Uncertainty Characterization and Modeling using Positive-definite Random Matrix Ensembles and Polynomial Chaos Expansions

Dr. Sonjay Das

Abstract:  The scientific community is quite familiar with random variables, or more precisely, scalar-valued random variables. They are used to characterize uncertainties in physical and model parameters of stochastic systems. In general, these parameters cannot be characterized, estimated, or chosen independently. In many applications, several of these parameters are often required to be mapped to a space of matrices that may need to preserve symmetry and positive-definiteness, resulting in matrix-valued random model parameters. This ensures that the underlying theoretical principles; physics-based criteria; or empirical evidence, that play critical roles in modeling and analyzing the stochastic system, are not violated. Furthermore, numerical discretization of governing equations of the stochastic system in many applications yields system-level symmetric and positive-definite random matrices that have much larger dimension (often in the order of hundreds or tens of thousands or even more). Random matrix theory provides a significant computational and experimental advantage in characterizing and modeling uncertainties of the matrix-valued random model parameters and large-dimensional system-level random matrices, particularly, for complex and large-scale applications that involve several hundreds or thousands of statistically dependent scalar-valued random variables. In the first half of this presentation, we would discuss a few relevant symmetric and positive-definite random matrices, e.g., Wishart/Gamma distribution, Beta Type 1 distribution, and Kummer-Beta distribution. We would also highlight certain limitations of this class of random matrices and introduce two new positive-definite random matrix ensembles that can provide more flexibility in characterizing high level of uncertainties.

The second half of this talk will digress slightly and focus on construction of polynomial chaos (PC) representations directly from the experimental measurements. The PC formalism provides a theoretically sound backbone that facilitates efficient and systematic construction of probabilistic descriptions of conventional model-based predictions in diverse fields of applications. The proposed techniques will be illustrated by characterizing a set of oceanographic temperature records obtained from a shallow-water acoustics transmission experiment that exhibits strong non-stationary and non-Gaussian features.

Speaker’s Bio: Since Feb 2019, Dr. Sonjoy Das is affiliated with the Department of Mathematics at SUNY Buffalo State College, Buffalo, New York, as an Adjunct Associate Professor. Prior to this position, he was an Assistant Professor in the Department of Mechanical and Aerospace Engineering, University at Buffalo (MAE/UB), New York, from December 2010—January 2019. Before joining MAE/UB, he completed his postdoctoral training at the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts, and at the Department of Aerospace and Mechanical Engineering, University of Southern California (USC), Los Angeles, California. He obtained his PhD from USC in 2008. He has two MS degrees – one from the Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland, and another from the Indian Institute of Science, Bangalore, India in Structural Engineering. As a faculty member, he advised 2 PhD students, 4 MS students, and mentored a few undergraduate and high school students. His general research interest revolves around uncertainty quantification, data analysis and machine learning, and their implementations through computer algorithms for a variety of scientific and engineering problems with particular focus on multiscale material modeling, nonlinear mechanics, computational stochastic mechanics, and multi-physics simulations such as brain mechanics and 3D printing. His research was funded by National Science Foundation and Industry.

Last Updated: November 5, 2020 - 2:12 pm