Geometric subspace updates with applications to online adaptive nonlinear model reduction

Ralf Zimmermann, Benjamin Peherstorfer, Karen Willcox

Research output: Contribution to journalArticlepeer-review

Abstract

In many scientific applications, including model reduction and image processing, subspaces are used as ansatz spaces for the low-dimensional approximation and reconstruction of the state vectors of interest. We introduce a procedure for adapting an existing subspace based on information from the least-squares problem that underlies the approximation problem of interest such that the associated least-squares residual vanishes exactly. The method builds on a Riemmannian optimization procedure on the Grassmann manifold of low-dimensional subspaces, namely the Grassmannian Rank-One Update Subspace Estimation (GROUSE). We establish for GROUSE a closed-form expression for the residual function along the geodesic descent direction. Specific applications of subspace adaptation are discussed in the context of image processing and model reduction of nonlinear partial differential equation systems.

Original languageEnglish (US)
Pages (from-to)234-261
Number of pages28
JournalSIAM Journal on Matrix Analysis and Applications
Volume39
Issue number1
DOIs
StatePublished - 2018

Keywords

  • Dimension reduction
  • Discrete empirical interpolation method (DEIM)
  • Gappy proper orthogonal decomposition (POD)
  • Grassmann manifold
  • Grassmannian Rank-One Update Subspace Estimation (GROUSE)
  • Image processing
  • Least-squares
  • Masked projection
  • Online adaptive model reduction
  • Optimization on manifolds
  • Rank-one updates
  • Subspace fitting

ASJC Scopus subject areas

  • Analysis

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