Surface reconstruction of anatomical structures is an integral part of medical modeling. Contour information is extracted from serial cross-sections of tissue data and is stored as "slice" files. Although there are several reasonably efficient triangulation algorithms that reconstruct surfaces from slice data, the models generated from them have a jagged or faceted appearance due to the large inter-slice distance created by the sectioning process. Moreover, inconsistencies in user input aggravate the problem. So, we created a method that reduces inter-slice distance, as well as ignores the inconsistencies in the user input. Our method called the piecewise weighted implicit functions, is based on the approach of weighting smaller implicit functions. It takes only a few slices at a time to construct the implicit function. This method is based on a technique called variational interpolation. <p> Other approaches based on variational interpolation have the disadvantage of becoming unstable when the model is quite large with more than a few thousand constraint points. Furthermore, tracing the intermediate contours becomes expensive for large models. Even though some fast fitting methods handle such instability problems, there is no apparent improvement in contour tracing time, because, the value of each data point on the contour boundary is evaluated using a single large implicit function that essentially uses all constraint points. Our method handles both these problems using a sliding window approach. As our method uses only a local domain to construct each implicit function, it achieves a considerable run-time saving over the other methods. The resulting software produces interpolated models from large data sets in a few minutes on an ordinary desktop computer.

We study computational theory and methods for finding multiple unstable solutions (corresponding to saddle points) to three types of nonlinear variational elliptic systems: cooperative, noncooperative, and Hamiltonian. We first propose a new Lorthogonal selection in a product Hilbert space so that a solution manifold can be defined. Then, we establish, respectively, a local characterization for saddle points of finite Morse index and of infinite Morse index. Based on these characterizations, two methods, called the local min-orthogonal method and the local min-max-orthogonal method, are developed and applied to solve those three types of elliptic systems for multiple solutions. Under suitable assumptions, a subsequence convergence result is established for each method. Numerical experiments for different types of model problems are carried out, showing that both methods are very reliable and efficient in computing coexisting saddle points or saddle points of infinite Morse index. We also analyze the instability of saddle points in both single and product Hilbert spaces. In particular, we establish several estimates of the Morse index of both coexisting and non-coexisting saddle points via the local min-orthogonal method developed and propose a local instability index to measure the local instability of both degenerate and nondegenerate saddle points. Finally, we suggest two extensions of an L-orthogonal selection for future research so that multiple solutions to more general elliptic systems such as nonvariational elliptic systems may also be found in a stable way.

Principal components analysis (PCA) has been widely used as a statistical tool for the dimension reduction of multivariate data in various application areas and extensively studied in the long history of statistics. One of the limitations of PCA machinery is that PCA can be applied only to the continuous type variables. Recent advances of information technology in various applied areas have created numerous large diverse data sets with a high dimensional feature space, including high dimensional binary data. In spite of such great demands, only a few methodologies tailored to such binary dataset have been suggested. The methodologies we developed are the model-based approach for generalization to binary data. We developed a statistical model for binary PCA and proposed two stable estimation procedures using MM algorithm and variational method. By considering the regularization technique, the selection of important variables is automatically achieved. We also proposed an efficient algorithm for model selection including the choice of the number of principal components and regularization parameter in this study.

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 paper, we establish several strong convergence results of general approximate proximal algorithm and general Bregman-function-based approximate proximal algorithm for solving the generalized variational inequality problem with pseudomonotone multifunction.

This dissertation developed a method that can accurately and efficiently capture the response of a structure by rigorous combination of a reduced-dimensional beam finite element model with a model based on full two-dimensional (2D) or three-dimensional (3D) finite elements. As a proof of concept, a joint 2D-beam approach is studied for planar-inplane deformation of strip-beams. This approach is developed for obtaining understanding needed to do the joint 3D-beam model. A Matlab code is developed to solve achieve this 2D-beam approach. For joint 2D-beam approach, the static response of a basic 2D-beam model is studied. The whole beam structure is divided into two parts. The root part where the boundary condition is applied is constructed as a 2D model. The free end part is constructed as a beam model. To assemble the two different dimensional model, a transformation matrix is used to achieve deflection continuity or load continuity at the interface. After the transformation matrix from deflection continuity or from load continuity is obtained, the 2D part and the beam part can be assembled together and solved as one linear system. For a joint 3D-beam approach, the static and dynamic response of a basic 3D-beam model is studied. A Fortran program is developed to achieve this 3D-beam approach. For the uniform beam constrained at the root end, similar to the joint 2D-beam analysis, the whole beam structure is divided into two parts. The root part where the boundary condition is applied is constructed as a 3D model. The free end part is constructed as a beam model. To assemble the two different dimensional models, the approach of load continuity at the interface is used to combine the 3D model with beam model. The load continuity at the interface is achieved by stress recovery using the variational-asymptotic method. The beam properties and warping functions required for stress recovery are obtained from VABS constitutive analysis. After the transformation matrix from load continuity is obtained, the 3D part and the beam part can be assembled together and solved as one linear system. For a non-uniform beam example, the whole structure is divided into several parts, where the root end and the non-uniform parts are constructed as 3D models and the uniform parts are constructed as beams. At all the interfaces, the load continuity is used to connect 3D model with beam model. Stress recovery using the variational-asymptotic method is used to achieve the load continuity at all interfaces. For each interface, there is a transformation matrix from load continuity. After we have all the transformation matrices, the 3D parts and the beam parts are assembled together and solved as one linear system.

