The linear algebra of interpolation with finite applications giving computational methods for multivariate polynomials

Thesis (Ph.D.) University of Alaska Fairbanks, 1988Linear representation and the duality of the biorthonormality relationship express the linear algebra of interpolation by way of the evaluation mapping. In the finite case the standard bases relate the maps to Gramian matrices. Five equivalent conditions on these objects are found which characterize the solution of the interpolation problem. This algebra succinctly describes the solution space of ordinary linear initial value problems. Multivariate polynomial spaces and multidimensional node sets are described by multi-index sets. Geometric considerations of normalization and dimensionality lead to cardinal bases for Lagrange interpolation on regular node sets. More general Hermite functional sets can also be solved by generalized Newton methods using geometry and multi-indices. Extended to countably infinite spaces, the method calls upon theorems of modern analysis

Doctor of Philosophy

dissertationPlatelet aggregation, an important part of the development of blood clots, is a complex process involving both mechanical interaction between platelets and blood, and chemical transport on and o the surfaces of those platelets. Radial Basis Function (RBF) interpolation is a meshfree method for the interpolation of multidimensional scattered data, and therefore well-suited for the development of meshfree numerical methods. This dissertation explores the use of RBF interpolation for the simulation of both the chemistry and mechanics of platelet aggregation. We rst develop a parametric RBF representation for closed platelet surfaces represented by scattered nodes in both two and three dimensions. We compare this new RBF model to Fourier models in terms of computational cost and errors in shape representation. We then augment the Immersed Boundary (IB) method, a method for uid-structure interaction, with our RBF geometric model. We apply the resultant method to a simulation of platelet aggregation, and present comparisons against the traditional IB method. We next consider a two-dimensional problem where platelets are suspended in a stationary fluid, with chemical diusion in the fluid and chemical reaction-diusion on platelet surfaces. To tackle the latter, we propose a new method based on RBF-generated nite dierences (RBF-FD) for solving partial dierential equations (PDEs) on surfaces embedded in 2D domains. To robustly tackle the former, we remove a limitation of the Augmented Forcing method (AFM), a method for solving PDEs on domains containing curved objects, using RBF-based symmetric Hermite interpolation. Next, we extend our RBF-FD method to the numerical solution of PDEs on surfaces embedded in 3D domains, proposing a new method of stabilizing RBF-FD discretizations on surfaces. We perform convergence studies and present applications motivated by biology. We conclude with a summary of the thesis research and present an overview of future research directions, including spectrally-accurate projection methods, an extension of the Regularized Stokeslet method, RBF-FD for variable-coecient diusion, and boundary conditions for RBF-FD

The Dynamics Of A Mantle With A Plume-Fed Asthenosphere -- Method Development And Numerical Experimental Studies

This dissertation is composed of three studies addressing different but related problems on dynamic processes occurring in a Plume-fed Asthenosphere system, as well as techniques for improving numerical models of mantle convection. The first paper, 2D Numerical Experiments on a Plume-fed Asthenosphere: Necessary Preconditions and Implications for Geoid and Dynamic Topography, performs a suite of 2D finite element-based experiments that explore what conditions are needed so that mantle flow includes a plume-fed asthenosphere (PFA) as a key part of its flow pattern. We find that a plume flux ~1.2 times big as the slab flux is needed for a persistent PFA. The numerical experiments also demonstrate that, instead of generating dynamic topography on the sea floor, flow-induced dynamic relief due to sub-asthenospheric density anomalies will preferentially form at the base of a buoyant asthenosphere, which is a promising mechanism to explain why Earth's ±100m Geoid variations are associated with much less than ~2km of dynamic topography at Earth's surface. The second paper, A Quasi-Cspline Interpolation Algorithm for Data on Unstructured Triangular and Tetrahedral Meshes, develops a quasi-cubic Hermite spline interpolation algorithm for 2D and 3D scattered data, fitting both nodal values and slopes to the edges of triangular or tetrahedral cubic serendipity elements. This explicit recipe for 2D and 3D interpolation has been tested in vectorized and parallelized Matlab code, and has been used in both 2D and 3D large numerical simulations using unstructured triangular and tetrahedral meshes. The third paper, Plumeasthenosphere-lithosphere Interactions Within a Mantle with a Plume-fed Asthenosphere: Implications for Hawaii- and Iceland-type Plume Dynamics, studies the effects of on- and off-axis deep-mantle plumes with thermally controlled density and viscosity variations, assuming that thermal expansion controls density and that viscosity is governed by a temperature-dependent Arrhenius-type relation. The code we use is a parallel Matlab-based 3-D Finite Element code that we have developed, which utilizes unstructured tetrahedral meshes, and which can handle large and abrupt (6 orders of magnitude) viscosity contrasts (Hasenclever, PhD Dissertation 2010). In this paper, We show the results of: 1) the necessary conditions (plume flux, density contrast, viscosity contrast) for the existence of a PFA system with an on/off-axis plume; 2) resulting 3-D flow patterns in the asthenosphere, and the dynamic topography that is associated with them; 3) the decoupling effect of a buoyant and less viscous asthenosphere layer to the underlying mantle, and how this helps lead to relatively fixed hot spots

