标题： 多级随机准蒙特卡罗估计量用于嵌套积分 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$f$而言，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）方法广泛用于单积分，但在处理嵌套积分的复杂性时可能效率低下。 本研究引入了一种新颖的多级估计器，结合确定性和随机准蒙特卡罗（rQMC）方法，以高效处理嵌套积分问题。 在此背景下，内部样本数量和内部被积函数评估的离散化精度构成了层级。 我们对估计器进行了全面的理论分析，推导出误差界，表明与标准方法相比，偏差和方差有显著减少。 所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的层级而不牺牲精度。 我们通过数值实验验证了我们方法的性能，重点是估计实验的期望信息增益。 我们进一步引入了一种截断方案，以解决实验噪声的最终无界性。 当应用于估计器中的高斯噪声时，这种截断方案使计算复杂度与有界噪声情况下的计算复杂度相同，仅在乘法对数项范围内。 结果表明，所提出的多级rQMC估计器优于现有的MC和rQMC方法，在计算成本方面有显著降低，并为处理跨各个领域的复杂嵌套积分问题的从业者提供了一个强大的工具。 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. We further introduce a truncation scheme to address the eventual unboundedness of the experimental noise. When applied to Gaussian noise in the estimator, this truncation scheme renders 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.