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Optimized multilevel Monte Carlo methods in Banach spaces

by K. Kirchner and F. Nobile and Ch. Schwab and T. Vanzan

(Report number 2026-21)

Abstract
We present a comprehensive theoretical and numerical analysis of Monte Carlo methods for the estimation of statistical moments of random variables \(X\colon\Omega\rightarrow E\) 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~[25]. Moreover, we show that, in favorable cases, the resulting error-vs.-work bound s are independent of the Rademacher type of \(E\). We then focus on \(L^p(S)\)-valued random variables for a \(\sigma\)-finite measure space \((S,\mathcal{S},\mu)\) satisfying certain approximation properties, and prove that for a random variable \(X\in L^q(\Omega;L^p(S))\cap L^p(S;L^q(\Omega))\), with \(q\in (1,\infty)\) and \(p\in [1,\infty)\), the \(L^q\)-convergence rate of a multilevel 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 \(L^q(\Omega;L^p(S))\)-norm while \(X\) possesses additional regularity, \(X\in L^{\widetilde{q}}(\Omega;L^p(S))\cap L^p(S;L^{\widetilde{q}}(\Omega))\) with \(\widetilde{q}\in [q,\infty)\). 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 in estimating both first and second moments

Keywords: Banach space valued random variable, injective tensor product, Monte Carlo estimation, multilevel methods, Rademacher averages, type of Banach space

@Techreport{KNSV26_1176,
  author = {K. Kirchner and F. Nobile and Ch. Schwab and T. Vanzan},
  title = {Optimized multilevel Monte Carlo methods in Banach spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-21.pdf },
  year = {2026}
}

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