Existing multilevel quasi-Monte Carlo (MLQMC) methods often rely on multiple independent randomizations of a low-discrepancy (LD) sequence to estimate statistical errors on each level. While this approach is standard, it can be less efficient than simply increasing the number of points from a single LD sequence. However, a single LD sequence does not permit statistical error estimates in the current framework. We propose to recast the MLQMC problem in a Bayesian cubature framework, which uses a single LD sequence and quantifies numerical error through the posterior variance of a Gaussian process (GP) model. When paired with certain LD sequences, GP regression and hyperparameter optimization can be carried out at only $\mathcal{O}(n \log n)$ cost, where $n$ is the number of samples. Building on the adaptive sample allocation used in traditional MLQMC, where the number of samples is doubled on the level with the greatest expected benefit, we introduce a new Bayesian utility function that balances the computational cost of doubling against the anticipated reduction in posterior uncertainty. We also propose a new digitally-shift-invariant (DSI) kernel of adaptive smoothness, which combines multiple higher-order DSI kernels through a weighted sum of smoothness parameters, for use with fast digital net GPs. A series of numerical experiments illustrate the performance of our fast Bayesian MLQMC method and error estimates for both single-level problems and multilevel problems with a fixed number of levels. The Bayesian error estimates obtained using digital nets are found to be reliable, although, in some cases, mildly conservative.

翻译：现有的多层拟蒙特卡洛（MLQMC）方法通常依赖于对低差异（LD）序列进行多次独立随机化，以估计每一层的统计误差。尽管该方法为标准做法，但其效率可能低于直接增加单一LD序列的采样点数。然而，在当前框架下，单一LD序列无法提供统计误差估计。我们提出将MLQMC问题重构于贝叶斯求积框架中，该框架使用单一LD序列，并通过高斯过程（GP）模型的后验方差来量化数值误差。当与特定LD序列结合时，GP回归与超参数优化的计算成本仅为$\mathcal{O}(n \log n)$，其中$n$为样本数量。基于传统MLQMC中使用的自适应样本分配策略（即在预期收益最大的层级上将样本数加倍），我们引入了一种新的贝叶斯效用函数，该函数在加倍的计算成本与预期后验不确定性的减少之间进行权衡。我们还提出了一种新的自适应平滑度的数字平移不变（DSI）核函数，该核通过加权求和多个高阶DSI核的平滑度参数，适用于快速数字网GP。一系列数值实验展示了我们提出的快速贝叶斯MLQMC方法在单层问题及固定层数多层问题中的性能与误差估计结果。使用数字网获得的贝叶斯误差估计被证明是可靠的，尽管在某些情况下略显保守。