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.
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