Desirability Functions continue to attract attention of scientists and researchers working in the area of multi-response optimization. There are many versions of such functions, differing mainly in formulations of individual and overall desirability functions. Derringer and Suich&rsquo / s desirability functions being used throughout this thesis are still the most preferred ones in practice and many other versions are derived from these. On the other hand, they have a drawback of containing nondifferentiable points and, hence, being nonsmooth. Current approaches to their optimization, which are based on derivative-free search techniques and modification of the functions by higher-degree polynomials, need to be diversified considering opportunities offered by modern nonlinear (global) optimization techniques and related softwares. A first motivation of this work is to develop a new efficient solution strategy for the maximization of overall desirability functions which comes out to be a nonsmooth composite constrained optimization problem by nonsmooth optimization methods. We observe that individual desirability functions used in practical computations are of mintype, a subclass of continuous selection functions. To reveal the mechanism that gives rise to a variation in the piecewise structure of desirability functions used in practice, we concentrate on a component-wise and generically piecewise min-type functions and, later on, max-type functions. It is our second motivation to analyze the structural and topological properties of desirability functions via piecewise max-type functions. In this thesis, we introduce adjusted desirability functions based on a reformulation of the individual desirability functions by a binary integer variable in order to deal with their piecewise definition. We define a constraint on the binary variable to obtain a continuous optimization problem of a nonlinear objective function including nondifferentiable points with the constraints of bounds for factors and responses. After describing the adjusted desirability functions on two well-known problems from the literature, we implement modified subgradient algorithm (MSG) in GAMS incorporating to CONOPT solver of GAMS software for solving the corresponding optimization problems. Moreover, BARON solver of GAMS is used to solve these optimization problems including adjusted desirability functions. Numerical applications with BARON show that this is a more efficient alternative solution strategy than the current desirability maximization approaches. We apply negative logarithm to the desirability functions and consider the properties of the resulting functions when they include more than one nondifferentiable point. With this approach we reveal the structure of the functions and employ the piecewise max-type functions as generalized desirability functions (GDFs). We introduce a suitable finite partitioning procedure of the individual functions over their compact and connected interval that yield our so-called GDFs. Hence, we construct GDFs with piecewise max-type functions which have efficient structural and topological properties. We present the structural stability, optimality and constraint qualification properties of GDFs using that of max-type functions. As a by-product of our GDF study, we develop a new method called two-stage (bilevel) approach for multi-objective optimization problems, based on a separation of the parameters: in y-space (optimization) and in x-space (representation). This approach is about calculating the factor variables corresponding to the ideal solutions of each individual functions in y, and then finding a set of compromised solutions in x by considering the convex hull of the ideal factors. This is an early attempt of a new multi-objective optimization method. Our first results show that global optimum of the overall problem may not be an element of the set of compromised solution. The overall problem in both x and y is extended to a new refined (disjunctive) generalized semi-infinite problem, herewith analyzing the stability and robustness properties of the objective function. In this course, we introduce the so-called robust optimization of desirability functions for the cases when response models contain uncertainty. Throughout this thesis, we give several modifications and extensions of the optimization problem of overall desirability functions.

In this thesis we present a numerical algorithm to solve the singular Inverse Sturm-Liouville problems with symmetric potential functions. The singularity, which comes from the unbounded domain of the problem, is treated by considering the limiting case of the associated problem on the symmetric finite interval. In contrast to regular problems which are considered on a finite interval the singular inverse problem has an ill-conditioned structure despite of the limiting treatment. We use the regularization techniques to overcome the ill-posedness difficulty. Moreover, since the problem is nonlinear the iterative solution procedures are needed. Direct computation of the eigenvalues in iterative solution is handled via psoudespectral methods. The numerical examples of the considered problem are given to illustrate the accuracy and convergence behaviour.

