  There is an intimate connection between numerical upscaling of multiscale
PDEs and scattered data approximation of heterogeneous functions: the coarse
variables selected for deriving an upscaled equation (in the former) correspond
to the sampled information used for approximation (in the latter). As such,
both problems can be thought of as recovering a target function based on some
coarse data that are either artificially chosen by an upscaling algorithm, or
determined by some physical measurement process. The purpose of this paper is
then to study that, under such a setup and for a specific elliptic problem, how
the lengthscale of the coarse data, which we refer to as the subsampled
lengthscale, influences the accuracy of recovery, given limited computational
budgets. Our analysis and experiments identify that, reducing the subsampling
lengthscale may improve the accuracy, implying a guiding criterion for
coarse-graining or data acquisition in this computationally constrained
scenario, especially leading to direct insights for the implementation of the
Gamblets method in the numerical homogenization literature. Moreover, reducing
the lengthscale to zero may lead to a blow-up of approximation error if the
target function does not have enough regularity, suggesting the need for a
stronger prior assumption on the target function to be approximated. We
introduce a singular weight function to deal with it, both theoretically and
numerically. This work sheds light on the interplay of the lengthscale of
coarse data, the computational costs, the regularity of the target function,
and the accuracy of approximations and numerical simulations.
