TY - JOUR

T1 - The price of pessimism for multidimensional quadrature

AU - Hickernell, Fred J.

AU - Woźniakowski, Henryk

N1 - Funding Information:
Multidimensional 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 LQm, the second is the spectral radius of the same matrix LQm, and the third is the trace of the matrix SLQm for a hermitian and nonnegative matrix S with trace (S)=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. © 2001 Elsevier Science 1This research was supported in part by Hong Kong Research Grants Council Grant RGC/HKBU/2030/99P and Hong Kong Baptist University Grant FRG/97-98/II-99. 2This author was supported in part by the National Science Foundation.

PY - 2001/12

Y1 - 2001/12

N2 - Multidimensional 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 ΛI μ, the second is the spectral radius of the same matrix Λμ, and the third is the trace of the matrix ΣΛI μ 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.

AB - Multidimensional 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 ΛI μ, the second is the spectral radius of the same matrix Λμ, and the third is the trace of the matrix ΣΛI μ 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.

UR - http://www.scopus.com/inward/record.url?scp=0035700367&partnerID=8YFLogxK

U2 - 10.1006/jcom.2001.0593

DO - 10.1006/jcom.2001.0593

M3 - Journal article

AN - SCOPUS:0035700367

SN - 0885-064X

VL - 17

SP - 625

EP - 659

JO - Journal of Complexity

JF - Journal of Complexity

IS - 4

ER -