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In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice.

We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.

The objective of this thesis is to evaluate different models and methods for website classification. The websites are classified based on their functionality, in this case specifically whether they are forums, news sites or blogs. The analysis aims at solving a search engine problem, which means that it is interesting to know from which categories in a information search the results come. The data consists of two datasets, extracted from the web in January and April 2013. Together these data sets consist of approximately 40.000 observations, with each observation being the extracted text from the website. Approximately 7.000 new word variables were subsequently created from this text, as were variables based on Latent Dirichlet Allocation. One variable (the number of links) was created using the HTML-code for the web site. These data sets are used both in multinomial logistic regression with Lasso regularization, and to create a Naive Bayes classifier. The best classifier for the data material studied was achieved when using Lasso for all variables with multinomial logistic regression to reduce the number of variables. The  accuracy of this model is 99.70 %. When time dependency of the models is considered, using the first data to make the model and the second data for testing, the accuracy, however, is only 90.74 %. This indicates that the data is time dependent and that websites topics change over time.

Many problems in the natural world have high contrast properties, like transport in composites, fluid in porous media and so on. These problems have huge numerical difficulties because of the singularities of their solutions. It may be really expensive to solve these problems directly by traditional numerical methods. It is necessary and important to understand these problems more in mathematical aspect first, and then using the mathematical results to simplify the original problems or develop more efficient numerical methods. In this thesis we are going to approximate the Dirichlet to Neumann map for the high contrast two phase composites. The mathematical formulation of our problem is to approximate the energy for an elliptic equation with arbitrary boundary conditions. The boundary conditions may have highly oscillations, which makes our problems very interesting and difficult. We developed a method to divide the domain into two different subdomains, one is close to and the other one is far from the boundary, and we can approximate the energy in these two subdomains separately. In the subdomain far from the boundary, the energy is not influenced that much by the boundary conditions. Methods for approximation of the energy in this subdomain are studied before. In the subdomain near the boundary, the energy depends on the boundary conditions a lot. We used a new method to approximate the energy there such that it works for any kind of boundary conditions. By this way, we can have the approximation for the total energy of high contrast problems with any boundary conditions. In other words, we can have a matrix up to any dimension to approximate the continuous Dirichlet to Neumann map of the high contrast composites. Then we will use this matrix as a preconditioner in domain decomposition methods, such that our numerical methods are very efficient to solve the problems in high contrast composites.

<p>We propose nonparametric Bayesian models for supervised dimension</p><p>reduction and regression problems. Supervised dimension reduction is</p><p>a setting where one needs to reduce the dimensionality of the</p><p>predictors or find the dimension reduction subspace and lose little</p><p>or no predictive information. Our first method retrieves the</p><p>dimension reduction subspace in the inverse regression framework by</p><p>utilizing a dependent Dirichlet process that allows for natural</p><p>clustering for the data in terms of both the response and predictor</p><p>variables. Our second method is based on ideas from the gradient</p><p>learning framework and retrieves the dimension reduction subspace</p><p>through coherent nonparametric Bayesian kernel models. We also</p><p>discuss and provide a new rationalization of kernel regression based</p><p>on nonparametric Bayesian models allowing for direct and formal</p><p>inference on the uncertain regression functions. Our proposed models</p><p>apply for high dimensional cases where the number of variables far</p><p>exceed the sample size, and hold for both the classical setting of</p><p>Euclidean subspaces and the Riemannian setting where the marginal</p><p>distribution is concentrated on a manifold. Our Bayesian perspective</p><p>adds appropriate probabilistic and statistical frameworks that allow</p><p>for rich inference such as uncertainty estimation which is important</p><p>for measuring the estimates. Formal probabilistic models with</p><p>likelihoods and priors are given and efficient posterior sampling</p><p>can be obtained by Markov chain Monte Carlo methodologies,</p><p>particularly Gibbs sampling schemes. For the supervised dimension</p><p>reduction as the posterior draws are linear subspaces which are</p><p>points on a Grassmann manifold, we do the posterior inference with</p><p>respect to geodesics on the Grassmannian. The utility of our</p><p>approaches is illustrated on simulated and real examples.</p> / Dissertation

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 162-169). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 104-109). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 162-169). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 104-109). Also available on the Internet.