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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 language | English |
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Pages (from-to) | 1655-1676 |
Number of pages | 22 |
Journal | Mathematics of Computation |
Volume | 93 |
Issue number | 348 |
Early online date | 26 Oct 2023 |
DOIs | |
Publication status | Published - Jul 2024 |
MoE publication type | A1 Journal article-refereed |
Keywords
- Gaussian Sobolev space
- Trapezoidal rule
- Lower bound
- Randomized setting
- Root- mean-squared error
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Hyvönen Nuutti: New frontiers in Bayesian optimal design for applied inverse problems
Hyvönen, N. (Principal investigator), Jääskeläinen, A. (Project Member), Suzuki, Y. (Project Member), Hirvensalo, M. (Project Member) & Puska, J.-P. (Project Member)
01/09/2022 → 31/08/2026
Project: Academy of Finland: Other research funding