This dissertation studies the problem of approximating functions of d variables in a separable Banach space Fd. In particular we are... Show moreThis dissertation studies the problem of approximating functions of d variables in a separable Banach space Fd. In particular we are interested in convergence and tractability results in the worst case setting and in the average case setting. The symmetric positive definite kernel in both settings is of a product form Kd(x, t) := d =1 1 − α2 + α2 Kγ (x , t ) for all x, t ∈ Rd. The kernel Kd generalizes the anisotropic Gaussian kernel, whose tractability properties have been established in the literature. For a fixed d, we study rates of convergence, which indicate how quickly approximation errors decay. Since rates of convergence can deteriorate quickly as d increases, it is desirable to have dimension-independent convergence rates, which corresponds to the concept of strong polynomial tractability. We present sufficient conditions on {α }∞ =1 and {γ }∞ =1 under which strong polynomial tractability holds for function approximation problems in Fd. Numerical examples are presented to support the theory and guaranteed automatic algorithms are provided to solve the function approximation problem in a straightforward and efficient way. viii Ph.D. in Applied Mathematics, December 2015 Show less
We live in an age of instant communication, rapid transportation, and smartphones; an age where the answer to a question can be prompted... Show moreWe live in an age of instant communication, rapid transportation, and smartphones; an age where the answer to a question can be prompted through voice command. With today’s unparalleled increase in digital information, big data is continuously adapting and developing our society with the support of emerging technologies. This has led to the rise and growth of our virtual communities as communication is primarily done through social media networks. The demise of our physical communities has reduced social interaction within the built environment. Despite this forward thinking and the ability to translate big data into architectural solutions, our urban environments have yet to reflect this. Patterns of human interaction within our cities can be transformed by incorporating and visualizing big data within public infrastructure. As a result, the architectural design process is due for an update. This research project explores the use of a high-speed rail station as a hybrid space for virtual and physical communities by providing an interface for users to interact with data streamed in real-time. With the use of data-driven design, new social links are formed and powered by technology as the city becomes a digital public space. M.S. in Architecture, December 2016 Show less
