  Traditional low-rank approximation is a powerful tool to compress the huge
data matrices that arise in simulations of partial differential equations
(PDE), but suffers from high computational cost and requires several passes
over the PDE data. The compressed data may also lack interpretability thus
making it difficult to identify feature patterns from the original data. To
address these issues, we present an online randomized algorithm to compute the
interpolative decomposition (ID) of large-scale data matrices {\em in situ}.
Compared to previous randomized IDs that used the QR decomposition to determine
the column basis, we adopt a streaming ridge leverage score-based column subset
selection algorithm that dynamically selects proper basis columns from the data
and thus avoids an extra pass over the data to compute the coefficient matrix
of the ID. In particular, we adopt a single-pass error estimator based on the
non-adaptive Hutch++ algorithm to provide real-time error approximation for
determining the best coefficients. As a result, our approach only needs a
single pass over the original data and thus is suitable for large and
high-dimensional matrices stored outside of core memory or generated in PDE
simulations. A strategy to improve the accuracy of the reconstructed data
gradient, when desired, within the ID framework is also presented. We provide
numerical experiments on turbulent channel flow and ignition simulations, and
on the NSTX Gas Puff Image dataset, comparing our algorithm with the offline ID
algorithm to demonstrate its utility in real-world applications.
