Print Email Facebook Twitter 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 Mεk[ξ] of a Banach space valued random variable ξ:Ω→E, interpreted as an element of the k-fold injective tensor product space ⊗εkE. For the standard Monte Carlo estimator of Mεk[ξ], we prove the k-independent convergence rate [Formula presented] in the Lq(Ω;⊗εkE)-norm, provided that (i) ξ∈Lkq(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 Lq(Ω;⊗εkE)-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. Subject Injective tensor productMultilevel Monte CarloRademacher averagesType of Banach space To reference this document use: http://resolver.tudelft.nl/uuid:8ebfa298-5734-45f5-b5d0-60d9f45b7352 DOI https://doi.org/10.1016/j.jfa.2023.110218 ISSN 0022-1236 Source Journal of Functional Analysis, 286 (3) Part of collection Institutional Repository Document type journal article Rights © 2024 K. Kirchner, Christoph Schwab Files PDF 1_s2.0_S0022123623003750_main.pdf 1.03 MB Close viewer /islandora/object/uuid:8ebfa298-5734-45f5-b5d0-60d9f45b7352/datastream/OBJ/view