  Presented is an algorithm based on dynamic mode decomposition (DMD) for
acceleration of the power method (PM). The power method is a simple technique
for determining the dominant eigenmode of an operator $\mathbf{A}$, and
variants of the power method are widely used in reactor analysis. Dynamic mode
decomposition is an algorithm for decomposing a time-series of
spatially-dependent data and producing an explicit-in-time reconstruction for
that data. By viewing successive power-method iterates as snapshots of a
time-varying system tending toward a steady state, DMD can be used to predict
that steady state using (a sometime surprisingly small) $n$ iterates. The
process of generating snapshots with the power method and extrapolating forward
with DMD can be repeated. The resulting restarted, DMD-accelerated power method
(or DMD-PM($n$)) was applied to the two-dimensional IAEA diffusion benchmark
and compared to the unaccelerated power method and Arnoldi's method. Results
indicate that DMD-PM($n$) can reduce the number of power iterations required by
approximately a factor of 5. However, Arnoldi's method always outperformed
DMD-PM($n$) for an equivalent number of matrix-vector products $\mathbf{Av}$.
In other words, DMD-PM($n$) cannot compete with leading eigensolvers if one is
not limited to snapshots produced by the power method. Contrarily, DMD-PM($n$)
can be readily applied as a post process to existing power-method applications
for which Arnoldi and similar methods are not directly applicable. A slight
variation of the method was also found to produce reasonable approximations to
the first and second harmonics without substantially affecting convergence of
the dominant mode.
