An average discrepancy for optimal vertex-modified number-theoretic rules (Q1968621)

In the first part of this paper, the authors consider the approximation of multidimensional integrals in which the integrands are not periodic. Vertex modified rules are introduced. Also, error bounds for such rules are obtained. In the second part, they construct vertex-modified rules which are optimally so that the discrepancy is minimized. An expression for the squared \(L_2\) discrepancy of optimal vertext-modified rules is obtained. This expression is used to derive an expression for the average of the squared \(L_2\) discrepancy for optimal vertex-modified number-theoretic rules. Finally, some numerical results are given.