Monte Carlo convergence rates for kth moments in Banach spaces

Kirchner, K.; Schwab, Christoph

doi:10.1016/j.jfa.2023.110218

Monte Carlo convergence rates for kth moments in Banach spaces

Title

Monte Carlo convergence rates for kth moments in Banach spaces

Author

Kirchner, K. (TU Delft Analysis)
Schwab, Christoph (ETH Zürich)

Date

2024

Abstract

We formulate standard and multilevel Monte Carlo methods for the kth moment M_ε^k[ξ] of a Banach space valued random variable ξ:Ω→E, interpreted as an element of the k-fold injective tensor product space ⊗_ε^kE. For the standard Monte Carlo estimator of M_ε^k[ξ], we prove the k-independent convergence rate [Formula presented] in the L_q(Ω;⊗_ε^kE)-norm, provided that (i) ξ∈L_kq(Ω;E) and (ii) q∈[p,∞), where p∈[1,2] is the Rademacher type of E. By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian's inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the L_q(Ω;⊗_ε^kE)-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space 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.