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Monte Carlo convergence rates for kth moments in Banach spaces

by K. Kirchner and Ch. Schwab

(Report number 2022-46)

Abstract
We formulate standard and multilevel Monte Carlo methods for the \(k\)th moment \(\mathbb{M}^k_\varepsilon[\xi]\) of a Banach space valued random variable \(\xi\colon\Omega\to E\), interpreted as an element of the \(k\)-fold injective tensor product space~\(\otimes^k_\varepsilon E\). For the standard Monte Carlo estimator of \(\mathbb{M}^k_\varepsilon[\xi]\), we prove the \(k\)-independent convergence rate \(1-\tfrac{1}{p}\) in the \(L_q(\Omega;\otimes^k_\varepsilon E)\)-norm, provided that (i) \(\xi\in L_{kq}(\Omega;E)\) and (ii) \(q\in[p,\infty)\), where \(p\in[1,2]\) is the Rademacher type of \(E\). We moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the \(L_q(\Omega;\otimes^k_\varepsilon E)\)-norm and the optimization of the computational cost for a given accuracy. Whenever the type of \(E\) is \(p=2\), our findings coincide with known results for Hilbert space valued random variables. We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type \(p<2\), are indicated.

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

@Techreport{KS22_1034,
  author = {K. Kirchner and Ch. Schwab},
  title = {Monte Carlo convergence rates for kth moments in Banach spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-46},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-46.pdf },
  year = {2022}
}

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