This talk will address the issue of closure in reduced order models (ROMs) and large eddy simulations (LES), leveraging ideas from non-equilibrium statistical mechanics. The approach is based on the Variational Multi-Scale method (VMS) and the Mori-Zwanzig (M-Z) formalism, which provides a framework to perform formal scale separation and re-cast a high-dimensional dynamical system into an equivalent, lower-dimensional system. In this reduced system, which is in the form of a generalized Langevin equation (GLE), the effect of the unresolved modes on the resolved modes appears as a convolution integral (which is sometimes referred to as memory). The M-Z formalism alone does not lead to a reduction in computational complexity as it requires the solution of the orthogonal dynamics PDE. A model for the memory is constructed by assuming that memory effects have a finite temporal support and by exploiting scale similarity. We discover that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and inter-element jumps (in the case of discontinuous finite elements). It is further shown that an MZ-based finite memory model is a variant of the well-known adjoint-stabilization method. For hyperbolic equations, this stabilization is shown to have the form of an artificial viscosity term. We further establish connections between the memory kernel and approximate Riemann solvers. In the context of ROMs, this model is shown to yield a Petrov-Galerkin projection. Several applications in ROMs and LES ranging from simple scalar PDEs to Magneto-hydro-dynamic turbulence will be presented.
Bio: Karthik Duraisamy is an Associate Professor of Aerospace Engineering at the University of Michigan, Ann Arbor. He obtained a doctorate in aerospace engineering and masters in applied mathematics from the University of Maryland, College Park. He is the director of the Center for Data-driven Computational Physics and the associate director of the Michigan Institute of Computational Discovery and Engineering (MICDE) at the Univ of Michigan. His research interests are in data-driven and reduced order modeling, turbulence modeling and simulations, and numerical methods for PDEs.