Spectral tensor-train decomposition

The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT decomposition and analyze its properties. We obtain results on the convergence of the decomposition, revealing links between the regularity of the function, the dimension of the input space, and the TT ranks. We also show that the regularity of the target function is preserved by the univariate functions (i.e., the "cores") comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting \textit{spectral tensor-train decomposition} combines the favorable dimension-scaling of the TT decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of tensors resulting from suitable discretizations of the target function. We assess the performance of the method on a range of numerical examples: a modifed set of Genz functions with dimension up to $100$, and functions with mixed Fourier modes or with local features. We observe significant improvements in performance over an anisotropic adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online.Comment: 33 pages, 19 figure

ShapeWright--finite element based free-form shape design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1990.Includes bibliographical references (p. 179-192).by George Celniker.Ph.D

Analysis and new constructions of generalized barycentric coordinates in 2D

Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer graphics and computational mechanics. There exist many coordinate functions that satisfy all the basic properties of barycentric coordinates, but they differ by a number of other properties. We start by providing an extensive comparison of all existing coordinate functions and pointing out which important properties of generalized barycentric coordinates are not satisfied by these functions. This comparison shows that not all of existing coordinates have fully investigated properties, and we complete such a theoretical analysis for a particular one-parameter family of generalized barycentric coordinates for strictly convex polygons. We also perform numerical analysis of this family and show how to avoid computational instabilities near the polygon’s boundary when computing these coordinates in practice. We conclude this analysis by implementing some members of this family in the Computational Geometry Algorithm Library. In the second half of this dissertation, we present a few novel constructions of non-negative and smooth generalized barycentric coordinates defined over any simple polygon. In this context, we show that new coordinates with improved properties can be obtained by taking convex combinations of already existing coordinate functions and we give two examples of how to use such convex combinations for polygons without and with interior points. These new constructions have many attractive properties and perform better than other coordinates in interpolation and image deformation applications

Proceedings of the NASA Workshop on Surface Fitting

Surface fitting techniques and their utilization are addressed. Surface representation, approximation, and interpolation are discussed. Along with statistical estimation problems associated with surface fitting

Fast Fourier Transform at Nonequispaced Nodes and Applications

The direct computation of the discrete Fourier transform at arbitrary nodes requires O(NM) arithmetical operations, too much for practical purposes. For equally spaced nodes the computation can be done by the well known fast Fourier transform (FFT) in only O(N log N) arithmetical operations. Recently, the fast Fourier transform for nonequispaced nodes (NFFT) was developed for the fast approximative computation of the above sums in only O(N log N + M log 1/e), where e denotes the required accuracy. The principal topics of this thesis are generalizations and applications of the NFFT. This includes the following subjects: - Algorithms for the fast approximative computation of the discrete cosine and sine transform at nonequispaced nodes are developed by applying fast trigonometric transforms instead of FFTs. - An algorithm for the fast Fourier transform on hyperbolic cross points with nonequispaced spatial nodes in 2 and 3 dimensions based on the NFFT and an appropriate partitioning of the hyperbolic cross is proposed. - A unified linear algebraic approach to recent methods for the fast computation of matrix-vector-products with special dense matrices, namely the fast multipole method, fast mosaic-skeleton approximation and H-matrix arithmetic, is given. Moreover, the NFFT-based summation algorithm by Potts and Steidl is further developed and simplified by using algebraic polynomials instead of trigonometric polynomials and the error estimates are improved. - A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on hard thresholding complex ridgelet coefficients combined with total variation minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the NFFT and then applying a dual-tree complex wavelet transform. - A new robust local scattered data approximation method is introduced. It is an advancement of the moving least squares approximation (MLS) and generalizes an approach of van den Boomgard and van de Weijer to scattered data. In particular, the new method is space and data adaptive