Medical image segmentation is a challenging problem, and is studied widely. In this thesis, the main goal is to develop automatic segmentation techniques of human mimic muscles and to compare them with ground truth data in order to determine the method that provides best segmentation results. The segmentation methods are based on Bayesian with Markov Random Field (MRF) and Level Set (Active Contour) models. Proposed segmentation methods are multi step processes including preprocess, main muscle segmentation step and post process, and are applied on three types of data: Magnetic Resonance Imaging (MRI) data, Computerized Tomography (CT) data and unified data, in which case, information coming from both modalities are utilized. The methods are applied both in three dimensions (3D) and two dimensions (2D) data cases. A simulation data and two patient data are utilized for tests. The patient data results are compared statistically with ground truth data which was labeled by an expert radiologist.

Optimal control of PDEs has a crucial place in many parts of sciences and industry. Over the last decade, there have been a great deal in, especially, control problems of elliptic problems. Optimal control problems of Burgers equation that is as a simplifed model for turbulence and in shock waves were recently investigated both theoretically and numerically. In this thesis, we analyze the space-time simultaneous discretization of control problem for Burgers equation. In literature, there have been two approaches for discretization of optimization problems: optimize-then-discretize and discretize-then-optimize. In the first part, we follow optimize-then-discretize appoproach. It is shown that both distributed and boundary time dependent control problem can be transformed into an elliptic pde. Numerical results obtained with adaptive and non-adaptive elliptic solvers of COMSOL Multiphysics are presented for both the unconstrained and the control constrained cases. As for second part, we consider discretize-then-optimize approach. Discrete adjoint concept is covered. Optimality conditions, KKT-system, lead to a saadle point problem. We investigate the numerical treatment for the obtained saddle point system. Both direct solvers and iterative methods are considered. For iterative mehods, preconditioners are needed. The structures of preconditioners for both distributed and boundary control problems are covered. Additionally, an a priori error analysis for the distributed control problem is given. We present the numerical results at the end of each chapter.

It is obvious that differential equations can be used to model real-life problems. Although it is possible to obtain analytical solutions of some of them, it is in general difficult to find closed form solutions of differential equations. Finding thus approximate solutions has been the subject of many researchers from different areas. In this thesis, we propose a new approach to Variational Iteration Method (VIM) to obtain the solutions of systems of first-order differential equations. The main contribution of the thesis to VIM is that proposed approach uses restricted variations only for the nonlinear terms and builds up a matrix-valued Lagrange multiplier that leads to the extension of the method (EVIM). Close relation between the matrix-valued Lagrange multipliers and fundamental solutions of the differential equations highlights the relation between the extended version of the variational iteration method and the classical variation of parameters formula. It has been proved that the exact solution of the initial value problems for (nonhomogenous) linear differential equations can be obtained by such a generalisation using only a single variational step. Since higher-order equations can be reduced to first-order systems, the proposed approach is capable of solving such equations too / indeed, without such a reduction, variational iteration method is also extended to higher-order scalar equations. Further, the close connection with the associated first-order systems is presented. Such extension of the method to higher-order equations is then applied to solve boundary value problems: linear and nonlinear ones. Although the corresponding Lagrange multiplier resembles the Green&rsquo / s function, without the need of the latter, the extended approach to the variational iteration method is systematically applied to solve boundary value problems, surely in the nonlinear case as well. In order to show the applicability of the method, we have applied the EVIM to various real-life problems: the classical Sturm-Liouville eigenvalue problems, Brusselator reaction-diffusion, and chemical master equations. Results show that the method is simple, but powerful and effective.

