2017/07/18 by Michael B. Giles, Mario Hefter, Giles, Michael B. +5 · 1 citation
Mathematics · #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1707.05723
We study the approximation of expectations E(f(X)) for Gaussian random\nelements X with values in a separable Hilbert space H and Lipschitz\ncontinuous functionals f colon H \→ R. We consider restricted Monte Carlo\nalgorithms, which may only use random bits instead of random numbers. We\ndetermine the asymptotics (in some cases sharp up to multiplicative constants,\nin the other cases sharp up to logarithmic factors) of the corresponding n-th\nminimal error in terms of the decay of the eigenvalues of the covariance\noperator of X. It turns out that, within the margins from above, restricted\nMonte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms,\nand suitable random bit multilevel algorithms are optimal. The analysis of this\nproblem leads to a variant of the quantization problem, namely, the optimal\napproximation of probability measures on H by uniform distributions supported\nby a given, finite number of points. We determine the asymptotics (up to\nmultiplicative constants) of the error of the best approximation for the\none-dimensional standard normal distribution, for Gaussian measures as above,\nand for scalar autonomous SDEs.\n