Investigations are proposed at the overlap of computational mathematics and applied dynamical systems, including improving our understanding of the role of chaotic dynamics in computer simulations, and new computational methods for inferring information about dynamical structure from time series data. The work on simulation is an ongoing study of large statistical errors occurring in deterministic modeling, including computational models of complex systems. The second major area is work by the investigator on fundamental questions in the interpretation of experimental multivariate time series, including data from physical and biological experiments. Particular attention will be paid to complex deterministic time series from physical, chemical, engineering and biological/medical settings that are produced by network dynamics. We propose new methods to solve discrimination problems, infer network structure and dynamics, and to track and reduce noise from trajectories using data assimilation methods in cases where the structure is known.
The project focuses on the development of new approaches to study computer simulation validity, and to the interpretation of experimental data. Computer simulation is a critical ingredient of modern science. Simulations that depend on the solution of differential equations are subject to small modeling and truncation errors. The proposal builds a foundation for analyzing the possible effect of the errors on long-term simulation results, for physically relevant models. The second major focus of this project is the interpretation of data collected from experimental systems and nature when only a partial mathematical model is available. We will investigate a range of techniques which, depending on the completeness of the available model, can be used to attempt to reconstruct the model and dynamical behavior of the process, with potential to predict or control the process. Both parts of the project have implications in many areas of sciences and engineering, and special relevance to biological systems.
