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.

This thesis concerns the approximation of optimally controlled partial differential equations for inverse problems in optimal design. Important examples of such problems are optimal material design and parameter reconstruction. In optimal material design the goal is to construct a material that meets some optimality criterion, e.g. to design a beam, with fixed weight, that is as stiff as possible. Parameter reconstrucion concerns, for example, the problem to find the interior structure of a material from surface displacement measurements resulting from applied external forces. Optimal control problems, particularly for partial differential equations, are often ill-posed and need to be regularized to obtain good approximations. We here use the theory of the corresponding Hamilton-Jacobi-Bellman equations to construct regularizations and derive error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method where the first, analytical, step is to regularize the Hamiltonian. Next its Hamiltonian system is computed efficiently with the Newton method using a sparse Jacobian. An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L² projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems. Another treated issue is the relevance of input data for parameter reconstruction problems, where the goal is to determine a spacially distributed coefficient of a partial differential equation from partial observations of the solution. It is here shown that the choice of input data, that generates the partial observations, affects the reconstruction, and that it is possible to formulate meaningful optimality criteria for the input data that enhances the quality of the reconstructed coefficient. In the thesis we present solutions to various applications in optimal material design and reconstruction. / Denna avhandling handlar om approximation av optimalt styrda partiella differentialekvationer för inversa problem inom optimal design. Viktiga exempel på sådana problem är optimal materialdesign och parameterskattning. Inom materialdesign är målet att konstruera ett material som uppfyller vissa optimalitetsvillkor, t.ex. att konstruera en så styv balk som möjligt under en given vikt, medan ett exempel på parameterskattning är att hitta den inre strukturen hos ett material genom att applicera ytkrafter och mäta de resulterande förskjutningarna. Problem inom optimal styrning, speciellt för styrning av partiella differentialekvationer,är ofta illa ställa och måste regulariseras för att kunna lösas numeriskt. Teorin för Hamilton-Jacobi-Bellmans ekvationer används här för att konstruera regulariseringar och ge feluppskattningar till problem inom optimaldesign. Den konstruerade Pontryaginmetoden är en enkel och generell metod där det första analytiska steget är att regularisera Hamiltonianen. I nästa steg löses det Hamiltonska systemet effektivt med Newtons metod och en gles Jacobian. Vi härleder även en feluppskattning för skillnaden mellan den exakta och den approximerade målfunktionen. Denna uppskattning beror endast på skillnaden mellan den sanna och den regulariserade, ändligt dimensionella, Hamiltonianen, båda utvärderade längst lösningsbanan och dessL²-projektion. Felet beror alltså ej på skillnaden mellan den exakta och denapproximativa lösningen till det Hamiltonska systemet. Ett annat fall som behandlas är frågan hur indata ska väljas för parameterskattningsproblem. För sådana problem är målet vanligen att bestämma en rumsligt beroende koefficient till en partiell differentialekvation, givet ofullständiga mätningar av lösningen. Här visas att valet av indata, som genererarde ofullständiga mätningarna, påverkar parameterskattningen, och att det är möjligt att formulera meningsfulla optimalitetsvillkor för indata som ökar kvaliteten på parameterskattningen. I avhandlingen presenteras lösningar för diverse tillämpningar inom optimal materialdesign och parameterskattning. / QC 20100712

Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第17511号 / 工博第3670号 / 新制||工||1558(附属図書館) / 30277 / 京都大学大学院工学研究科機械理工学専攻 / (主査)教授 西脇 眞二, 教授 田畑 修, 教授 蓮尾 昌裕 / 学位規則第4条第1項該当

<p> This thesis presents an optimal design method for MRI (Magnetic Resonance Imaging) teardrop gradient waveforms in two and three dimensions. Teardrop in two dimensions was introduced at ISMRM 2001 by Anand et al. to address the need for a high efficiency balanced k-space trajectory for real-time cardiac SSFP (Steady State Free Precession) imaging.</p> <p> We have modeled 2D and 3D teardrop gradient waveform design as nonlinear convex optimization problems with a variety of constraints including global constraints (e.g., moment nulling for motion insensitivity). Commercial optimization solvers can solve the models efficiently. The implementation of AMPL models and numerical testing results with the solver MOSEK are provided. This optimal design procedure produces physically realizable teardrop waveforms which enable real-time cardiac imaging with equipment otherwise incapable of doing it, and optimally achieves the maximum resolution and motion artifact reduction goals. The research may encompass other waveform design problems in MRI and has built a good foundation for further research in this area.</p> / Thesis / Master of Science (MSc)

Microarray experiments are used to perform gene expression profiling on a large scale. E- and A-optimality of mixed designs was established for experiments with up to 26 different varieties and with the restriction that the number of arrays available is equal to the number of varieties. Because the IBD setting only allows for a single blocking factor (arrays), the search for optimal designs was extended to the Row-Column Design (RCD) setting with blocking factors dye (row) and array (column). Relative efficiencies of these designs were further compared under analysis of variance (ANOVA) models. We also compared the performance of classification analysis for the interwoven loop and the replicated reference designs under four scenarios. The replicated reference design was favored when gene-specific sample variation was large, but the interwoven loop design was preferred for large variation among biological replicates. We applied mixed model methodology to detection and estimation of gene differential expression. For identification of differential gene expression, we favor contrasts which include both variety main effects and variety by gene interactions. In terms of t-statistics for these contrasts, we examined the equivalence between the one- and two-step analyses under both fixed and mixed effects models. We analytically established conditions for equivalence under fixed and mixed models. We investigated the difference of approximation with the two-step analysis in situations where equivalence does not hold. The significant difference between the one- and two-step mixed effects model was further illustrated through Monte Carlo simulation and three case studies. We implemented the one-step analysis for mixed models with the ASREML software. / Ph. D.

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; &#x00A2; &#x00A2; &#x00A2; ; 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&#x00A4;D and C&#x00A4;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.