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Mathematics Department

Upcoming Talks

This is a list of all upcoming talks for the next two weeks. Talks are from 3:45-4:45 p.m. in the Colloquium or Seminar Room, unless otherwise specified.

  • Oct
    27
  • Statistical reduced-order models and machine learning-based closure strategies for turbulent dynamical systems
    Di Qi
    Purdue University
    Location: https://meet.google.com/bkq-sumw-gxe
    Time: 03:45 PM
    Colloquium Series

    View Abstract

    The capability of using imperfect statistical reduced-order models to capture crucial statistics in complex turbulent systems is investigated. Much simpler and more tractable block-diagonal models are proposed to approximate the complex and high-dimensional turbulent dynamical equations using both parameterization and machine learning strategies. A systematic framework of correcting model errors with empirical information theory is introduced, and optimal model parameters under this unbiased information measure can be achieved in a training phase before the prediction. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the reduced-order model in various dynamical regimes of the flow field with distinct statistical structures. In addition, new machine learning strategies are proposed to learn the expensive unresolved processes directly from data.
  • Nov
    03
  • RONS: Reduced-order nonlinear solutions for PDEs with conserved quantities
    Mohammad Farazmand
    North Carolina State University
    Location: https://meet.google.com/bkq-sumw-gxe
    Time: 03:45 PM
    Colloquium Series

    View Abstract

    Reduced-order models of time-dependent partial differential equations (PDEs) where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time-dependent variables have thus far been derived in an ad hoc manner. I introduce Reduced-order Nonlinear Solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent variables. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The variables are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. This results in a set of explicit ordinary differential equations on the tangent bundle of the manifold. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems. I demonstrate its applications on a few examples including the nonlinear Schrodinger equation and Euler's equation for ideal fluids.
  • Nov
    05
  • TBA
    Kostya Medynets
    USNA Math
    Time: 12:00 PM
    Applied Math Seminar
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