OPTIMIZING OBLIQUE PROJECTIONS FOR NONLINEAR SYSTEMS USING TRAJECTORIES

Samuel E. Otto, Alberto Padovan, Clarence W. Rowley

Research output: Contribution to journalArticlepeer-review

Abstract

Reduced-order modeling techniques, including balanced truncation and H2-optimal model reduction, exploit the structure of linear dynamical systems to produce models that accurately capture the dynamics. For nonlinear systems operating far away from equilibria, on the other hand, current approaches seek low-dimensional representations of the state that often neglect low-energy features that have high dynamical significance. For instance, low-energy features are known to play an important role in fluid dynamics, where they can be a driving mechanism for shear-layer instabilities. Neglecting these features leads to models with poor predictive accuracy despite being able to accurately encode and decode states. In order to improve predictive accuracy, we propose to optimize the reduced-order model to fit a collection of coarsely sampled trajectories from the original system. In particular, we optimize over the product of two Grassmann manifolds defining Petrov–Galerkin projections of the full-order governing equations. We compare our approach with existing methods including proper orthogonal decomposition, balanced truncation-based Petrov–Galerkin projection, quadratic-bilinear balanced truncation, and the quadratic-bilinear iterative rational Krylov algorithm. Our approach demonstrates significantly improved accuracy both on a nonlinear toy model and on an incompressible (nonlinear) axisymmetric jet flow with 105 states.

Original languageEnglish (US)
Pages (from-to)A1681-A1702
JournalSIAM Journal on Scientific Computing
Volume44
Issue number3
DOIs
StatePublished - 2022

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Grassmann manifold
  • Petrov–Galerkin projection
  • Riemannian optimization
  • adjoint sensitivity method
  • model reduction
  • nonlinear systems

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