CAREER: A Path Integral Methodology for Accurate and Computationally Efficient Stochastic Analysis of Diverse Dynamical Systems

这个教师早期职业发展计划（Career）项目将支持那些将为分析具有随机行为的复杂工程系统的行为提供新知识的研究，这样做将促进科学的进步和促进国家的繁荣。为了分析和最终设计结构系统，需要强大的数学工具来解释复杂的响应行为以及建模过程中存在的不确定性。当前最先进的分析技术要么表现出高精度，要么表现出计算效率，但不能两者兼而有之。这不足以进行适当的系统分析、设计和优化。该方法的独特之处在于，它显示出相当高的准确性和计算效率，将把当前随机分析的能力推向前所未有的水平。这将导致从纳米机械振荡器到振动能量采集器和低成本民用基础设施系统的高质量多样化工程系统优化和设计的范式转变。这项研究的结果将有利于美国经济和整个社会。多样化的教育和推广计划还将通过教学创新影响下一代研究人员和实践工程师，以及学生和教育工作者的进步。用于求解控制方程和确定动力系统随机响应的可用技术大致可分为两类：a)那些表现出高精度，但由于计算成本过高而只能处理极少数随机维度的技术；b)那些可以很容易地处理高维系统，但只能提供低阶响应统计量（例如平均值和标准差）的可靠估计的技术。与目前最先进的技术相比，路径积分方法将表现出优越的精度和计算效率。为了达到较高的精度，研究方法是在相关的路径积分展开中考虑高阶项（波动）。同时对方法误差进行量化。为了获得较高的计算效率，并考虑到大量的随机维度（100），研究方法是探索系统响应的高度稀疏表示，并结合适当的优化算法。该方法也将是通用的，考虑复杂的随机激励和系统建模的情况，包括分数阶导数，非高斯，非线性和滞后响应行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This Faculty Early Career Development Program (CAREER) project will support research that will contribute new knowledge for analyzing behavior of complex engineering systems that have random behavior and, in doing so, will promote both the progress of science and advance national prosperity. To analyze and eventually design structural systems requires potent mathematical tools to account for complex response behaviors as well as for the presence of uncertainties in the modeling process. The current state-of-the-art analysis techniques exhibit either high accuracy or computational efficiency but not both. This is inadequate for proper system analysis, design and optimization. The unique feature of the approach, that exhibits both considerable accuracy and computational efficiency, will push the current capabilities of stochastic analysis to unprecedented levels. This will lead to a paradigm shift in the optimization and design of higher quality diverse engineering systems ranging from nano-mechanical oscillators, to vibratory energy harvesters and civil infrastructure systems at reduced cost. The results from this research will benefit the U.S. economy and society in general. The diverse education and outreach plan will also impact the advancement of next-generation researchers and practicing engineers, as well as students and educators via the teaching and learning innovations. Available techniques for solving the governing equations and determining the stochastic response of dynamical systems can be broadly divided into two categories: a) those that exhibit high accuracy, but can handle a very small number of stochastic dimensions due to prohibitive computational cost, and b) those that can readily treat high-dimensional systems, but provide reliable estimates for low-order response statistics only (e.g. mean and standard deviation). In comparison to the current state-of-the-art techniques, the path integral methodology will exhibit superior accuracy and computational efficiency. To achieve high accuracy the research approach is to account for higher order terms (fluctuations) in related path integral expansions. At the same time the error of the methodology will be quantified. To achieve high computational efficiency and account for a large number of stochastic dimensions (100), the research approach is to explore highly sparse representations for the system response in conjunction with appropriate optimization algorithms. The methodology will also be versatile, accounting for cases of complex stochastic excitation and system modeling including fractional derivatives, and non-Gaussian, nonlinear, and hysteretic response behaviors.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.