A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations Scott W. Mosher 110 Pages Directed by Dr. Farzad Rahnema It seems very likely that the next generation of reactor analysis methods will be based largely on neutron transport theory, at both the assembly and core levels. Signifi-cant progress has been made in recent years toward the goal of developing a transport method that is applicable to large, heterogeneous coarse-meshes. Unfortunately, the ma-jor obstacle hindering a more widespread application of transport theory to large-scale calculations is still the computational cost. In this dissertation, a variational heterogeneous coarse-mesh transport method has been extended from one to two-dimensional Cartesian geometry in a practical fashion. A generalization of the angular flux expansion within a coarse-mesh was developed. This allows a far more efficient class of response functions (or basis functions) to be employed within the framework of the original variational principle. New finite element equations were derived that can be used to compute the expansion coefficients for an individual coarse-mesh given the incident fluxes on the boundary. In addition, the non-variational method previously used to converge the expansion coefficients was developed in a new and more thorough manner by considering the implications of the fission source treat-ment imposed by the response expansion. The new coarse-mesh method was implemented for both one and two-dimensional (2-D) problems in the finite-difference, multigroup, discrete ordinates approximation. An efficient set of response functions was generated using orthogonal boundary conditions constructed from the discrete Legendre polynomials. Several one and two-dimensional heterogeneous light water reactor benchmark problems were studied. Relatively low-order response expansions were used to generate highly accurate results using both the variational and non-variational methods. The expansion order was found to have a far more significant impact on the accuracy of the results than the type of method. The varia-tional techniques provide better accuracy, but at substantially higher computational costs. The non-variational method is extremely robust and was shown to achieve accurate re-sults in the 2-D problems, as long as the expansion order was not very low.

In this paper, we report existing convergence results on monotone variational inequalities where the governing monotone operators are either strongly monotone or inverse strongly monotone. We reformulate the variational inequality problem as an equivalent fixed point problem and then use fixed point iteration method to solve the original variational inequality problem. In the case of strong monotonicity case we use the Banach¡¦s contraction principle to define out iteration sequence; while in the case of inverse strong monotonicity we use the technique of averaged mappings to define our iteration sequence. In both cases we prove strong convergence for our iteration methods. An application to a minimization problem is also included.

A novel video observational method for the space-time stereoscopic reconstruction of dynamic surfaces representable as graphs, such as ocean waves, is developed. Variational optimization algorithms combining image processing, computer vision and partial differential equations are designed to address the problem of the recovery of the shape of an object's surface from sequences of synchronized multi-view images. Several theoretical and numerical paths are discussed to solve the problem. The variational stereo method developed in this thesis has several advantages over existing 3-D reconstruction algorithms. Our method follows a top-down approach or object-centered philosophy in which an explicit model of the target object in the scene is devised and then related to image measurements. The key advantages of our method are the coherence (smoothness) of the reconstructed surface caused by a coherent object-centered design, the robustness to noise due to a generative model of the observed images, the ability to handle surfaces with smooth textures where other methods typically fail to provide a solution, and the higher resolution achieved due to a suitable graph representation of the object's surface. The method provides competitive results with respect to existing variational reconstruction algorithms. However, our method is based upon a simplified but complete physical model of the scene that allows the reconstruction process to include physical properties of the object's surface that are otherwise difficult to take into account with existing reconstruction algorithms. Some initial steps are taken toward incorporating the physics of ocean waves in the stereo reconstruction process. The developed method is applied to empirical data of ocean waves collected at an off-shore oceanographic platform located off the coast of Crimea, Ukraine. An empirically-based physical model founded upon current ocean engineering standards is used to validate the results. Our findings suggest that this remote sensing observational method has a broad impact on off-shore engineering to enrich the understanding of sea states, enabling improved design of off-shore structures. The exploration of ways to incorporate dynamical properties, such as the wave equation, in the reconstruction process is discussed for future research.

The adoption of plug-in electric vehicles (PEV) requires research for models and algorithms tracing the vehicle assignment incorporating PEVs in the transportation network so that the traffic pattern can be more precisely and accurately predicted. To attain this goal, this dissertation is concerned with developing new formulations for modeling travelling behavior of electric vehicle drivers in a mixed flow traffic network environment. Much of the work in this dissertation is motivated by the special features of PEVs (such as range limitation, requirement of long electricity-recharging time, etc.), and the lack of tools of understanding PEV drivers traveling behavior and learning the impacts of charging infrastructure supply and policy on the network traffic pattern. The essential issues addressed in this dissertation are: (1) modeling the spatial choice behavior of electric vehicle drivers and analyzing the impacts from electricity-charging speed and price; (2) modeling the temporal and spatial choices behavior of electric vehicle drivers and analyzing the impacts of electric vehicle range and penetration rate; (3) and designing the optimal charging infrastructure investments and policy in the perspective of revenue management. Stochastic traffic assignment that can take into account for charging cost and charging time is first examined. Further, a quasi-dynamic stochastic user equilibrium model for combined choices of departure time, duration of stay and route, which integrates the nested-Logit discrete choice model, is formulated as a variational inequality problem. An extension from this equilibrium model is the network design model to determine an optimal charging infrastructure capacity and pricing. The objective is to maximize revenue subject to equilibrium constraints that explicitly consider the electric vehicle drivers’ combined choices behavior. The proposed models and algorithms are tested on small to middle size transportation networks. Extensive numerical experiments are conducted to assess the performance of the models. The research results contain the author’s initiative insights of network equilibrium models accounting for PEVs impacted by different scenarios of charging infrastructure supply, electric vehicles characteristics and penetration rates. The analytical tools developed in this dissertation, and the resulting insights obtained should offer an important first step to areas of travel demand modeling and policy making incorporating PEVs. / text

In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero. We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.

