Nesse trabalho estudamos questões de existência, multiplicidade e concentração de soluções de uma classe de problemas elípticos biharmônicos. Nos três primeiros capítulos são utilizados métodos variacionais para estudar a existência, multiplicidade e comportamento assintótico das soluções fracas não-triviais de equações de Schrödinger estacionárias biharmônicas com diferentes hipóteses sobre o potencial e sobre a não-linearidade. No último capítulo, o método de decomposição em cones duais é empregado para obter a existência de três soluções (positiva, negativa e nodal) para uma equação biharmônica / In this work we study some problems on existence, multiplicity and concentration of solutions of biharmonic elliptic equtions. In the first three chapters, variational methods are used to study the existence, multiplicity and the asymptotic behavior of weak nontrivial solutions of stationary Schrödinger biharmonic equations under certain assumptions on the potential function and the nonlinearity. In the last chapter we use variational methods again and also the dual decomposition method to get existence of positive, negative and sign-changing solutions for a biharmonic equation

Gaussian processes can be used as priors on functions. The need for a flexible, principled, probabilistic model of functional relations is common in practice. Consequently, such an approach is demonstrably useful in a large variety of applications. Two challenges of Gaussian process modelling are often encountered. These are dealing with the adverse scaling with the number of data points and the lack of closed form posteriors when the likelihood is non-Gaussian. In this thesis, we study variational inference as a framework for meeting these challenges. An introductory chapter motivates the use of stochastic processes as priors, with a particular focus on Gaussian process modelling. A section on variational inference reviews the general definition of Kullback-Leibler divergence. The concept of prior conditional matching that is used throughout the thesis is contrasted to classical approaches to obtaining tractable variational approximating families. Various theoretical issues arising from the application of variational inference to the infinite dimensional Gaussian process setting are settled decisively. From this theory we are able to give a new argument for existing approaches to variational regression that settles debate about their applicability. This view on these methods justifies the principled extensions found in the rest of the work. The case of scalable Gaussian process classification is studied, both for its own merits and as a case study for non-Gaussian likelihoods in general. Using the resulting algorithms we find credible results on datasets of a scale and complexity that was not possible before our work. An extension to include Bayesian priors on model hyperparameters is studied alongside a new inference method that combines the benefits of variational sparsity and MCMC methods. The utility of such an approach is shown on a variety of example modelling tasks. We describe GPflow, a new Gaussian process software library that uses TensorFlow. Implementations of the variational algorithms discussed in the rest of the thesis are included as part of the software. We discuss the benefits of GPflow when compared to other similar software. Increased computational speed is demonstrated in relevant, timed, experimental comparisons.

First, we study the associated regularized gap functions and the D-gap functions and compute their Clarke-Rockafellar directional derivatives and the Clarke generalized gradients. Second, using these tools and extending the works of Fukushima and Pang (who studied the case when F is smooth), we present results on the relationship between minimizing sequences and stationary sequences of the D-gap functions, regardless the existence of solutions of (VIP). Finally, as another application, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the (VIP). / In this thesis, we investigate a nonsmooth variational inequality problem (VIP) defined by a locally Lipschitz function F which is not necessarily differentiable or monotone on its domain which is a closed convex set in an Euclidean space. / Tan Lulin. / "December 2005." / Adviser: Kung Fu Ng. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6444. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 79-84) and index. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

A faithful depiction of the tropical atmosphere requires three-dimensional sets of observations. Despite the increasing amount of observations presently available, these will hardly ever encompass the entire atmosphere and, in addition, observations have errors. Additional (background) information will always be required to complete the picture. Valuable added information comes from the physical laws governing the flow, usually mediated via a numerical weather prediction (NWP) model. These models are, however, never going to be error-free, why a reliable estimate of their errors poses a real challenge since the whole truth will never be within our grasp. The present thesis addresses the question of improving the analysis procedures for NWP in the tropics. Improvements are sought by addressing the following issues: - the efficiency of the internal model adjustment, - the potential of the reliable background-error information, as compared to observations, - the impact of a new, space-borne line-of-sight wind measurements, and - the usefulness of multivariate relationships for data assimilation in the tropics. Most NWP assimilation schemes are effectively univariate near the equator. In this thesis, a multivariate formulation of the variational data assimilation in the tropics has been developed. The proposed background-error model supports the mass-wind coupling based on convectively-coupled equatorial waves. The resulting assimilation model produces balanced analysis increments and hereby increases the efficiency of all types of observations. Idealized adjustment and multivariate analysis experiments highlight the importance of direct wind measurements in the tropics. In particular, the presented results confirm the superiority of wind observations compared to mass data, in spite of the exact multivariate relationships available from the background information. The internal model adjustment is also more efficient for wind observations than for mass data. In accordance with these findings, new satellite wind observations are expected to contribute towards the improvement of NWP and climate modeling in the tropics. Although incomplete, the new wind-field information has the potential to reduce uncertainties in the tropical dynamical fields, if used together with the existing satellite mass-field measurements. The results obtained by applying the new background-error representation to the tropical short-range forecast errors of a state-of-art NWP model suggest that achieving useful tropical multivariate relationships may be feasible within an operational NWP environment.

