论文标题
$ k $ th th Moments in Banach空间的蒙特卡洛收敛率
Monte Carlo convergence rates for $k$th moments in Banach spaces
论文作者
论文摘要
我们为$ k $ th Moment $ \ mathbb {m}^k_ \ varepsilon [ξ] $制定标准和多层次的蒙特卡洛方法。对于$ \ Mathbb {M}^k_ \ Varepsilon [ξ] $的标准蒙特卡洛估计器，我们证明了$ k $ - 独立的收敛速度$ 1- \ frac {1} {p} {p} {p} {p} $ l_ {kq}（ω; e）$和（ii）$ q \ in [p，\ infty）$，其中$ p \ in [1,2] $是$ e $的rademacher类型。 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.每当Banach Space $ E $的类型为$ P = 2 $时，我们的发现与Hilbert Space有价值的随机变量的已知结果一致。 我们通过三个模型问题说明了抽象结果：二阶椭圆PDE具有随机强迫或随机系数以及随机演化方程。在这些情况下，溶液过程自然会在非希尔伯特式的Banach空间中采用值。预示着进一步的应用程序建模约束在类型$ p <2 $的Banach空间中施加设置的应用程序。
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.
