show that derivative free methods are highly demanded by researches for solving optimization problems in various practical contexts. Although well-known optimization methods that employ derivative information can be very effcient, a derivative free method will be more effcient in cases where the objective function is nondifferentiable, the derivative information is not available or is not reliable. Derivative Free Optimization (DFO) is developed for solving small dimensional problems (less than 100 variables) in which the computation of an objective function is relatively expensive and the derivatives of the objective function are not available. Problems of this nature more and more arise in modern physical, chemical and econometric measurements and in engineering applications, where computer simulation is employed for the evaluation of the objective functions. In this thesis, we give an example of the implementation of DFO in an approach for optimizing stirrer configurations, including a parametrized grid generator, a flow solver, and DFO. A derivative free method, i.e., DFO is preferred because the gradient of the objective function with respect to the stirrer&rsquo / s design variables is not directly available. This nonlinear objective function is obtained from the flow field by the flow solver. We present and interpret numerical results of this implementation. Moreover, a contribution is given to a survey and a distinction of DFO research directions, to an analysis and discussion of these. We also state a derivative free algorithm used within a clustering algorithm in combination with non-smooth optimization techniques to reveal the effectiveness of derivative free methods in computations. This algorithm is applied on some data sets from various sources of public life and medicine. We compare various methods, their practical backgrounds, and conclude with a summary and outlook. This work may serve as a preparation of possible future research.