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 indicate how to combine this method with importance sampling to remedy the measure concentration arising in the inner integral. We verify the estimator quality through numerical experiments in the context of expected information gain estimation. We compare 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：嵌套积分由通过非线性函数连接的外层积分和内层积分表征，是工程与数学金融等领域的挑战性问题。基于蒙特卡洛（MC）方法的现有嵌套积分数值方法因误差从内层积分向外层积分传播而可能成本过高。先前利用拟蒙特卡洛（QMC）或随机拟蒙特卡洛（rQMC）方法提升近似效率的研究仅聚焦于内层或外层积分的单一近似。本文提出一种新型嵌套rQMC方法，可同时处理内层与外层积分的近似问题。该方法利用嵌套积分的独特结构，提供更高效的近似机制。通过引入Owen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，从而获得理论可靠的误差估计。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，以及实现这些界所需的正则性条件。此外，我们为支撑所提方法数值实现的rQMC近似提供了近似最优样本量。同时，我们阐述了如何将该方法与重要性抽样结合，以修正内层积分中出现的测度集中现象。通过预期信息增益估计框架下的数值实验验证了估计器质量。我们针对热力学和药代动力学两个案例，比较了嵌套rQMC方法与标准嵌套MC积分的计算效率。这些实例凸显了所提方法的计算节约性与增强的适用性。