Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions
Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions
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DOI：
10.1002/nme.4437
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发表时间：
2013-04
影响因子：
2.9
通讯作者：
V. Yadav;S. Rahman
中科院分区：
文献类型：
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作者：
V. Yadav;S. Rahman
本文的中心主题是求解高维随机问题的乘性多项式维分解(PDD)方法。当随机响应主要是乘性响应时，基于加性函数分解的标准PDD近似可能不能提供足够精确的复杂系统的概率解。为了绕过这个问题，开发了两种乘法版本的PDD，称为因式分解PDD和对数PDD。这两个版本都涉及多元函数的分层乘法分解，用于分量函数的傅里叶多项式展开的广泛的正交化多项式基，以及用于估计展开系数的降维或采样技术。解决了三个涉及数学函数或不确定动态系统的数值问题，以证实乘法PDD如何以及何时比加法PDD更有效或更准确。结果表明，当随机响应的隐含乘性结构存在时，因子分解的PDD近似和对数PDD近似都能有效地利用它。由于乘法PDD循环使用加法PDD的相同分量函数，因此不会产生额外的成本。最后，对包含40个随机变量的运动型多功能车的随机本征解进行了评估，验证了新方法解决工业规模问题的能力。版权所有©2013 John Wiley&Sons，Ltd.
The central theme of this paper is multiplicative polynomial dimensional decomposition (PDD) methods for solving high‐dimensional stochastic problems. When a stochastic response is dominantly of multiplicative nature, the standard PDD approximation, predicated on additive function decomposition, may not provide sufficiently accurate probabilistic solutions of a complex system. To circumvent this problem, two multiplicative versions of PDD, referred to as factorized PDD and logarithmic PDD, were developed. Both versions involve a hierarchical, multiplicative decomposition of a multivariate function, a broad range of orthonormal polynomial bases for Fourier‐polynomial expansions of component functions, and a dimension‐reduction or sampling technique for estimating the expansion coefficients. Three numerical problems involving mathematical functions or uncertain dynamic systems were solved to corroborate how and when a multiplicative PDD is more efficient or accurate than the additive PDD. The results show that indeed, both the factorized and logarithmic PDD approximations can effectively exploit the hidden multiplicative structure of a stochastic response when it exists. Since a multiplicative PDD recycles the same component functions of the additive PDD, no additional cost is incurred. Finally, the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated, demonstrating the ability of the new methods to solve industrial‐scale problems. Copyright © 2013 John Wiley & Sons, Ltd.