The price of pessimism for multidimensional quadrature

Fred J. Hickernell*, Henryk Woźniakowski

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

18 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)625-659
Number of pages35
JournalJournal of Complexity
Volume17
Issue number4
DOIs
Publication statusPublished - Dec 2001
Externally publishedYes

Scopus Subject Areas

  • Algebra and Number Theory
  • Statistics and Probability
  • Numerical Analysis
  • Mathematics(all)
  • Control and Optimization
  • Applied Mathematics

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