  The characterization of intermittent, multiscale and transient dynamics using
data-driven analysis remains an open challenge. We demonstrate an application
of the Dynamic Mode Decomposition (DMD) with sparse sampling for the diagnostic
analysis of multiscale physics. The DMD method is an ideal spatiotemporal
matrix decomposition that correlates spatial features of computational or
experimental data to periodic temporal behavior. DMD can be modified into a
multiresolution analysis to separate complex dynamics into a hierarchy of
multiresolution timescale components, where each level of the hierarchy divides
dynamics into distinct background (slow) and foreground (fast) timescales. The
multiresolution DMD is capable of characterizing nonlinear dynamical systems in
an equation-free manner by recursively decomposing the state of the system into
low-rank spatial modes and their temporal Fourier dynamics. Moreover, these
multiresolution DMD modes can be used to determined sparse sampling locations
which are nearly optimal for dynamic regime classification and full state
reconstruction. Specifically, optimized sensors are efficiently chosen using QR
column pivots of the DMD library, thus avoiding an NP-hard selection process.
We demonstrate the efficacy of the method on several examples, including global
sea-surface temperature data, and show that only a small number of sensors are
needed for accurate global reconstructions and classification of El Ni\~no
events.
