Stochastic properties of quadrature formulas
Stochastic properties of several quadrature formulas are discussed. Let \(F\subset L_ 1([0,1]\beta d,\lambda \beta d)\) be a given set of functions and let \(S(f)=\int_{[0,1]\beta d} f(x)dx\) and \(\tilde S(f)=\phi(a_ 1,a_ 2,...,a_ n)\) be a quadrature method only usingthe function values at n knots \(a_ 1,a_ 2,...,a_ n\in [0,1]\beta d\). The maximal error of a method \(\tilde S\) is given by \(\Delta_{\max}(S)=\sup \| S(f)-S(f)\|,\) while the average error is given by \(\Delta_{\mu}(S)=\int_{F}| S(f)-\tilde S(f)| d\mu(f),\) where \(\mu\) is a Borel probability measure on \((F,\| \|_{\infty})\). It is shown that stochastic error bounds, i.e., the error bounds of the average error, for quadratic formulas are much smaller than deterministic ones (bound for the maximal error) in many cases, though it dependson the class of F on functions to be considered. Moreover, nonlinear methods, adaptive methods or even methods with varying cardinality are shown not be significantly better than the simple linear method with respect to certain stochastic error bounds.
scientific article; zbMATH DE number 4070162