多索引蒙特卡洛方法：当稀疏性遇见采样
Multi-Index Monte Carlo: When Sparsity Meets Sampling
AI总结：
本文提出并分析了一种新的多索引蒙特卡洛方法，用于弱逼近由随机测度或随机系数驱动的微分方程模型。通过使用高阶混合差分减少方差，改进了收敛性结果，并在高维问题中实现了更精确的解。
摘要：
我们提出并分析了一种新颖的多索引蒙特卡洛（MIMC）方法，用于弱逼近由随机测度或随机系数驱动的微分方程模型。MIMC方法是Zenger、Griebel等引入的组合技术的随机版本，也是Heinrich和Giles首次描述的多级蒙特卡洛（MLMC）方法的扩展。受Giles的开创性工作启发，我们使用高阶混合差分代替MLMC中的第一阶差分，从而大幅减少层级差的方差。这导致了新的改进的复杂性结果，这些结果是Giles的MLMC分析的自然推广，并扩大了能够实现最优收敛（O(TOL^{-2})）的问题参数域。此外，在MIMC中，所需内存随TOL增加的速率与方向数量无关（至多一个对数项），这允许在高维问题中计算比MLMC更精确的解。我们通过首先聚焦于一个简单的全张量索引集来说明MIMC的设置，然后提出了一种系统构造MIMC最优索引集的方法，基于适当定义的利润，这些利润取决于样本成本平均值和相应的弱误差和方差。在标准假设下，最优索引集被证明为总次数（TD）类型。在某些情况下，使用最优索引集，MIMC在计算复杂性方面比使用全张量索引集时的对应速率更优。
英文摘要：
We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients. The MIMC method is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Inspired by Giles's seminal work, we use in MIMC high-order mixed differences instead of using first-order differences as in MLMC to reduce the variance of the hierarchical differences dramatically. This in turn yields new and improved complexity results, which are natural generalizations of Giles's MLMC analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence, $\mathcal{O}(\text{TOL}^{-2}).$ Moreover, in MIMC, the rate of increase of required memory with respect to $\text{TOL}$ is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using MLMC. We motivate the setting of MIMC by first focusing on a simple full tensor index set. We then propose a systematic construction of optimal sets of indices for MIMC based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance. Under standard assumptions on the convergence rates of the weak error, variance and work per sample, the optimal index set turns out to be the total degree (TD) type. In some cases, using optimal index sets, MIMC achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets...
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