A general frame for uncertainty propagation under multimodally distributed random variables
A general frame for uncertainty propagation under multimodally distributed random variables
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DOI：
10.1016/j.cma.2020.113109
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发表时间：
2020-08
影响因子：
7.2
通讯作者：
Xianghua Meng;Jie Liu;Lixiong Cao;Zhongbo Yu;Dongmin Yang
中科院分区：
文献类型：
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作者：
Xianghua Meng;Jie Liu;Lixiong Cao;Zhongbo Yu;Dongmin Yang
多模态分布随机变量下的不确定性传播，简称多模态分布传播，由于随机变量的概率密度函数复杂，是一个具有挑战性的问题。提出了一种基于导数λ概率密度函数和多项式混沌展开方法构造的有限混合模型（λ MM）的通用框架，有效地解决了多峰分布传播问题。首先，提出了一种具有高精度和广泛适用性的λ MM来表示任意的单峰分布和多峰分布。其次，提出了一种严格证明的伪EM方法来估计该λ MM的参数。第三，从数学上推导了多项式混沌展开法响应的统计矩。由于新的混合模型可以分解为多个单峰概率分布，因此可以将多峰分布的传播转化为多个单峰分布的传播，进而可以用多项式混沌展开法求解。最后，采用最大熵原理求出各单峰分布传播结果的概率密度函数，并将其叠加到系统响应的最终概率密度函数中。该框架只需要前四个统计矩来估计概率密度函数，避免了计算更高的统计矩，特别是对于系统响应的概率分布是多模态的情况。四个例子来验证所提出的一般框架的不确定性传播的高精度和效率。
Uncertainty propagation under multimodally distributed random variables, called multimodal distribution propagation for short, is a challenging problem due to the complicate probability density function of the random variables. In this paper, a general frame based on a new finite mixture model (λ MM) constructed by derivative lambda probability density function and polynomial chaos expansion method is put forward to efficiently solve the multimodal distribution propagation problem. First, λ MM with high accuracy and extensive applicability is proposed to represent the arbitrary unimodal distributions and multimodal distributions. Second, a pseudo EM method proved strictly is proposed to estimate the parameters of this λ MM. Third, the statistical moments of the response by polynomial chaos expansion method is derived mathematically. Since the new mixture model can be decomposed into several unimodal probability distributions, the multimodal distribution propagation can be successfully converted into several unimodal distribution propagations, which further can be easily solved by polynomial chaos expansion method. Finally, the maximum entropy principle is adopted to evaluate the probability density function of the result of every unimodal distribution propagation, which is then superimposed into the final probability density function of the system response. The proposed frame only requires the first four statistical moments to evaluate the probability density function and avoid calculating the higher statistical moments, especially for the situation that the probability distribution of the system response is multimodal. Four examples are presented to verify the high accuracy and efficiency of the proposed general frame for uncertainty propagation.