This thesis presents and develops efficient and effective methodologies for optimal hydraulic fracture design and fracture performance evaluation. These methods incorporate algorithms that simultaneously optimize all of the treatment parameters while accounting for required constraints. Damage effects, such as closure stress, gel damage and non-Darcy flow, are also considered in the optimal design and evaluation algorithms. Two user-friendly program modules, which are active server page (ASP) based, were developed to implement the utility of the methodologies. Case analysis was executed to demonstrate the workflow of the two modules. Finally, to validate the results from the two modules, results were compared to those from a 3D simulation program. The main contributions of this work are: An optimal fracture design methodology called unified fracture design (UFD) is presented and damage effects are considered in the optimal design calculation. As a by-product of UFD, a fracture evaluation methodology is proposed to conduct well stimulation performance evaluation. The approach is based on calculating and comparing the actual dimensionless productivity index of fractured wells with the benchmark which has been developed for optimized production. To implement the fracture design and evaluation methods, two web ASP based user interfaces were developed; one is called Frac Design (Screening), and the other is Frac Evaluation. Both modules are built to hold the following features. o Friendly web ASP based user interface o Minimum user input o Proppant type and mesh size selection o Damage effects consideration options o Convenient on-line help.

This thesis represents an attempt to generalize the classical Theory of Optimal Design to popular regression models, based on Rational and Spline approximations. The problem of finding optimal designs for such models can be reduced to solving certain minimax problems. Explicit solutions to such problems can be obtained only in a few selected models, such as polynomial regression. Even when an optimal design can be found, it has, from the point of view of modern nonparametric regression, certain drawbacks. For example, in the polynomial regression case, the optimal design crucially depends on the degree m of approximating polynomial. Hence, it can be used only when such degree is given/known in advance. We present a partial, but practical, solution to this problem. Namely, the so-called Super Chebyshev Design has been found, which does not depend on the degree m of the underlying polynomial regression in a large range of m, and at the same time is asymptotically more than 90% efficient. Similar results are obtained in the case of rational regression, even though the exact form of optimal design in this case remains unknown. Optimal Designs in the case of Spline Interpolation are also currently unknown. This problem, however, has a simple solution in the case of Cardinal Spline Interpolation. Until recently, this model has been practically unknown in modern nonparametric regression. We demonstrate the usefulness of Cardinal Kernel Spline Estimates in nonparametric regression, by proving their asymptotic optimality, in certain classes of smooth functions. In this way, we have found, for the first time, a theoretical justification of a well known empirical observation, by which cubic splines suffice, in most practical applications. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-07-18 16:06:06.767

An aneurysm is a balloon-like bulge in the wall of blood vessels, occurring in major arteries from the heart and brain. Biodegradable stent-assisted coiling is expected to be the ideal treatment of wide-neck complex aneurysms. A number of biodegradable stents are promising, but also with issues and/or several limitations to be addressed. From the design point of view, biodegradable stents are typically designed without structure optimization. The drawbacks of these stents often cause weaker mechanical properties than native arterial vessels. From the fabrication point of view, the conventional methods of the fabricating stent are time-consuming and expensive, and also lack precise control over the stent microstructure. As an emerging fabrication technique, dispensing-based rapid prototyping (DBRP) allows for more accurate control over the scaffold microstructure, thus facilitating the fabrication of stents as designed. This thesis is aimed at developing methods for fabrication and optimal design of biodegradable stents for treating aneurysms. Firstly, a method was developed to fabricate biodegradable stents by using the DBRP technique. Then, a compression test was carried out to characterize the radial deformation of the stents fabricated. The results illustrated the stent with a zigzag structure has a higher radial stiffness than the one with a coil structure. On this basis, the stent with a zigzag structure was chosen to develop a finite element model for simulating the real compression tests. The result showed the finite element model of biodegradable stents is acceptable within a range of radial deformation around 20%. Furthermore, an optimization of the zigzag structure was performed with ANSYS DesignXplorer, and the results indicated that the total deformation could be decreased by 35% by optimizing the structure parameters, which would represent a significant advance of the radial stiffness of biodegradable stents. Finally, the optimized stent was used to investigate its deformation in a blood vessel. The deformation is found to be 0.25 mm in the simulation, and the rigidity of biodegradable stents is 7.22%, which is able to support the blood vessel all. It is illustrated that the finite element analysis indeed helps in designing stents with new structures and therefore improved mechanical properties.

For current status data, the failure time of interest may not be observed exactly. The type of this data consists only of a monitoring time and knowledge of whether the failure time occurred before or after the monitoring time. In order to be able to obtain more information from this data, so the monitoring time is very important. In this work, the optimal designs for determining the monitoring times such that maximum information may be obtained in bivariate copula model (Clayton) are investigated. Here, the D- optimal criterion is used to decide the best monitoring time Ci (i = 1; ¢ ¢ ¢ ; n), then use these monitoring times Ci to estimate the unknown parameters simultaneously by maximizing the corresponding likelihood function. Ds-optimal designs for estimation of association parameter in the copula model are also discussed. Simulation studies are presented to compare the performance of using monitoring time C¤D and C¤Ds to do the estimation.