In this study, the application of a Monte Carlo method (MCM) for radiative heat transfer in three-dimensional rectangular enclosures was investigated. The study covers the development of the method from simple surface exchange problems to enclosure problems containing absorbing, emitting and isotropically/anisotropically scattering medium. The accuracy of the MCM was first evaluated by applying the method to cubical enclosure problems. The first one of the cubical enclosure problems was prediction of radiative heat flux vector in a cubical enclosure containing purely, isotropically and anisotropically scattering medium with non-symmetric boundary conditions. Then, the prediction of radiative heat flux vector in an enclosure containing absorbing, emitting, isotropically and anisotropically scattering medium with symmetric boundary conditions was evaluated. The predicted solutions were compared with the solutions of method of lines solution (MOL) of discrete ordinates method (DOM). The method was then applied to predict the incident heat fluxes on the freeboard walls of a bubbling fluidized bed combustor, and the solutions were compared with those of MOL of DOM and experimental measurements. Comparisons show that MCM provides accurate and computationally efficient solutions for modelling of radiative heat transfer in 3-D rectangular enclosures containing absorbing, emitting and scattering media with isotropic and anisotropic scattering properties.

An expansion method for the stationary Schrodinger equation of a particle moving freely in an arbitrary axisymmeric three dimensional region defined by an analytic function is introduced. The region is transformed into the unit ball by means of coordinate substitution. As a result the Schrodinger equation is considerably changed. The wavefunction is expanded into a series of spherical harmonics, thus, reducing the transformed partial differential equation to an infinite system of coupled ordinary differential equations. A Fourier-Bessel expansion of the solution vector in terms of Bessel functions with real orders is employed, resulting in a generalized matrix eigenvalue problem. The method is applied to two particular examples. The first example is a prolate spheroidal billiard which is also treated by using an alternative method. The numerical results obtained by using both the methods are compared. The second exampleis a billiard family depending on a parameter. Numerical results concerning the second example include the statistical analysis of the eigenvalues.

In this thesis, fatigue life of a stationary component of F110-GE-100 jet engine is assessed. Three-dimensional finite element model of the component itself and the neighboring components are modeled by using a finite element package program, ANSYS, in order to perform thermal, stress and fracture mechanics analyses. Coupled-field (thermal-stress) analysis is performed to identify fracture-critical locations and to describe the stress histories of the components. After determining the critical location, fracture mechanics calculations are performed by modeling a crack of various lengths at the critical locations with FRANC3D in order to calculate mode I and II stress intensity factors and geometry factors beta. Combining the outputs of coupled-field and fracture mechanics analyses, fatigue lives and creep rupture times are calculated with a crack growth life prediction program, AFGROW. A linear damage summation method is used to assess the fatigue life of the component of interest.

This thesis analyzes the probable outcome of the fictitious failure of a dam under a set of pre-defined scenarios, within the framework of a case study, the case subject being the Cinarcik Dam located within Bursa Province of Turkey. The failure of the dam is not analyzed neither structural nor hydraulic-wise but is assumed to be triggered when certain critical criteria are exceeded. Hence, the analyses focus on the aftermath of the failure and strive to anticipate the level of inundation downstream of the dam itself. For the purpose of the analyses, the FLDWAV software developed by the National Weather Service of USA is used to spatially and temporally predict the flow profiles, water surface elevations and discharges occurring downstream of the &Ccedil / inarcik Dam under the defined set of scenarios. Based on these analyses, indicative inundation maps and settlements under risk will be identified, and the thesis study will further address some available pre-event measures that may be taken in advance.

In this thesis, Finite Volume Weighted Essentially Non-Oscillatory (FV-WENO) codes for one and two-dimensional discretised Euler equations are developed. The construction and application of the FV-WENO scheme and codes will be described. Also the effects of the grid coefficients as well as the effect of the Gaussian Quadrature on the solution have been tested and discussed. WENO schemes are high order accurate schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the high approximation level, where a convex combination of all the candidate stencils is used with certain weights. Those weights are used to eliminate the stencils, which contain discontinuity. WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures. The applications tested in this thesis are the Diverging Nozzle, Shock Vortex Interaction, Supersonic Channel Flow, Flow over Bump, and supersonic Staggered Wedge Cascade. The numerical solutions for the diverging nozzle and the supersonic channel flow are compared with the analytical solutions. The results for the shock vortex interaction are compared with the Roe scheme results. The results for the bump flow and the supersonic staggered cascade are compared with results from literature.