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
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.