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

Abstract

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.

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