Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

• Published in: Foundations of computational mathematics Vol. 19; no. 1; pp. 205 - 238
• Main Authors: Giles, Michael B., Hefter, Mario, Mayer, Lukas, Ritter, Klaus
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2019; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applications of Mathematics; Approximation; Asymptotic properties; Computer Science; Continuity (mathematics); Covariance; Economics; Eigenvalues; Gaussian distribution; Hilbert space; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix Theory; Normal distribution; Numerical Analysis; Numerical integration; Random numbers; Statistical analysis; United States; 65D30; Integration; Stochastic differential equations; 60H10; Random bits; Gaussian measures on Hilbert spaces; Quantization; Approximation of probability measures; 65C05; 60G15; Multilevel Monte Carlo algorithms; 60H35
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-018-9382-3

We study the approximation of expectations E ( f ( X ) ) for Gaussian random elements X with values in a separable Hilbert space H and Lipschitz continuous functionals f : H → 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.