This paper presents a multilevel quasi-Monte Carlo method for interval analysis, as a computationally efficient method for high-dimensional linear models. Interval analysis typically requires a global optimization procedure to calculate the interval bounds on the output side of a computational model. The main issue of such a procedure is that it requires numerous full-scale model evaluations. Even when simplified approaches such as the vertex method are applied, the required number of model evaluations scales combinatorially with the number of input intervals. This increase in required model evaluations is especially problematic for highly detailed numerical models containing thousands or even millions of degrees of freedom. In the context of probabilistic forward uncertainty propagation, multifidelity techniques such as multilevel quasi-Monte Carlo show great potential to reduce the computational cost. However, their translation to an interval context is not straightforward due to the fundamental differences between interval and probabilistic methods. In this work, we introduce a multilevel quasi-Monte Carlo framework. First, the input intervals are transformed to Cauchy random variables. Then, based on these Cauchy random variables, a multilevel sampling is designed. Finally, the corresponding model responses are post-processed to estimate the intervals on the output quantities with high accuracy. Two numerical examples show that the technique is very efficient for a medium to a high number of input intervals. This is in comparison with traditional propagation approaches for interval analysis and with results well within a predefined tolerance.

中文翻译：

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用于区间分析的多级准蒙特卡罗

本文提出了一种用于区间分析的多级准蒙特卡罗方法，作为高维线性模型的计算有效方法。区间分析通常需要全局优化过程来计算计算模型输出端的区间界限。这种程序的主要问题是它需要大量的全尺寸模型评估。即使应用了简化方法（例如顶点方法），所需的模型评估数量也会与输入间隔的数量相结合。对于包含数千甚至数百万自由度的高度详细的数值模型来说，所需模型评估的增加尤其成问题。在概率前向不确定性传播的背景下，多保真技术（例如多级准蒙特卡罗）显示出降低计算成本的巨大潜力。然而，由于区间和概率方法之间的根本差异，将它们转换为区间上下文并不简单。在这项工作中，我们引入了一个多级准蒙特卡罗框架。首先，输入区间被转换为柯西随机变量。然后，基于这些柯西随机变量，设计了多级抽样。最后，对相应的模型响应进行后处理，以高精度估计输出量的区间​​。两个数值示例表明，该技术对于中等到大量的输入间隔非常有效。