Geometric subspace updates with applications to online adaptive nonlinear model reduction

Ralf Zimmermann, Benjamin Peherstorfer, Karen Willcox

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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 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
JournalSIAM Journal on Matrix Analysis and Applications
Volume39
Issue number1
Pages (from-to)234–261
ISSN0895-4798
DOIs
Publication statusPublished - 2018

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Keywords

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

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