Computing Discrepancies of Smolyak Quadrature Rules

Computing Discrepancies of Smolyak Quadrature Rules

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as a possible competitor to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadra-ture formulas consists in computing their L 2-discrepancy. Especially for larger dimensions, such computations are...

Computing Discrepancies Related to Spaces of Smooth Periodic Functions

A notion of discrepancy is introduced, which represents the integration error on spaces of r-smooth periodic functions. It generalizes the diaphony and constitutes a periodic counterpart to the classical L2-discrepancy as well as r-smooth versions of it introduced recently by Paskov Pas93]. Based on previous work FH96], we develop an eecient algorithm for computing periodic discrepancies for qu...

Quadrature Rules for Fuzzy Integrals

Weighted quadrature rules with binomial nodes

In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...

Anti-Szego quadrature rules

Szegő quadrature rules are discretization methods for approximating integrals of the form ∫ π −π f(e it)dμ(t). This paper presents a new class of discretization methods, which we refer to as anti-Szegő quadrature rules. AntiSzegő rules can be used to estimate the error in Szegő quadrature rules: under suitable conditions, pairs of associated Szegő and anti-Szegő quadrature rules provide upper a...