Collocation with cubic splines is used as a method for solving Linear second order parabolic partial differential equations. The collocation method is shown to be equivalent to a finite difference method that is consistent with the differential equation and stable in the sense of Von Neumann. Results of numerical computations are given, as well as an application of the method to a moving boundary problem for the heat equation. / Science, Faculty of / Mathematics, Department of / Graduate

Let S31(?) be the bivariate C1-cubic spline space over a triangulated quadrangulation ?. In this paper, an explicit representation of a locally supported basis of S31(?) is given using the interpolation conditions at vertices.

Quantile function modeling is a more robust, comprehensive, and flexible method of statistical analysis than the commonly used mean-based methods. More and more data are collected in the form of multivariate, functional, and multivariate functional data, for which many aspects of quantile analysis remain unexplored and challenging. This thesis presents a set of quantile analysis methods for multivariate data and multivariate functional data, with an emphasis on environmental applications, and consists of four significant contributions. Firstly, it proposes bivariate quantile analysis methods that can predict the joint distribution of bivariate response and improve on conventional univariate quantile regression. The proposed robust statistical techniques are applied to examine barley plants grown in saltwater and freshwater conditions providing interesting insights into barley’s responses, informing future crop decisions. Secondly, it proposes modeling and visualization of bivariate functional data to characterize the distribution and detect outliers. The proposed methods provide an informative visualization tool for bivariate functional data and can characterize non-Gaussian, skewed, and heavy-tailed distributions using directional quantile envelopes. The radiosonde wind data application illustrates our proposed quantile analysis methods for visualization, outlier detection, and prediction. However, the directional quantile envelopes are convex by definition. This feature is shared by most existing methods, which is not desirable in nonconvex and multimodal distributions. Thirdly, this challenge is addressed by modeling multivariate functional data for flexible quantile contour estimation and prediction. The estimated contours are flexible in the sense that they can characterize non-Gaussian and nonconvex marginal distributions. The proposed multivariate quantile function enjoys the theoretical properties of monotonicity, uniqueness, and the consistency of its contours. The proposed methods are applied to air pollution data. Finally, we perform quantile spatial prediction for non-Gaussian spatial data, which often emerges in environmental applications. We introduce a copula-based multiple indicator kriging model, which makes no distributional assumptions on the marginal distribution, thus offers more flexibility. The method performs better than the commonly used variogram approach and Gaussian kriging for spatial prediction in simulations and application to precipitation data.

The recent development of the impulse/summation approach for efficient B-spline computation in the discrete domain should increase the use of B-splines in many applications. Because we show here how the impulse/summation approach can also be used for constructing polynomials, the approach with a search table approach for the inverse square root operation allows an efficient shading algorithm for rendering an image in a computer graphics system. The approach reduces the number of multiplies and makes it possible for the entire rendering process to be implemented using an integer processor. In many applications, Chebyshev approximation with polynomials and splines is useful in representing a stream of data or a function. Because the impulse/summation approach is developed for discrete systems, some aspects of traditional continuous approximation are not applicable. For example, the lack of the continuity concept in the discrete domain affects the definition of the local extrema of a function. Thus, the method of finding the extrema must be changed. Both forward differences and backward differences must be checked to find extrema instead of using the first derivative in the continuous domain approximation. Polynomial Chebyshev approximation in the discrete domain, just as in the continuous domain, forms a Chebyshev system. Therefore, the Chebyshev approximation process always produces a unique best approximation. Because of the non-linearity of free knot polynomial spline systems, there may be more than one best solution and the convexity of the solution space cannot be guaranteed. Thus, a Remez Exchange Algorithm may not produce an optimal approximation. However, we show that the discrete polynomial splines approximate a function using a smaller number of parameters (for a similar minimax error) than the discrete polynomials do. Also, the discrete polynomial spline requires much less computation and hardware than the discrete polynomial for curve generation when we use the impulse/summation approach. This is demonstrated using two approximated FIR filter implementations. / Ph. D.

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Generating and rendering ribbons from a stream of particles is common in games and VFX to visualize flow. To create detailed smooth particle ribbons this work combines tessellation and B-splines, which gives smooth surfaces with contiguous normals. The traditional pipeline for doing hardware tessellation with hull- and domain shaders is limited to a maximum tessellation level, which limits how much detail a ribbon can get. In Nvidia’s Turing architecture, mesh- and task shaders were introduced, which provide support for tessellating geometry while not having any constraints on the amount of geometry that can be generated. This work evaluates three implementations for generating and drawing smooth particle ribbons by measuring performance in terms of execution time. The implementations are based on mesh shaders, mesh + task shaders and the traditional pipeline. In turn, three optimizations are proposed. The optimizations evaluated are adaptive level of detail, culling and Gouraud shading. The results show that the traditional pipeline is faster than using mesh shader based tessellation up to its maximum tessellation factor. For larger tessellation factors, when combining adaptive LOD, culling and Gouraud shading, mesh and task shaders combined can generate ribbons at higher quality than the traditional tessellation pipeline, with comparable performance. / Att generera och rendera band från partikelströmmar är vanligt i spel och VFX för att visualisera flöden. För att skapa detaljrika mjuka partikelband kombinerar det här arbetet tessellering och B-splines, vilket ger mjuka ytor med kontinuerliga normaler. Den traditionella pipelinen för att göra hårdvarutessellering med hull- och domain shaders är begränsade till en maximal tesselleringsnivå, vilket begränsar hur detaljrik ett band kan bli. Med Nvidias turingarkitektur introducerades mesh- och task shaders, vilket också stödjer tessellering av geometri med fördelen att inte vara begränsade till en maximal tesselleringsnivå. Det här arbetet evaluerar tre implementationer av mjuka partikelband när de genereras och ritas genom att mäta prestanda genom exekveringstid. Implementationerna är baserade på mesh shaders, mesh + task shaders och den traditionella pipelinen. Utöver detta föreslås tre optimeringar. Dessa optimeringar är adaptiv LOD, culling och Gouraud shading. Resultaten visar att den traditionella pipelinen är snabbare än att använda mesh shaderbaserad tessellering upp till och med dess maximala tesselleringsfaktor. För större tesselleringsfaktorer, när adaptiv LOD, culling och Gouraud shading kombineras, kan mesh och task shaders tillsammans generera band med högre kvalitet än den traditionella tesselleringspipelinen, med jämförbar prestanda.

An optimal ordering scheme for the capacitated, finite-horizon, multi-period, multiproduct newsvendor problem was proposed by cite {shao06} using a hedging point policy. This solution requires the calculation of a central curve that divides the different ordering regions and a vector that defines the target inventory levels. The central curve is a nonlinear curve that determines the optimal order quantities as a function of the initial inventory levels. In this paper we propose a method for calculating this curve and vector using spline functions, infinitesimal perturbation analysis (IPA), and convex optimization. Using IPA the derivatives of the cost with respect to the variables that determine the spline function are efficiently calculated. A convex optimization algorithm is used to optimize the spline function, resulting in a optimal policy. We present the mathematical derivations and simulation results validating this solution.

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