Browsing by Author "Kaarnioja, Vesa"

This thesis is an introduction to the theoretical foundation and practical usage of the Smolyak quadrature rule, which is used to evaluate high-dimensional integrals over regions of Euclidean spaces. Given a sequence of univariate quadrature rules, the Smolyak construction is defined in terms of tensor products taken over the univariate rules' consecutive differences. The evaluation points of the resulting multivariate quadrature rule are distributed more sparsely than those of e.g. tensor product quadrature. It can be shown that a multivariate quadrature rule formulated in this way inherits many useful properties of the underlying sequence of univariate quadrature rules, such as the polynomial exactness. The original formulation of the Smolyak rule is prone to a copious amount of cancellation of terms in practice. This issue can be circumvented by isolating the occurrence of duplicates to a separate term, which can be computed a priori. The resulting combination method forms the basis for a numerical implementation of the Smolyak quadrature rule, which we have provided using the MATLAB scripting language. Our tests suggest that the Smolyak rule provides a competitive alternative in the realm of multidimensional integration routines saturated by the stochastic Monte Carlo method and the deterministic Quasi-Monte Carlo method. This statement is especially valid in the case of smooth integrands and it is backed by the error analysis developed in the second chapter of this thesis. The classical convergence rate is also derived for integrands of sufficient smoothness in the case of a bounded integration region. The third chapter serves as a qualitative approach to generalized sparse grid quadrature. Especially of interest is the dimension-adaptive construction. While it lacks the theoretical foundation of the Smolyak quadrature rule, it has the added benefit of adapting to the spatial structure of the integrand. A MATLAB implementation of this routine is presented vis-à-vis the Smolyak quadrature rule.