Analysis of Stochastic Dynamical Systems
Stochastic dynamical systems (SDSs) generally involve processes where intrinsic variability affects system behavior, for example due to phenomena on a nanoscale, molecular level, where thermal noise introduces variability in molecular interactions. The intrinsic noise in these SDSs requires them to be modeled as continuous time, discrete-state Markov processes. As the models for such systems are often inferred from noisy observations of a subset of the system state at sparse instances in time, these SDSs generally also have large levels of parametric and model uncertainties, further complicating their simulation and analysis. Despite considerable advances in the formulation and simulation of such systems, more effective tools are needed for their analysis in terms of sensitivity, uncertainty, and dynamics.
Various methods are being developed to analyze SDSs. One class of approaches relies on sampled solutions of the chemical master equation, which governs stochastic dynamical systems. Polynomial chaos (PC) representations are used to represent the effects of intrinsic noise or parametric uncertainty on these systems. Significant advances were made in the areas of data partitioning methods, relying on clustering algorithms to improve the handling of multimodal data distributions. Other approaches, which are the subject of ongoing work, focus on direct solutions of the chemical master equation.
Overall, the advanced simulation and analysis methods developed in this work will enable detailed studies of SDSs to better understand the underlying systems, their dynamics, and the associated uncertainties. Such insight is essential to utilize stochastic processes in, for example, gene regulation, cell signaling, and interfacial electrochemistry; relevant to bioremediation and bioenergy (bacterial behavior), biomedicine (immune system signaling), and electrical storage (electrodes).