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Mathematics Department

Colloquium Series

Fall 2021

All talks are 3:45-4:45 p.m. online at this Google Meet link

  • Nov
    03
  • RONS: Reduced-order nonlinear solutions for PDEs with conserved quantities
    Mohammad Farazmand
    North Carolina State University
    Location: Virtual Talk
    Time: 03:45 PM

    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.
  • Oct
    27
  • Statistical reduced-order models and machine learning-based closure strategies for turbulent dynamical systems
    Di Qi
    Purdue University
    Location: Virtual Talk
    Time: 03:45 PM

    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.
  • Oct
    13
  • Frameworks in motion: theory, design, and fabrication
    Jessica Sidman
    Mount Holyoke College
    Location: Virtual Talk
    Time: 03:45 PM

    View Abstract

    What do your umbrella, a folding gate, and a scissor lift have in common? They are all made of rigid parts attached at joints designed to yield a structure that can move with one degree of freedom and then locked in a rigid state to perform a useful function. In 1981, famed architect Santiago Calatrava wrote a PhD thesis, "Concerning the Foldability of Space Frames," consisting of a systematic exploration of the geometry and design of foldable frameworks. I'll use his thesis as a jumping off point to explore the fundamentals of rigidity theory and share some ongoing work on the design of a tent framework in collaboration with architect Naomi Darling and Mount Holyoke students Sohini Bhatia, Stephanie Einstein, Nana Aba Turkson, and Zainab Umar.
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