Randomizing the Trapezoidal Rule Gives the Optimal Rmse Rate in Gaussian Sobolev Spaces

Takashi Goda, Yoshihito Kazashi, Yuya Suzuki

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)

Abstract

Randomized quadratures for integrating functions in Sobolev spaces of order α ≥ 1, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of O(n α 1/ 2) is proven, where n denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory.

Original languageEnglish
Pages (from-to)1655-1676
Number of pages22
JournalMathematics of Computation
Volume93
Issue number348
Early online date26 Oct 2023
DOIs
Publication statusPublished - Jul 2024
MoE publication typeA1 Journal article-refereed

Keywords

  • Gaussian Sobolev space
  • Trapezoidal rule
  • Lower bound
  • Randomized setting
  • Root- mean-squared error

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