Characterized by an outer integral connected to an inner integral through a nonlinear function, nested integration is a challenging problem in various fields, such as engineering and mathematical finance. The available numerical methods for nested integration based on Monte Carlo (MC) methods can be prohibitively expensive owing to the error propagating from the inner to the outer integral. Attempts to enhance the efficiency of these approximations using the quasi-MC (QMC) or randomized QMC (rQMC) method have focused on either the inner or outer integral approximation. This work introduces a novel nested rQMC method that simultaneously addresses the approximation of the inner and outer integrals. This method leverages the unique nested integral structure to offer a more efficient approximation mechanism. By incorporating Owen's scrambling techniques, we address integrands exhibiting infinite variation in the Hardy--Krause sense, enabling theoretically sound error estimates. As the primary contribution, we derive asymptotic error bounds for the bias and variance of our estimator, along with the regularity conditions under which these bounds can be attained. In addition, we provide nearly optimal sample sizes for the rQMC approximations underlying the numerical implementation of the proposed method. Moreover, we derive a truncation scheme to make our estimator applicable in the context of expected information gain estimation and indicate how to use importance sampling to remedy the measure concentration arising in the inner integral. We verify the estimator quality through numerical experiments by comparing the computational efficiency of the nested rQMC method against standard nested MC integration for two case studies: one in thermomechanics and the other in pharmacokinetics. These examples highlight the computational savings and enhanced applicability of the proposed approach.

翻译：嵌套积分问题在工程和数学金融等多个领域中普遍存在，其特点是通过非线性函数将外层积分与内层积分相连接。基于蒙特卡洛（MC）方法的现有数值方法由于误差从内层积分传播至外层积分，计算成本往往过高。使用拟蒙特卡洛（QMC）或随机拟蒙特卡洛（rQMC）方法提升这些近似效率的尝试通常仅聚焦于内层或外层积分的近似。本研究提出了一种新颖的嵌套rQMC方法，可同时处理内层与外层积分的近似。该方法利用嵌套积分特有的结构，提供了一种更高效的近似机制。通过引入Owen的置乱技术，我们处理了在Hardy–Krause意义下具有无限变差的被积函数，从而获得了理论严密的误差估计。作为主要贡献，我们推导了估计器偏差与方差的渐近误差界，以及达到这些误差界所需的正则性条件。此外，我们为该方法数值实现所依赖的rQMC近似给出了近乎最优的样本量配置方案。进一步，我们提出了一种截断策略，使估计器能够适用于期望信息增益估计的场景，并说明了如何利用重要性采样缓解内层积分中出现的测度集中问题。通过两个案例研究（热力学和药代动力学），我们将嵌套rQMC方法与标准嵌套MC积分法的计算效率进行对比，验证了估计器的性能。这些算例凸显了所提方法在计算效率上的优势及其更广泛的应用潜力。