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Adaptive Monte Carlo analysis for strongly nonlinear stochastic systems

机译：强非线性随机系统的自适应蒙特卡洛分析

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This paper compares space-filling and importance sampling (IS)-based Monte Carlo sample designs with those derived for optimality in the error of stratified statistical estimators. Space-filling designs are shown to be optimal for systems whose response depends linearly on the input random variables. They are, however, shown to be far from optimal when the system is nonlinear. To achieve optimality, it is shown that samples should be placed densely in regions of large variation (sparsely in regions of small variation). This notion is shown to be subtly, but importantly, different from other non-space-filling designs, particularly IS. To achieve near-optimal sample designs, the adaptive Gradient Enhanced Refined Stratified Sampling (GE-RSS) is proposed that sequentially refines the probability space in accordance with stratified sampling. The space is refined according to the estimated local variance of the system computed from gradients using a surrogate model. The method significantly reduces the error in stratified Monte Carlo estimators for strongly nonlinear systems, outperforms both space-filling methods and IS-based methods, and is simple to implement. Numerical examples on strongly nonlinear systems illustrate the improvement over space-filling and IS designs. The method is applied to study the probability of shear band formation in a bulk metallic glass. (C) 2018 Elsevier Ltd. All rights reserved.
机译：本文将基于空间填充和重要性抽样（IS）的蒙特卡洛样本设计与分层统计估计误差的最优性进行了比较。对于响应线性依赖于输入随机变量的系统，显示空间填充设计是最佳的。但是，当系统为非线性时，它们显示出远非最佳。为了达到最佳效果，表明应该将样本密集地放置在变化较大的区域中（稀疏放置在变化较小的区域中）。该概念显示出与其他非空间填充设计（尤其是IS）有微妙但重要的区别。为了实现接近最佳的样本设计，提出了一种自适应梯度增强细化分层采样（GE-RSS），该方法根据分层采样顺序细化了概率空间。根据使用代理模型从梯度计算得出的系统估计局部方差来精简空间。该方法显着减少了强非线性系统的分层蒙特卡洛估计器中的误差，优于空间填充方法和基于IS的方法，并且易于实现。关于强非线性系统的数值示例说明了对空间填充和IS设计的改进。该方法用于研究大块金属玻璃中剪切带形成的可能性。 （C）2018 Elsevier Ltd.保留所有权利。

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