Monte Carlo convergence rates for kth moments in Banach spaces

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.