Numerical Analysis and Scientific Computing Seminar | Hessam Babaee, On-the-fly Reduced Order Modeling with Time Dependent Subspaces

For Zoom Link please contact ddelreyfernandez@uwaterloo.ca  

Speaker

Hessam Babaee, University of Pittsburgh Department of Mechanical Engineering and Materials Science

Title

On-the-fly Reduced Order Modeling with Time Dependent Subspaces

Abstract

Many important problems in fluid mechanics are described by high-dimensional partial differential equations (PDEs). The computational cost of solving these problems using classical discretization techniques increases exponentially with respect to the number of dimensions –– a fundamental challenge that is dubbed as the curse of dimensionality. On the other hand, many of these high-dimensional problems have a much lower intrinsic dimensionality, that if discovered,  can mitigate the curse of dimensionality.  This calls for techniques that extract and exploit correlated structures directly from the PDE. This approach is in direct contrast to classical discretization techniques that disregard multi-dimensional correlations and result in inefficient solutions for high-dimensional problems. While there are numerous data-driven dimension reduction techniques that can extract these correlated structures by solving the full-dimensional PDE, these techniques are only feasible for lower-dimensional PDEs (e.g., 2D/3D). This same workflow is impracticable for many high-dimensional PDEs as computing the solution of the full-dimensional PDE is the very problem we cannot afford to solve. To this end, we present a reduced order modeling framework, in which the correlated structures are extracted directly from the PDE –– bypassing the need to generate data.  These structures are exploited by building on-the-fly reduced order models (ROM). The correlated structures are represented by a set of time-dependent orthonormal bases and their evolution is prescribed by the physics of the problem.  We present several demonstration cases including reduced order modeling of reactive species transport equation in turbulent combustion as well as sensitivity analysis and uncertainty quantification in fluid dynamics problems.