Certain Thoughts on Uncertainty Analysis for Dynamical Systems

The main difficulty in the understanding of complex physical dynamic systems is due to uncertainties in the models and measurements. These systems are often studied via numerical simulations with very high dimensional input parameter space. The most critical challenge here is to provide a quantitative assessment of how closely these simulations reflect reality in the presence of model uncertainty, discretization errors, as well as measurement errors. These issues affect the way in which models are constructed, how models are used for performance analysis, and how data is integrated with models for prediction.

This talk will focus on recent development of mathematical and algorithmic fundamentals for uncertainty characterization, forecasting, and data assimilation for nonlinear systems. The central idea is to replace evolution of initial conditions for a large dynamical system by evolution of probability density functions (pdf) for state variables. The use of Fokker-Planck-Kolmogorov equation (FPKE) and Chapman-Kolmogorov equation (CKE) to determine evolution of state pdf due to probabilistic uncertainty in initial or boundary conditions, model parameters and forcing function will be discussed. Recently developed Conjugate Unscented Transformation (CUT) methodology will be presented to compute multi-dimensional expectation integrals. By accurately characterizing the uncertainty associated with both process and measurement models, this work offers systematic design of low-complexity data assimilation algorithms with significant improvement in nominal performance and computational effort. The applicability and feasibility of these new ideas will be demonstrated on benchmark problems and some real world problems such as tracking resident space objects and forecasting volcano ash footprints.

Dr. Puneet Singla is an Associate Professor of Mechanical & Aerospace engineering at the University at Buffalo. He received his bachelor's degree in Aerospace Engineering from Indian Institute of Technology, Kanpur, India in 2000 and earned his doctoral degree in Aerospace Engineering from Texas A&M University in 2006. Dr. Singla's research focus pertains to uncertainty propagation through nonlinear systems, nonlinear filtering, stochastic control, autonomous systems, and modeling of large-scale dynamical systems. He has authored over 90 papers to-date including one book and 18 journal articles covering a wide array of problems. He is the recipient of NSF CAREER award for his work on toxic plume forecasting and AFOSR young investigator award for information collection and fusion for space situational awareness.