Monte Carlo convergence rates for $k$th moments in Banach spaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911351033561088 |
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| author | Kirchner, Kristin Schwab, Christoph |
| author_facet | Kirchner, Kristin Schwab, Christoph |
| contents | We formulate standard and multilevel Monte Carlo methods for the $k$th moment $\mathbb{M}^k_\varepsilon[ξ]$ of a Banach space valued random variable $ξ\colonΩ\to E$, interpreted as an element of the $k$-fold injective tensor product space $\otimes^k_\varepsilon E$. For the standard Monte Carlo estimator of $\mathbb{M}^k_\varepsilon[ξ]$, we prove the $k$-independent convergence rate $1-\frac{1}{p}$ in the $L_q(Ω;\otimes^k_\varepsilon E)$-norm, provided that (i) $ξ\in L_{kq}(Ω;E)$ 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 $L_q(Ω;\otimes^k_\varepsilon E)$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_03797 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Monte Carlo convergence rates for $k$th moments in Banach spaces Kirchner, Kristin Schwab, Christoph Numerical Analysis Functional Analysis Probability 65C05 (Primary) 46A32, 60B11, 60H35 (Secondary) We formulate standard and multilevel Monte Carlo methods for the $k$th moment $\mathbb{M}^k_\varepsilon[ξ]$ of a Banach space valued random variable $ξ\colonΩ\to E$, interpreted as an element of the $k$-fold injective tensor product space $\otimes^k_\varepsilon E$. For the standard Monte Carlo estimator of $\mathbb{M}^k_\varepsilon[ξ]$, we prove the $k$-independent convergence rate $1-\frac{1}{p}$ in the $L_q(Ω;\otimes^k_\varepsilon E)$-norm, provided that (i) $ξ\in L_{kq}(Ω;E)$ 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 $L_q(Ω;\otimes^k_\varepsilon E)$-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. |
| title | Monte Carlo convergence rates for $k$th moments in Banach spaces |
| topic | Numerical Analysis Functional Analysis Probability 65C05 (Primary) 46A32, 60B11, 60H35 (Secondary) |
| url | https://arxiv.org/abs/2212.03797 |