Multilevel randomized quasi-Monte Carlo estimator for nested integration

TL;DR

This paper introduces a multilevel randomized quasi-Monte Carlo estimator for nested integrals, significantly improving efficiency and accuracy in high-dimensional, complex integration problems common in scientific and engineering applications.

Contribution

It presents a novel multilevel estimator combining deterministic and randomized quasi-Monte Carlo methods, with theoretical error bounds and demonstrated superior performance over existing approaches.

Findings

Reduces bias and variance compared to standard methods

Outperforms existing MC and rQMC approaches in numerical experiments

Effective in scenarios with approximate integrand evaluations

Abstract

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. When applied to Gaussian noise in the experiment, a truncation scheme ensures finite error bounds at 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.