Monte Carlo convergence rates for $k$th moments in Banach spaces

Abstract

We formulate standard and multilevel Monte Carlo methods for the $k$th moment ${\mathbb{M}}_{\epsilon }^k[\xi ]$ of a Banach space valued random variable $\xi :\Omega \to E$, interpreted as an element of the $k$-fold injective tensor product space ${\otimes }_{\epsilon }^{k}E$. For the standard Monte Carlo estimator of ${\mathbb{M}}_{\epsilon }^k[\xi ]$, we prove the $k$-independent convergence rate $1-\frac{1}{p}$ in the $L_q(\Omega ;{\otimes }_{\epsilon }^{k}E)$-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 $E$. 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 $L_q(\Omega ;{\otimes }_{\epsilon }^{k}E)$-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.

47 pages