MULTILEVEL RANDOMIZED QUASI-MONTE CARLO ESTIMATOR FOR NESTED INTEGRATION

作     者：Bartuska, Arved Carlon, André Gustavo Espath, Luis Krumscheid, Sebastian Tempone, Raúl 

作者机构：Department of Mathematics RWTH Aachen University Gebäude-1953 1.OG Pontdriesch 14-16 161 Aachen52062 Germany  Thuwal23955-6900 Saudi Arabia School of Mathematical Sciences University of Nottingham NottinghamNG7 2RD United Kingdom RWTH Aachen University Germany Scientific Computing Center Institute for Applied and Numerical Mathematics Karlsruhe Institute of Technology Germany 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Risk assessment 

摘      要：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 R f (R g(y, x) dx) dy, 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 *** Codes 62F15, 65C05, 65D30, 65D32 © 2024, CC0.