DynSyst_Special_Topics: Collaborative Research: Reduced Dynamical Descriptions of Infinite-Dimensional Nonlinear systems via a-Priori Basis Functions from Upper Bound Theories

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The aim of this interdisciplinary collaborative research project is to develop a novel model reduction technique for forced dissipative infinite-dimensional dynamical systems by employing basis functions computed using upper bound theories. Like popular Proper Orthogonal Decomposition (POD) based methods, this approach associates the condensed variables needed for model reduction with coherent structures and captures nonlinear interactions between these linear modes via Galerkin projection and finite-dimensional truncation. Unlike empirical POD methods, however, this new method does not require extensive data sets from experiments or direct numerical simulations of the governing partial differential equations (PDEs) and thus yields truly predictive reduced models. The theoretical and computational methodology will be developed in the context of a particular physical system, thermal convection in fluid saturated porous media, that is of considerable environmental and technological importance and an ideal testbed for new ideas.This research will contribute to the development of a general methodology for deriving simplified mathematical models of highly complex dynamical systems arising in diverse areas of science and engineering. In many applications of interest (e.g., control of various fluid flows to achieve drag reduction for oil pumped in pipelines or for air flowing past commercial jets, or for estimation of carbon dioxide sequestration by porous rock material for reducing global warming), direct numerical simulations based on the complete governing mathematical equations are infeasible using even the world's fastest high-performance supercomputers. This project will address these challenges using novel mathematical techniques to derive simplified equations directly from the governing physical laws that are amenable to practical computation and analysis.