Why simple quadrature is just as good as Monte Carlo

Vanslette Kevin (), Al Alsheikh Abdullatif () and Youcef-Toumi Kamal ()
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Vanslette Kevin: Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
Al Alsheikh Abdullatif: King Abdulaziz City for Science and Technology, Riyadh, Saudi Arabia
Youcef-Toumi Kamal: Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Monte Carlo Methods and Applications, 2020, vol. 26, issue 1, 1-16

Abstract: We motive and calculate Newton–Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton–Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor ∝N-12{\propto N^{-\frac{1}{2}}}. This dimension independent factor is validated in our simulations.

Keywords: Monte Carlo integration; quadrature integration; Bayesian; frequentist; probability (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1515/mcma-2020-2055

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