  In many applications it is important to estimate a fluid flow field from
limited and possibly corrupt measurements. Current methods in flow estimation
often use least squares regression to reconstruct the flow field, finding the
minimum-energy solution that is consistent with the measured data. However,
this approach may be prone to overfitting and sensitive to noise. To address
these challenges we instead seek a sparse representation of the data in a
library of examples. Sparse representation has been widely used for image
recognition and reconstruction, and it is well-suited to structured data with
limited, corrupt measurements. We explore sparse representation for flow
reconstruction on a variety of fluid data sets with a wide range of complexity,
including vortex shedding past a cylinder at low Reynolds number, a mixing
layer, and two geophysical flows. In addition, we compare several measurement
strategies and consider various types of noise and corruption over a range of
intensities. We find that sparse representation has considerably improved
estimation accuracy and robustness to noise and corruption compared with least
squares methods. We also introduce a sparse estimation procedure on local
spatial patches for complex multiscale flows that preclude a global sparse
representation. Based on these results, sparse representation is a promising
framework for extracting useful information from complex flow fields with
realistic measurements.
