Monte Carlo convergence rates for kth moments in Banach spaces
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We formulate standard and multilevel Monte Carlo methods for the kth moment 𝕄^k_ε[ξ] of a Banach space valued random variable ξΩ→ E, interpreted as an element of the k-fold injective tensor product space ⊗^k_ε E. For the standard Monte Carlo estimator of 𝕄^k_ε[ξ], we prove the k-independent convergence rate 1-1/p in the L_q(Ω;⊗^k_ε E)-norm, provided that (i) ξ∈ L_kq(Ω;E) and (ii) q∈[p,∞), where p∈[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(Ω;⊗^k_ε 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.
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