Steuerungsprobleme treten in vielen Anwendungen in Naturwissenschaft und Technik auf. In dieser Arbeit werden Optimalsteuerungsprobleme mit semilinearen elliptischen partiellen Differentialgleichungen als Nebenbedingungen untersucht. Die Kontrolle wird durch Kontrollschranken als Ungleichungsnebenbedingungen eingeschränkt. Dabei ist die Zielfunktion quadratisch in der Kontrolle. Die Lösung des Optimierungsproblems kann dann durch die Projektionsbedingung mit Hilfe des adjungierten Zustandes dargestellt werden. Ein neuer Zugang ist die variationelle Diskretisierung. Bei dieser wird nur der Zustand und der adjungierte Zustand diskretisiert, nicht aber der Raum der Kontrollen. Dieser Zugang erlaubt höhere Konvergenzraten für die Kontrolle für kontrollrestingierte Probleme als bei einer Diskretisierung des Kontrollraumes. Die Projektionsbedingung für das variationell diskretisierte Problem ist dabei auf die gleiche zulässige Menge wie beim nicht diskretisierten Problem. In der vorliegenden Arbeit wird die Methode der variationellen Diskretisierung auf semilineare elliptische Optimalkontrollprobleme angewendet und Fehlerabschätzungen für die Kontrollen bewiesen. Dabei wird hauptsächlich auf die verteilte Steuerung Wert gelegt, aber auch die Neumann-Randsteuerung mitbehandelt. Nach einem Überblick über die Literatur wird die Aufgabenstellung mit den Voraussetzungen aufgeschrieben und die Optimalitätsbedingungen angegeben. Danach wird die Existenz einer Lösung, sowie die Konvergenz der diskreten Lösungen gegen eine kontinuierliche Lösung gezeigt. Außerdem werden Finite-Elemente-Konvergenzordnungen angegeben. Dann werden optimale Fehlerabschätzungen in verschiedenen Normen für die variationelle Kontrolle bewiesen. Insbesondere werden die Fehlerabschätzung in Abhängigkeit vom Finite-Elemente-Fehler des Zustandes und des adjungierten Zustandes angegeben. Dabei wird die nichtlineare Fixpunktgleichung mittels semismooth Newtonverfahrens linearisiert. Das Newtonverfahren wird auch für die numerische Lösung des Problems eingesetzt. Die Voraussetzung für die Konvergenzordnung ist dabei nicht die SSC, die hinreichende Bedingung zweiter Ordnung, welche eine lokale Konvexität in der Zielfunktion impliziert, sondern die Invertierbarkeit des Newtonoperators. Dies ist eine stationäre Bedingung in der optimalen Kontrolle. Dabei wird nur benötigt, dass der Rand der aktiven Menge eine Nullmenge ist und die Invertierbarkeit des Newtonoperators in der Optimallösung. Der Schaudersche Fixpunktsatz wird benutzt, um für die Newtongleichung die Existenz eines Fixpunktes innerhalb der gewünschten Umgebung zu beweisen. Außerdem wird die Eindeutigkeit eines solchen Fixpunktes für eine gegebene Triangulation bei hinreichend feiner Diskretisierung gezeigt. Das Ergebnis ist, dass die Konvergenzrate nur durch die Finite-Elemente-Konvergenzraten von Zustand und adjungiertem Zustand beschränkt wird. Diese Rate wird nicht nur durch die Ansatzfunktionen, sondern auch durch die Glattheit der rechten Seite beschränkt, so dass der Knick am Rand der aktiven Menge hier ein Grenze setzt. Außerdem wird die Implementation des semismooth Newtonverfahrens für den unendlichdimensionalen Kontrollraum für die variationelle Diskretisierung erläutert. Dabei wird besonders auf den zweidimensionalen verteilten Fall eingegangen. Es werden die bewiesenen Konvergenzraten an einigen semilinearen und linearen Beispielen mittels der variationellen Diskretisierung demonstriert. Es entsprechen sich die bei den analytische Beweisen und der numerischen Lösung eingesetzten Verfahren, die Fixpunktiteration sowie das nach Kontrolle oder adjungiertem Zustand aufgelöste Newtonverfahren. Dabei sind einige Besonderheiten bei der Implementation zu beachten, beispielsweise darf die Kontrolle nicht inkrementell mit dem Newtonverfahren oder der Fixpunktiteration aufdatiert werden, sondern muss in jedem Schritt neu berechnet werden.

