Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

  • Michael B. Giles
  • Mario Hefter
  • Lukas Mayer
  • Klaus Ritter
Article
  • 19 Downloads

Abstract

We study the approximation of expectations \({\text {E}}(f(X))\) for Gaussian random elements X with values in a separable Hilbert space H and Lipschitz continuous functionals \(f :H \rightarrow {{\mathbb {R}}}\). We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding n-th minimal error in terms of the decay of the eigenvalues of the covariance operator of X. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely the optimal approximation of probability measures on H by uniform distributions supported by a given finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs.

Keywords

Gaussian measures on Hilbert spaces Integration Approximation of probability measures Quantization Random bits Multilevel Monte Carlo algorithms Stochastic differential equations 

Mathematics Subject Classification

60G15 60H35 60H10 65D30 65C05 

Notes

Acknowledgements

The authors are grateful to Steffen Omland for many valuable discussions and contributions at an early stage of this project. We thank an anonymous referee for providing us with the reference [30]. Lukas Mayer is supported by the Deutsche Forschungsgemeinschaft (DFG) within the RTG 1932 “Stochastic Models for Innovations in the Engineering Sciences.”

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Copyright information

© SFoCM 2018

Authors and Affiliations

  • Michael B. Giles
    • 1
  • Mario Hefter
    • 2
  • Lukas Mayer
    • 2
  • Klaus Ritter
    • 2
  1. 1.Mathematical InstituteUniversity of OxfordOxfordEngland, UK
  2. 2.Fachbereich Mathematik, Technische Universität KaiserslauternKaiserslauternGermany

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