AI 中文总结
针对高维问题和高性能计算环境下多级采样方法扩展的挑战，提出预算约束的MLMC方法和MLSGD方法，在HPC系统上验证了其在不确定性量化和最优控制应用中的有效性。
AI 中文摘要
本文介绍资源约束下多级采样方法的最新进展。过去15年，多级方法已广泛应用于不确定性量化，但将其扩展到高维问题和高性能计算（HPC）环境仍具挑战性。本研究讨论两种解决该问题的算法：预算约束的多级蒙特卡洛（MLMC）方法和多级随机梯度下降（MLSGD）方法，在考虑可用计算资源的HPC系统上验证了它们在前向不确定性量化（UQ）和不确定性下的最优控制（OC）应用中的有效性。
英文摘要
We present recent developments in multilevel sampling methods under resource constraints. Over the past 15 years, multilevel methods have become widely used for uncertainty quantification. However, scaling them to high-dimensional problems and high-performance computing (HPC) environments remains challenging. In this work, we discuss two algorithms designed to address these issues: the budgeted Multilevel Monte Carlo (MLMC) method and the Multilevel Stochastic Gradient Descent (MLSGD) method. We demonstrate their effectiveness on HPC systems under consideration of the available computational resources for applications in forward uncertainty quantification (UQ) and optimal control (OC) under uncertainty.
