New Advances in Uncertainty Analysis and Estimation

s for Talks: Uncertainty Modeling: (Speaker: Raktim Bhattacharya) – There are many aspects of uncertainty in engineering systems that are broadly classified as aleatory or epistemic. Aleatory uncertainty includes uncertainty in initial conditions and parameters, which models structured uncertainty or known unknowns. Examples of epistemic uncertainty include unmodeled dynamics or incomplete knowledge of physical systems, or a source of uncertainty that has no known structure. We classify them as unknown unknowns. We will highlight these classes of uncertainties with examples from various engineering disciplines and discuss how they can be modeled for purposes of uncertainty quantification. Short Review of Probability and Stochastic Processes (Speaker: Puneet Singla) – We will review basic concepts related to random variables, random process, conditional probability density function, Bayesian inference, entropy, Kullback-Fisher information, continuous and discrete random process, Brownian motion, and white noise. These will provide the necessary mathematical background for the material presented later in the workshop. Propagation of PDF by solving Kolmogorov Equation (Speaker: Puneet Singla, Raktim Bhattacharya) – This talk will focus on recent development in computational methods for uncertainty characterization and forecasting 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. Analytical solutions for the FPKE/CKE exist only for a stationary pdf and are restricted to a limited class of dynamical systems. Traditional numerical approaches based upon variational formulation which discretize the space in which the pdf lies, suffer from the “curse of dimensionality.” In this talk, we will discuss that how one can make use of recent advances in approximation theory to not only break the “curse of dimensionality” but can also pose the the pdf evolution problem as a convex optimization problem with guaranteed convergence. In particular, the audience will be introduced to use of Gaussian mixture model based approaches to solve both the differential and integral form of the Kolmogorov equation. For systems with weak diffusion, the FPKE can be simplified to the continuity equation. This results in considerable simplification in the solution of the governing equation. The continuity equation being first order linear can be solved using method of characteristics and Rothe’s method. Additionally, we will describe use of maximum entropy basis functions to approximate the PDF evolution in a mesh less computational framework. Spectral Representation and Moment Propagation (Speaker: Raktim Bhattacharya) – We will discuss approximation of random processes using KL and polynomial chaos expansions, and describe when these approximation techniques can be applied. We will discuss how these techniques can be applied to both linear and nonlinear systems. We will also highlight the computational complexity associated with polynomial chaos approximation, in particular with Galerkin projections, and make a case for stochastic collocation techniques. Examples illustrating strengths and weakness of spectral approximations will also be presented. Implementation details and accuracy of approximations will be discussed using few numerical examples. Quadrature methods (Speaker: Puneet Singla) – This talk will introduce the theory of Gauss quadrature methods to evaluate expectation integral involving a generic density function. Quadrature methods involve an approximation of the expectation integral as a weighted sum of integrand values at specified points within the domain of integration. A quadrature rule is said to be exact to degree d, if it can only integrate all polynomials with degree ≤ d. For 1-D (1Dimensional) integrals, one needs N quadrature points according to the Gaussian quadrature scheme to exactly reproduce the expectation integrals of polynomials with degree 2N-1 or less, . However, in generic n-D, one needs to take the tensor product of 1-D quadrature points and hence would yield a total of N quadrature points. This is a non-trivial number of points that might make the calculation of the integral computationally expensive, especially when the evaluation of function at each cubature point itself can be an expensive procedure. This talk will introduce recently developed Conjugate Unscented Transformation (CUT) approach to accurately evaluate expectation integrals in high dimension space while minimizing the number of simulations. Rather than using tensor products as in Gauss quadrature, the CUT approach judiciously selects specific structures to extract symmetric quadrature points. Several benchmark problems will be considered to highlight relative merits of various algorithms and their use in stochastic collocation. Applications to Estimation & Filtering (Speakers: Puneet Singla and Raktim Bhattacharya) – This talk will introduce the concept of model-data fusion, which has its birth with the development of Kalman filter for linear system. We will discuss that how various uncertainty propagation methods introduced in prior sections can be used along with Bayes’ rule to find system state and parameter estimates. In addition, the concept of maximum likelihood estimation and best linear unbiased estimator will be discussed. Relative merits of different approaches will be discussed while considering various benchmark problems. Bio-sketches for Speakers: Puneet Singla: Dr. Puneet Singla is an Associate Professor of Mechanical & Aerospace engineering at the University at Buffalo (UB), the State University of New York. He received his bachelor’s degree in Aerospace Engineering from Indian Institute of Technology, Kanpur, India in 2000 and earned his master’s and doctoral degree in Aerospace Engineering from Texas A&M University, College Station in 2002 and 2006, respectively. His research work includes three thrusts: 1) characterization and propagation of uncertainties through dynamical systems, 2) design of computationally efficient data assimilation algorithms for large scale problems, and 3) design robust methodologies for optimal sensor management while taking into account the uncertainties in the system dynamics. The research layer surrounding the focus areas includes approximation theory, study of stochastic systems, nonlinear filtering and control. The theoretical developments form the framework for diverse problems such as dispersion & transport of toxic material clouds through the atmosphere, tracking resident space objects, tumor motion modeling for conformal radiation therapy, flow control and control of robotic systems. During his tenure at UB, he has secured several research grants as a PI or co-PI from the National Science Foundation (NSF), Air Force Office of Scientific Research (AFOSR), Air Force Research Laboratory (AFRL) and the National Geospatial Intelligence Agency (NGA). He is a recipient of the competitive NSF CAREER award for his work on Uncertainty Propagation and Data Assimilation for Toxic Cloud Prediction and the AFOSR Young Investigator Award for his work on Information Collection and Fusion for Space Situational Awareness. He has also been awarded the UB’s “Exceptional Scholar” Young Investigator Award in recognition of his scholarly activities. He has authored over 100 papers to-date including 25 journal articles covering a wide array of problems, including: attitude estimation, nonlinear estimation, dynamics and control, adaptive control, approximation theory, including novel methods for solving the Fokker-Planck-Kolmogorov equation (FPKE) for uncertainty propagation. He is the principal author of a new textbook entitled “Multi-Resolution Methods for Modeling and Control of Dynamical Systems,” (300 pages) published in August 2008 by CRC Press (Boca Raton, FL). He has received the best paper awards at the 2006 AIAA/AAS Astrodynamics Specialists Conference and 2009 International Information Fusion Conference for his work on uncertainty propagation. Raktim Bhattacharya received his M.S. and PhD in Aerospace Engineering from the University of Minnesota in 2000 and 2003 respectively. He was a postdoctoral researcher in Control & Dynamical Systems at Caltech from 2003 to 2004. He spent 2004-2005 at United Technologies Research Center, East Hartford, CT, as a research scientist in the Controls and Embedded Systems Group. He joined the Aerospace Engineering department at Texas A&M University on 2005, and is currently an associate professor. He has published several journal & conference papers and book chapters in the area of probabilistic robust control, nonlinear estimation, UQ in hypersonic flight problems, nonlinear trajectory generation, anytime control algorithms, and receding horizon control methodologies. NASA and NSF have funded his research.