Optimized multilevel Monte Carlo methods in Banach spaces
Provides sharper theoretical error bounds for Monte Carlo estimation in Banach spaces, benefiting researchers in numerical analysis and uncertainty quantification.
The paper develops refined error analysis for Monte Carlo methods in Banach spaces, accounting for dimension-dependent Rademacher type constants, leading to novel complexity bounds that in favorable cases are independent of the Rademacher type. For L^p(S)-valued random variables, the L^q-convergence rate is determined solely by min{q,2}, with no dependence on the Rademacher type of L^p(S).
We present a theoretical and numerical analysis of Monte Carlo methods for the estimation of statistical moments of random variables $X:Ω\rightarrow E$ 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 $σ$-finite measure space satisfying certain approximation properties, and prove that for a random variable $X\in L^q(Ω;L^p(S))\cap L^p(S;L^q(Ω))$, 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 $L^q(Ω;L^p(S))$-norm while $X$ possesses additional regularity, $X\in L^{\tilde{q}}(Ω;L^p(S))\cap L^p(S;L^{\tilde{q}}(Ω))$ 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.