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

Columbia University, New York NY

Investigators

Abstract

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.