Model, identification & analysis of complex stochastic systems: applications in stochastic partial differential equations and multiscale mechanics

UnrestrictedThis dissertation focusses on characterization, identification and analysis of stochastic systems. A stochastic system refers to a physical phenomenon with inherent uncertainty in it, and is typically characterized by a governing conservation law or partial differential equation (PDE) with some of its parameters interpreted as random processes, or/and a model-free random matrix operator. In this work, three data-driven approaches are first introduced to characterize and construct consistent probability models of non-stationary and non-Gaussian random processes or fields within the polynomial chaos (PC) formalism. The resulting PC representations would be useful to probabilistically characterize the system input-output relationship for a variety of applications. Second, a novel hybrid physics- and data-based approach is proposed to characterize a complex stochastic systems by using random matrix theory. An application of this approach to multiscale mechanics problems is also presented. In this context, a new homogenization scheme, referred here as "nonparametric" homogenization, is introduced. Also discussed in this work is a simple, computationally efficient and experiment-friendly coupling scheme based on frequency response function. This coupling scheme would be useful for analysis of a complex stochastic system consisting of several subsystems characterized by, e.g., stochastic PDEs or/and model-free random matrix operators. While chapter 1 sets up the stage for the work presented in this dissertation, further highlight of each chapter is included at the outset of the respective chapter