  This article discusses a newly developed online manifold learning method,
subspace iteration using reduced models (SIRM), for the dimensionality
reduction of dynamical systems. This method may be viewed as subspace iteration
combined with a model reduction procedure. Specifically, starting with a test
solution, the method solves a reduced model to obtain a more precise solution,
and it repeats this process until sufficient accuracy is achieved. The reduced
model is obtained by projecting the full model onto a subspace that is spanned
by the dominant modes of an extended data ensemble. The learning procedure is
computed in the online stage, as opposed to being computed offline, which is
used in many projection-based model reduction techniques that require prior
calculations or experiments. After providing an error bound of the classical
POD-Galerkin method in terms of the projection error and the initial condition
error, we prove that the sequence of approximate solutions converge to the
actual solution of the original system as long as the vector field of the full
model is locally Lipschitz on an open set that contains the solution
trajectory. Good accuracy of the proposed method has been demonstrated in two
numerical examples, from a linear advection-diffusion equation to a nonlinear
Burgers equation. In order to save computational cost, the SIRM method is
extended to a local model reduction approach by partitioning the entire time
domain into several subintervals and obtaining a series of local reduced models
of much lower dimensionality. The accuracy and efficiency of the local SIRM are
shown through the numerical simulation of the Navier--Stokes equation in a
lid-driven cavity flow problem.