Consider a linear regression model with a two-dimensional control vector (x_1, x_2) and an m-dimensional response vector y = (y_1, . . . , y_m). The components of y are correlated with a known covariance matrix. Based on the assumed regression model, there are two problems of interest. The first one is to estimate unknown control vector x_c corresponding to an observed y, where xc will be estimated by the classical estimator. The second one is to obtain a suitable estimation of the control vector x_T corresponding to a given target T = (T_1, . . . , T_m) on the expected responses. Consideration in this work includes the deviation of the expected response E(y_i) from its corresponding target value T_i for each component and defines the optimal control vector x, say x_T , to be the one which minimizes the weighted sum of squares of standardized deviations within the range of x. The objective of this study is to find c-optimal designs for estimating x_c and x_T , which minimize the mean squared error of the estimator of xc and x_T respectively. The comparison of the difference between the optimal calibration design and the optimal design for estimating x_T is provided. The efficiencies of the optimal calibration design relative to the uniform design are also presented, and so are the efficiencies of the optimal design for given target vector relative to the uniform design.

Bayesian optimal designs have received increasing attention in recent years, especially in biomedical and clinical trials. Bayesian design procedures can utilize the available prior information of the unknown parameters so that a better design can be achieved. However, a difficulty in dealing with the Bayesian design is the lack of efficient computational methods. In this research, a hybrid computational method, which consists of the combination of a rough global optima search and a more precise local optima search, is proposed to efficiently search for the Bayesian D-optimal designs for multi-variable generalized linear models. Particularly, Poisson regression models and logistic regression models are investigated. Designs are examined for a range of prior distributions and the equivalence theorem is used to verify the design optimality. Design efficiency for various models are examined and compared with non-Bayesian designs. Bayesian D-optimal designs are found to be more efficient and robust than non-Bayesian D-optimal designs. Furthermore, the idea of the Bayesian sequential design is introduced and the Bayesian two-stage D-optimal design approach is developed for generalized linear models. With the incorporation of the first stage data information into the second stage, the two-stage design procedure can improve the design efficiency and produce more accurate and robust designs. The Bayesian two-stage D-optimal designs for Poisson and logistic regression models are evaluated based on simulation studies. The Bayesian two-stage optimal design approach is superior to the one-stage approach in terms of a design efficiency criterion. / Ph. D.

Doctor of Philosophy / Department of Statistics / Shie-Shien Yang / When the primary research interest is in the marginal dependence between the response and the covariates, logistic GEE (Generalized Estimating Equation) models are often used to analyze clustered binary data. Relative to ordinary logistic regression, very little work has been done to assess the lack of fit of a logistic GEE model. A new method addressing the LOF of a logistic GEE model was proposed. Simulation results indicate the proposed method performs better than or as well as other currently available LOF methods for logistic GEE models. A SAS macro was developed to implement the proposed method. Nonlinear regression models are widely used in medical science. Before the models can be fit and parameters interpreted, researchers need to decide which design points in a prespecified design space should be included in the experiment. Careful choices at this stage will lead to efficient usage of limited resources. We proposed a cost efficient Bayesian optimal design method for nonlinear combinations of parameters in a nonlinear model with quantitative predictors. An R package was developed to implement the proposed method.

In this paper we study the loss of Fisher information in approximating a continous distribution by a multinominal distribution coming from a partition of the sample space into a finite number of intervals. We describe and characterize the Fisher information as a function of the partition chosen especially for location parameters. For a small number of intervals the consequences of the choice is demonstrated by instructive examples. For increasing number of individuals we give the asymptotically optimal partition. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Consider the multiresponse polynomial regression model with one control variable and arbitrary covariance matrix among responses. The present results complement solutions by Krafft and Schaefer (1992) and Imhof (2000), who obtained the n-point D-optimal designs for the multiresponse regression model with one linear and one quadratic. We will show that the D-optimal design is invariant under linear transformation of the control variable. Moreover, the most cases of the exact D-optimal designs on [-1,1] for responses consisting of linear and quadratic polynomials only are derived. The efficiency of the exact D-optimal designs for the univariate quadratic model to that for the above model are also discussed. Some conjectures based on intensively numerical results are also included.

The D-optimal design problems in polynomial regression models with a one-dimensional control variable and k-dimensional response variable Y=(Y_1,...,Y_k) where there are some common unknown parameters are discussed. The approximate D-optimal designs are shown to be independent of the covariance structure between the k responses when the degrees of the k responses are of the same order. Then, the exact n-point D-optimal designs are also discussed. Krafft and Schaefer (1992) and Imhof (2000) are useful in obtaining our results. We extend the proof of symmetric cases for k>= 2.