Cubature rules from a generalized Taylor perspective

Hanna, George T (2009) Cubature rules from a generalized Taylor perspective. PhD thesis, Victoria University.

Abstract

The accuracy and efficiency of computing multiple integrals is a very important problem that arises in many scientific, financial and engineering applications. The research conducted in this thesis is designed to build on past work and develop and analyze new numerical methods to evaluate double integrals efficiently. The fundamental aim is to develop and assess techniques for (numerically) evaluating double integrals with high accuracy. The general approach presented in this thesis involves the development of new multivariate approximations from a generalaised Taylor perspective in terms of Appell type polynomials and to study their use in multi-dimensional integration. The expectation is that the new methods will provide polynomial and polynomial-like approximations that can be used for application in a straight forward manner with better accuracy. That is, we aim to devise and investigate new multiple integration formulae and as well as provide information on a priori error bounds. A further major contribution of the work builds on the research conducted in the field of Grüss type inequalities and leads to a new approximation of the one and two dimensional finite Fourier transform. The approximations are in terms of the complex exponential mean and estimate of the error of approximation for different classes of functions of bounded variation defined on finite intervals. It is believed that this work will also have an impact in the area of numerical multidimensional integral evaluation for other integral operators.

Item type Thesis (PhD thesis)
URI http://vuir.vu.edu.au/id/eprint/1922
Subjects Historical > RFCD Classification > 230000 Mathematical Sciences
Historical > Faculty/School/Research Centre/Department > School of Engineering and Science
Keywords multiple integrals, Taylor perspective, multi-dimension integration, Fourier transforms, numerical methods
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