This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum. / Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus.

Rotor (helicopter/wind turbine) blades are typically slender structures that can be modeled as beams. Beam modeling, however, involves a substantial mathematical formulation that ultimately helps save computational costs. A beam theory for rotor blades must account for (i) initial twist and/or curvature, (ii) inclusion of composite materials, (iii) large displacements and rotations; and be capable of offering significant computational savings compared to a non-linear 3D FEA (Finite Element Analysis). The mathematical foundation of the current effort is the Variational Asymptotic Method (VAM), which is used to rigorously reduce the 3D problem into a 1D or beam problem, i.e., perform a cross-sectional analysis, without any ad hoc assumptions regarding the deformation. Since its inception, the VAM based cross-sectional analysis problem has been in a constant state of flux to expand its horizons and increase its potency; and this is precisely the target at which the objectives of this work are aimed. The problems addressed are the stress-strain-displacement recovery for spanwise non-uniform beams, analytical verification studies for the initial curvature effect, higher fidelity stress-strain-displacement recovery, oblique cross-sectional analysis, modeling of thin-walled beams considering the interaction of small parameters and the analysis of plates of variable thickness. The following are the chief conclusions that can be drawn from this work: 1. In accurately determining the stress, strain and displacement of a spanwise non-uniform beam, an analysis which accounts for the tilting of the normal and the subsequent modification of the stress-traction boundary conditions is required. 2. Asymptotic expansion of the metric tensor of the undeformed state and its powers are needed to capture the stiffnesses of curved beams in tune with elasticity theory. Further improvements in the stiffness matrix can be achieved by a partial transformation to the Generalized Timoshenko theory. 3. For the planar deformation of curved laminated strip-beams, closed-form analytical expressions can be generated for the stiffness matrix and recovery; further certain beam stiffnesses can be extracted not only by a direct 3D to 1D dimensional reduction, but a sequential dimensional reduction, the intermediate being a plate theory. 4. Evaluation of the second-order warping allows for a higher fidelity extraction of stress, strain and displacement with negligible additional computational costs. 5. The definition of a cross section has been expanded to include surfaces which need not be perpendicular to the reference line. 6. Analysis of thin-walled rotor blade segments using asymptotic methods should consider a small parameter associated with the wall thickness; further the analysis procedure can be initiated from a laminated shell theory instead of 3D. 7. Structural analysis of plates of variable thickness involves an 8×8 plate stiffness matrix and 3D recovery which explicitly depend on the parameters describing the thickness, in contrast to the simplistic and erroneous approach of replacing the thickness by its variation.

This thesis addresses the problem of inference in factor graphs, especially the LDPC codes, almost solved by message-passing algorithms. In particular, the Belief Propagation algorithm (BP) is investigated as a particular message-passing algorithm whose suboptimality is discussed in the case where the factor graph has a loop-like topology. From the equivalence between the BP and the Bethe approximation in statistical physics that is generalized to the region-based approximation, is detailed the Generalized Belief Propagation algorithm (GBP), a message-passing algorithm between clusters of the factor graph. It is experimentally shown to surpass the BP in the cases where the clustering deals with the harmful topological structures that prevents the BP from rightly decoding any LDPC code, namely the trapping sets. We do not only confront the BP and the GBP algorithms according to their performance from the point of view of the channel coding with the error-rate, but also according to their dynamical behaviors for non-trivial error-events for which both algorithms can exhibit chaotic beahviors. By means of classical and original dynamical quantifiers, it is shown that the GBP algorithm can overcome the BP algorithm.

Many problems in continuum mechanics, especially in the theory of elastic materials, lead to nonlinear partial differential equations. The nonconvexity of their underlying energy potential is a challenge for mathematical analysis, since convexity plays an important role in the classical theories of existence and regularity. In the last years one main point of interest was to develop techniques to circumvent these difficulties. One approach was to use different notions of convexity like quasi-- or polyconvexity, but most of the work was done only for static (time independent) equations. In this thesis we want to make some contributions concerning existence, regularity and numerical approximation of nonconvex dynamical problems.

Computer Vision aims at developing techniques to extract and exploit information from images. The successful applications of computer vision approaches are multiple and have benefited diverse fields such as manufacturing, medicine or defense. Some of the most challenging tasks performed by computer vision systems are arguably segmentation and tracking. Segmentation can be defined as the partitioning of an image into homogeneous or meaningful regions. Tracking also aims at extracting meaning or information from images, however, it is a dynamic task that operates on temporal (video) sequences. Active contours have been proven to be quite valuable at performing the two aforementioned tasks. The active contours framework is an example of variational approaches, in which a problem is compactly (and elegantly) described and solved in terms of energy functionals. The objective of the proposed research is to develop statistical and shape-based tools inspired from or completing the geometric active contours methodology. These tools are designed to perform segmentation and tracking. The approaches developed in the thesis make an extensive use of partial differential equations and differential geometry to address the problems at hand. Most of the proposed approaches are cast into a variational framework. The contributions of the thesis can be summarized as follows: 1. An algorithm is presented that allows one to robustly track the position and the shape of a deformable object. 2. A variational segmentation algorithm is proposed that adopts a shape-driven point of view. 3. Diverse frameworks are introduced for including prior knowledge on shapes in the geometric active contour framework. 4. A framework is proposed that combines statistical information extracted from images with shape information learned a priori from examples 5. A technique is developed to jointly segment a 3D object of arbitrary shape in a 2D image and estimate its 3D pose with respect to a referential attached to a unique calibrated camera. 6. A methodology for the non-deterministic evolution of curves is presented, based on the theory of interacting particles systems.