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.

翻译：嵌套积分问题广泛存在于科学与工程领域，如贝叶斯实验设计、金融风险评估及不确定性量化等。此类嵌套积分通常表现为 $\int f\left(\int g(\bs{y},\bs{x})\di{}\bs{x}\right)\di{}\bs{y}$ 的形式，其中 $f$ 为非线性函数，导致其计算极具挑战性，在高维情形下尤为显著。尽管传统蒙特卡洛方法在单重积分中应用广泛，但在处理嵌套积分复杂性时往往效率低下。本文提出一种新颖的多层估计器，结合确定性及随机化拟蒙特卡洛方法，以高效处理嵌套积分问题。在此框架下，内层样本数量与内层被积函数求值的离散精度共同构成层级。我们对该估计器进行了完整的理论分析，推导出的误差界表明其相较于标准方法能显著降低偏差与方差。该估计器在被积函数需近似求值的场景中表现尤为突出，因其能在不损失精度的前提下自适应不同分辨率层级。我们通过数值实验验证了方法的性能，重点关注实验期望信息增益的估计。此外，针对实验噪声可能无界的问题，我们提出了一种截断方案。当在估计器中应用于高斯噪声时，该截断方案在计算复杂度上（至多包含对数乘法项）与有界噪声情形保持一致。实验结果表明，所提出的多层随机拟蒙特卡洛估计器在计算成本上显著优于现有蒙特卡洛与随机拟蒙特卡洛方法，为各领域处理复杂嵌套积分问题的实践者提供了强有力的工具。