In this thesis, we use multitype Galton-Watson branching processes in random environments as individual-based models for the evolution of structured populations with both demographic stochasticity and environmental stochasticity, and investigate the phenotype allocation problem. We explore a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. When the population under consideration is large and the time scale is fast, we deduce the continuous approximation for multitype Markov branching processes in random environments. Many problems in evolutionary biology involve the allocation of some limited resource among several investments. It is often of interest to know whether, and how, allocation strategies can be optimized for the evolution of a structured population with randomness. In our work, the investments represent different types of offspring, or alternative strategies for allocations to offspring. As payoffs we consider the long-term growth rate, the expected number of descendants with some future discount factor, the extinction probability of the lineage, or the expected survival time. Two different kinds of population randomness are considered: demographic stochasticity and environmental stochasticity. In chapter 2, we solve the allocation problem w.r.t. the above payoff functions in three stochastic population models depending on different kinds of population randomness. Evolution is often understood as an optimization problem, and there is a long tradition to look at evolutionary models from a variational perspective. In chapter 3, we deduce a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process. Continuous approximation of branching processes is of both practical and theoretical interest. However, to our knowledge, there is no literature on approximation of multitype branching processes in random environments. In chapter 4, we firstly construct a multitype Markov branching process in a random environment. When conditioned on the random environment, we deduce the Kolmogorov equations and the mean matrix for the conditioned branching process. Then we introduce a parallel mutation-selection Markov branching process in a random environment and analyze its instability property. Finally, we deduce a weak convergence result for a sequence of the parallel Markov branching processes in random environments and give examples for applications.

This research seeks to expand the role of variational and adjoint processing methods into segments of the sonar, radar, and nondestructive testing communities where they have not yet been widely introduced. First, synthetic aperture reconstruction is expressed in terms of the adjoint operator. Many, if not all, practical imaging modalities can be traced back to this general result, as the adjoint is the foundation for backprojection-type algorithms. Next, active surfaces are developed in the context of the Helmholtz equation for the cases of opaque scatterers (i.e., with no interior field) embedded in free space, and penetrable scatterers embedded in a volume which may be bounded. The latter are demonstrated numerically using closed-form solutions based on spherical harmonics. The former case was chosen as the basis for a laboratory experiment using Lamb waves in an aluminum plate. Lamb wave propagation in plates is accurately described by the Helmholtz equation, where the field quantity is the displacement potential. However, the boundary conditions associated with the displacement potential formulation of Lamb waves are incompatible with the shape gradient derived for the Helmholtz equation, except for very long or very short wavelengths. Lastly, optical flow is used to solve a new and unique problem in the field of synthetic aperture sonar. Areas of acoustic focusing and dilution attributable to refraction can sometimes resemble the natural bathymetry of the ocean floor. The difference is often visually indistinguishable, so it is desirable to have a means of detecting these transient refractive effects without having to repeat the survey. Optical flow proved to be effective for this purpose, and it is shown that the parameters used to control the algorithm can be linked to known properties of the data collection and scattering physics.

Results from models that describe frequency chirps of toroidal Alfvén eigenmode excited by energetic particles are presented here. This structure forms in TAE gap and may or may not chirp into the continuum. Initial work described the particle wave interaction in terms of a generic Hamiltonian for the particle wave interaction, whose spatial dependence was xed in time. In addition, we have developed an improved adiabatic TAE model that takes into account the spatial prole variation of the mode and the nite orbit excursion from the resonant ux surfaces, for a wide range of toroidal mode numbers. We have shown for the generic xed prole model that the results from the adiabatic model agree very well with simulation result except when the adiabatic condition breaks down due to the rapid variations of the wave amplitude and chirping frequency. We have been able to solve the adiabatic problem in the case when the spatial prole is allowed to vary in time, in accord with the structure of the response functions, as a function of frequency. All the models predict that up-chirping holes do not penetrate into the continuum. On the other hand clump structures, which down chirp in frequency may, depending on detailed parameters, penetrate the continuum. The systematic theory is more restrictive than the generic theory, for the conditions that enable clump to penetrate into the continuum. In addition, the systematic theory predicts an important nite drift orbit width eect, which eventually limits and suppresses a down-chirping response in the lower continuum. This interruption of the chirping occurs when the trapped particles make a transition from intersecting both resonant points of the continuum to just one resonant point. / text