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.

Unlike published shell theories, the main two parts of this thesis are devoted to the asymptotic construction of a refined theory for composite laminated shells valid over a wide range of frequencies and wavelengths. The resulting theory is applicable to shells each layer of which is made of materials with monoclinic symmetry. It enables one to analyze shell dynamic responses within both long-wavelength, low- and high-frequency vibration regimes. It also leads to energy functionals that are both positive definiteness and sufficient simplicity for all wavelengths. This whole procedure was first performed analytically. From the insight gained from the procedure, a finite element version of the analysis was then developed; and a corresponding computer program, DVAPAS, was developed. DVAPAS can obtain the generalized 2-D constitutive law and recover accurately the 3-D results for stress and strain in composite shells. Some independent works will be needed to develop the corresponding 2-D surface analysis associated with the present theory and to continue towards full verification and validation of the present process by comparison with available published works.

In this thesis,we report recent results on existence for variational inequalities in infinite-dimensional spaces under generalized monotonicity.

Understand the behavior of queueing networks in heavy tra c is very important due to its importance in evaluating the network performance in related applications. However, in many cases, the stationary distributions of such networks are intractable. Based on di usion limits of queueing networks, we can use Re ected Brownian Motion (RBM) processes as reasonable approximations. As such, we are interested in obtaining the stationary distribution of RBM. Unfortunately, these distributions are also in most cases intractable. However, the tail behavior (large deviations) of RBM may give insight into the stationary distribution. Assuming that a large deviations principle holds, we need only solve the corresponding variational problem to obtain the rate function. Our research is mainly focused on how to solve variational problems in the case of rotationally symmetric (RS) data. The contribution of this dissertation primarily consists of three parts. In the rst part we give out the speci c stability condition for the RBM in the octant in the RS vi case. Although the general stability conditions for RBM in the octant has been derived previously, we simplify these conditions for the case we consider. In the second part we prove that there are only two types of possible solutions for the variational problem. In the last part, we provide a simple computational method. Also we give an example under which a spiral path is the optimal solution. / text

Digital media collections hold an unprecedented source of knowledge and data about the world. Yet, even at current scales, the data exceeds by many orders of magnitude the amount a single user could browse through in an entire lifetime. Making use of such data requires computational tools that can index, search over, and organize media documents in ways that are meaningful to human users, based on the meaning of their content. This dissertation develops an automated approach to analyzing digital media content based on topic models. Its primary contribution, the Infinite-Word Topic Model (IWTM), helps extend topic modeling to digital media domains by removing model assumptions that do not make sense for them -- in particular, the assumption that documents are composed of discrete, mutually-exclusive words from a fixed-size vocabulary. While conventional topic models like Latent Dirichlet Allocation (LDA) require that media documents be converted into bags of words, IWTM incorporates clustering into its probabilistic model and treats the vocabulary size as a random quantity to be inferred based on the data. Among its other benefits, IWTM achieves better performance than LDA while automating the selection of the vocabulary size. This dissertation contributes fast, scalable variational inference methods for IWTM that allow the model to be applied to large datasets. Furthermore, it introduces a new method, Incremental Variational Inference (IVI), for training IWTM and other Bayesian non-parametric models efficiently on growing datasets. IVI allows such models to grow in complexity as the dataset grows, as their priors state that they should. Finally, building on IVI, an active learning method for topic models is developed that intelligently samples new data, resulting in models that train faster, achieve higher performance, and use smaller amounts of labeled data. / text

Advances in machine autonomy hold great promise in advancing technology, economic markets, and general societal well-being. For example, the progression of unmanned air systems (UAS) research has demonstrated the effectiveness and reliability of these autonomous systems in performing complex tasks. UAS have shown to not only outperformed human pilots in some tasks, but have also made novel applications not possible for human pilots practical. Nevertheless, human pilots are still favored when performing specific challenging tasks. For example, transportation of suspended (sometimes called slung or sling) loads requires highly skilled pilots and has only been performed by UAS in highly controlled environments. The presented work begins to bridge this autonomy gap by proposing a trajectory optimization framework for operations involving autonomous rotorcraft with suspended loads. The framework generates optimized vehicle trajectories that are used by existing guidance, navigation, and control systems and estimates the state of the non-instrumented load using a downward facing camera. Data collected from several simulation studies and a flight test demonstrates the proposed framework is able to produce effective guidance during autonomous suspended load operations. In addition, variational integrators are extensively studied in this dissertation. The derivation of a stochastic variational integrator is presented. It is shown that the presented stochastic variational integrator significantly improves the performance of the stochastic differential dynamical programming and the extended Kalman filter algorithms. A variational integrator for the propagation of polynomial chaos expansion coefficients is also presented. As a result, the expectation and variance of the trajectory of an uncertain system can be accurately predicted.