Cubature reduction using the theory of inequalities

Hanna, George T (2002) Cubature reduction using the theory of inequalities. Research Master thesis, Victoria University of Technology.


This dissertation is a detailed analysis of two-dimensional integration providing a priori error bounds in a variety of measures of integrand derivatives. Cubature formulae involving both function evaluations and one-dimensional integration are furnished and numerical experiments to investigate the efficacy of the error formulae are performed. Product (and singular) double integration is investigated. Two-dimensional rectangular integral inequalities are constructed via embedding two one dimensional Peano kernels. In one dimension, linear kernels with a parametric discontinuity furnish "three point" rules where sampling occurs at the boundary and an interior point. The error is bounded in terms of the Lebesgue norms of the first derivative of the integrand. In two dimensions for a rectangular region, we find that the rule generalises to three "three point" rules in each dimension. That is nine sample points and six one dimensional integrals. The error bound is expressed in terms of norms of the first mixed partial derivative of the integrand. These results are further generalised to provide error bounds in terms an arbitrary order mixed partial derivative of the integrand. That is, error bounds in measures of δfn+m/δtnδsm for some integers n,m>0 where the integrand is f. In this case, we find that the rule involves both sample points and one-dimensional integrals involving all the partial derivatives of the integrand up to the stated order. Finally, we explore product integrands, where the weight ω(•,•) is positive and integrable. In this case, the rule and the error bound involve moments of the weight. Particular attention is applied to identifying a priori two dimensional grids for which the error bound is minimized. Various weights and weight null spaces are explored and cubature formulae providing "optimal" grids are given.

Additional Information

Master of Science

Item type Thesis (Research Master thesis)
Subjects Historical > FOR Classification > 0101 Pure Mathematics
Historical > FOR Classification > 0802 Computation Theory and Mathematics
Historical > Faculty/School/Research Centre/Department > School of Engineering and Science
Keywords Mathematics, computational analysis, cubature formulae, one-dimensional integral inequalities, two-dimensional integrals, techniques, weighted quadrature rules
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