Optimized multilevel Monte Carlo methods in Banach spaces
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| Format: | Preprint |
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2026
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| author | Kirchner, Kristin Nobile, Fabio Schwab, Christoph Vanzan, Tommaso |
| author_facet | Kirchner, Kristin Nobile, Fabio Schwab, Christoph Vanzan, Tommaso |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24620 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimized multilevel Monte Carlo methods in Banach spaces Kirchner, Kristin Nobile, Fabio Schwab, Christoph Vanzan, Tommaso Numerical Analysis Probability 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. |
| title | Optimized multilevel Monte Carlo methods in Banach spaces |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2605.24620 |