Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

  • Michael B. Giles
  • Mario Hefter
  • Lukas Mayer
  • Klaus Ritter


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.


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 



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.”


  1. 1.
    C. Brugger, C. De Schryver, N. Wehn, S. Omland, M. Hefter, K. Ritter, A. Kostiuk, R. Korn, Mixed precision multilevel Monte Carlo on hybrid computing systems, in: 2014 IEEE Conference on Computational Intelligence for Financial Engineering Economics (CIFEr), 2014, pp.  215–222.Google Scholar
  2. 2.
    J. Chevallier, Uniform decomposition of probability measures: quantization, classification, rate of convergence, arXiv:1801.02871 (2018).
  3. 3.
    J. Creutzig, S. Dereich, T. Müller-Gronbach, K. Ritter, Infinite-dimensional quadrature and approximation of distributions, Found. Comput. Math. 9 (2009), 391–429.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. Dereich, High Resolution Coding of Stochastic Processes and Small Ball Probabilities, Ph.D. Thesis, FU Berlin, 2003.Google Scholar
  5. 5.
    S. Dereich, The coding complexity of diffusion processes under supremum norm distortion, Stochastic Process. Appl. 118 (2008), 917–937.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    S. Dereich, The coding complexity of diffusion processes under \(L^p[0,1]\) -norm distortion, Stochastic Process. Appl. 118 (2008), 938–951.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Dereich, Asymptotic formulae for coding problems and intermediate optimization problems: a review, in: Trends in Stochastic Analysis (J. Blath, P. Moerters, M. Scheutzow, eds.), Cambridge Univ. Press, Cambridge, 2009, pp. 187–232.Google Scholar
  8. 8.
    S. Dereich, F. Fehringer, A. Matoussi, M. Scheutzow, On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces, J. Theor. Probab. 16 (2003), 249–265.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Dereich, M. Scheutzow, High-resolution quantization and entropy coding for fractional Brownian motion, Electron. J. Probab. 11 (2006), 700–722.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. Dereich, M. Scheutzow, R. Schottstedt, Constructive quantization: approximation by empirical measures, Ann. Inst. Henri Poincaré (B) 49 (2013), 1183–1203.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    W. Gao, P. Ye, H. Wang, Optimal error bound of restricted Monte Carlo in anisotropic Sobolev classes, Prog. Natur. Sci. 16 (2006), 588–593.CrossRefzbMATHGoogle Scholar
  12. 12.
    M. B. Giles, Multilevel Monte Carlo methods, Acta Numer. 24 (2015), 259–328.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Math. 1730, Springer, Berlin, 2000.Google Scholar
  14. 14.
    S. Heinrich, E. Novak, H. Pfeiffer, How many random bits do we need for Monte Carlo integration?, in: MCQMC 2002 (H. Niederreiter, ed.), Springer, Berlin, 2004, pp. 27–49.Google Scholar
  15. 15.
    A. Karol’, A. Nazarov, Y. Nikitin, Small ball probabilities for Gaussian random fields and tensor products of compact operators, Trans. Amer. Math. Soc. 360 (2008), 1443–1474.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Lifshits, Lectures on Gaussian Processes, Springer Briefs in Mathematics, Springer, Heidelberg, 2012.Google Scholar
  17. 17.
    H. Luschgy, G. Pagès, Functional quantization of Gaussian processes, J. Funct. Anal. 196 (2002), 486–531.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    H. Luschgy, G. Pagès, Sharp asymptotics for the functional quantization problem for Gaussian processes, Ann. Appl. Probab. 32 (2004), 1574–1599.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    H. Luschgy, G. Pagès, Functional quantization of a class of Brownian diffusion: a constructive approach, Stochastic Process. Appl. 116 (2006), 310–336.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    G. N. Milstein, M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004.CrossRefzbMATHGoogle Scholar
  21. 21.
    T. Müller-Gronbach, K. Ritter, A local refinement strategy for constructive quantization of scalar SDEs, Found. Comput. Math. 13 (2013), 1005–1033.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    E. Novak, Eingeschränkte Monte Carlo Verfahren zur numerischen Integration, in: Mathematical Statistics and Applications (W. Grossman et al., eds.), Reidel, Dordrecht, 1985, pp. 269–282.Google Scholar
  23. 23.
    E. Novak, Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Math. 1349, Springer, Berlin, 1988.Google Scholar
  24. 24.
    E. Novak, Quantum complexity of integration, J. Complexity 17 (2001), 2–16.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    E. Novak, H. Pfeiffer, Coin tossing algorithms for integral equations and tractability, Monte Carlo Methods Appl. 10 (2004), 491–498.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    S. Omland, Mixed Precision Multilevel Monte Carlo Algorithms for Reconfigurable Hardware Systems, Ph.D. Thesis, Verlag Dr. Hut, 2016.Google Scholar
  27. 27.
    S. Omland, M. Hefter, K. Ritter, C. Brugger, C. De Schryver, N. Wehn, A. Kostiuk, Exploiting mixed-precision arithmetics in a multilevel Monte Carlo approach on FPGAs, in: FPGA Based Accelerators for Financial Applications (C. De Schryver, ed.), Springer, Cham, 2015, pp. 191–220.Google Scholar
  28. 28.
    K. Ritter, Average-Case Analysis of Numerical Problems, Lecture Notes in Math. 1733, Springer, Berlin, 2000.Google Scholar
  29. 29.
    J. F. Traub, H. Woźniakowski, The Monte Carlo algorithm with a pseudorandom generator, Math. Comp. 58, (1992), 323–339.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    C. Xu, A. Berger, Best finite constrained approximations of one-dimensional probabilities, arXiv:1704.07871 (2017).
  31. 31.
    P. Ye, X. Hu, Optimal integration error on anisotropic classes for restricted Monte Carlo and quantum algorithms, J. Approx. Theory 150 (2008), 24–47.MathSciNetCrossRefzbMATHGoogle Scholar

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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