The Price of Pessimism for Multidimensional Quadrature

AbstractMultidimensional quadrature error for Hilbert spaces of integrands is studied in three settings: worst-case, random-case, and average-case. Explicit formulae are derived for the expected errors in each case. These formulae show the relative, pessimism of the three approaches. The first is the trace of a hermitian and nonnegative definite matrix ΛμQ, the second is the spectral radius of the same matrix ΛμQ, and the third is the trace of the matrix ΣΛμQ for a hermitian and nonnegative matrix Σ with trace (Σ)=1. Several examples are studied, including Monte Carlo quadrature and shifted lattice rules. Some of the results for Hilbert spaces of integrands can be extended to Banach spaces of integrands