Optimized multilevel Monte Carlo methods in Banach spaces

Abstract

We present a theoretical and numerical analysis of Monte Carlo methods for the estimation of statistical moments of random variables $X:\hat{I}\text{c◯}\to E$ taking values in a Banach space $E$. For practical computation, we consider finite-dimensional approximation subspaces $(E_{\ell }{)}_{\ell \in \mathbb{N}}\subset E$ of increasing dimension. We develop a refined error analysis that explicitly accounts for a dependence of the Rademacher type constants on the dimension of $E_{\ell }$, leading to novel complexity results for single- and multilevel Monte Carlo methods to estimate the mean and injective moments of arbitrary order, which are, in certain cases, sharper than those derived in [Kirchner, Schwab, J. Funct. Anal, 2024]. Moreover, we show that, in favorable cases, the resulting error-vs.-work bounds are independent of the Rademacher type of $E$. We then focus on $L^p(S)$-valued random variables for a $\ddot{I}ƒ$-finite measure space satisfying certain approximation properties, and prove that for a random variable $X\in L^q(\hat{I}\text{c◯};L^p(S))\cap L^p(S;L^q(\hat{I}\text{c◯}))$, with $q\in (1,\infty )$ and $p\in [1,\infty )$, the $L^q$-convergence rate of a Monte Carlo estimator is determined exclusively by the integrability parameter $\min \{q,2\}$, with no dependence on the Rademacher type $\min \{p,2\}$ of $L^p(S)$. We further investigate the impact of measuring the (multilevel) Monte Carlo error in the $L^q(\hat{I}\text{c◯};L^p(S))$-norm while $X$ possesses additional regularity, $X\in L^{\tilde{q}}(\hat{I}\text{c◯};L^p(S))\cap L^p(S;L^{\tilde{q}}(\hat{I}\text{c◯}))$ with $\tilde{q}\in [q,\infty )$. This analysis reveals an interplay between the sampling error and the strong approximation error, and leads to optimized error-vs.-work bounds for both single- and multilevel Monte Carlo methods. Numerical experiments confirm the sharpness of the analyses presented.

46 pages, 6 figures