Computational Methods and Data Assimilation in Nonlinear Dynamics

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Computer simulation is a critical ingredient of modern science. Simulations that depend on the solution of differential equations are subject to small modeling, truncation, and floating point arithmetic errors. A relatively new set of algorithms, called set-oriented methods, have been developed for the analysis of structures in computational dynamics. We propose to build a foundation for analyzing the possible effect of the errors on these techniques and others used in computational dynamics. 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 willinvestigate 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. Complex deterministic time series from physical, and biological/medical settings that are produced by network dynamics will be particular applications.The project focuses on two major areas: the development of new approaches to study computer simulation validity, and the interpretation of experimental data from nonlinear processes. Computer simulation is a critical ingredient of modern science. Simulations are subject to errors both in modeling and computation. The proposal builds a foundation for analyzing the possible effect of the errors on set-oriented 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 can be used to attempt to reconstruct the model and dynamical behavior, with potential to predict or control the process.