标题： 多层随机拟蒙特卡罗估计器用于嵌套积分 显示英文标题

标题： Multilevel randomized quasi-Monte Carlo estimator for nested integration

摘要： 嵌套积分问题出现在各种科学和工程应用中，包括贝叶斯实验设计、金融风险评估和不确定性量化。 这些嵌套积分的形式为 $\int f\left(\int g(\bs{y},\bs{x})\di{}\bs{x}\right)\di{}\bs{y}$，对于非线性 $f$，这使得它们在计算上具有挑战性，尤其是在高维情况下。 尽管传统蒙特卡罗（MC）方法广泛用于单个积分，但在处理嵌套积分的复杂性时效率可能较低。 本文介绍了一种新的多级估计器，结合确定性和随机化拟蒙特卡罗（rQMC）方法，以高效解决嵌套积分问题。 在此背景下，内层样本数量和内层被积函数评估的离散精度构成了级别。 我们对该估计器提供了全面的理论分析，推导出误差界限，显示出与标准方法相比显著减少的偏差和方差。 该提出的估计器在被积函数近似评估的情况下特别有效，因为它可以适应不同分辨率水平而不影响精度。 我们通过数值实验验证了该方法的性能，重点是估计实验的信息增益期望。 当应用于实验中的高斯噪声时，截断方案确保在相同的计算复杂度下（至多乘法对数项），误差界有限，与有界噪声情况一致。 结果表明，所提出的多级rQMC估计器优于现有的MC和rQMC方法，在提供显著降低计算成本的同时，也为处理各种领域中复杂的嵌套积分问题提供了强大的工具。 显示英文摘要

摘要： Nested integration problems arise in various scientific and engineering applications, including Bayesian experimental design, financial risk assessment, and uncertainty quantification. These nested integrals take the form $\int f\left(\int g(\bs{y},\bs{x})\di{}\bs{x}\right)\di{}\bs{y}$, for nonlinear $f$, making them computationally challenging, particularly in high-dimensional settings. Although widely used for single integrals, traditional Monte Carlo (MC) methods can be inefficient when encountering complexities of nested integration. This work introduces a novel multilevel estimator, combining deterministic and randomized quasi-MC (rQMC) methods to handle nested integration problems efficiently. In this context, the inner number of samples and the discretization accuracy of the inner integrand evaluation constitute the level. We provide a comprehensive theoretical analysis of the estimator, deriving error bounds demonstrating significant reductions in bias and variance compared with standard methods. The proposed estimator is particularly effective in scenarios where the integrand is evaluated approximately, as it adapts to different levels of resolution without compromising precision. We verify the performance of our method via numerical experiments, focusing on estimating the expected information gain of experiments. When applied to Gaussian noise in the experiment, a truncation scheme ensures finite error bounds at the same computational complexity as in the bounded noise case up to multiplicative logarithmic terms. The results reveal that the proposed multilevel rQMC estimator outperforms existing MC and rQMC approaches, offering a substantial reduction in computational costs and offering a powerful tool for practitioners dealing with complex, nested integration problems across various domains.