  This article presents a new method to compute matrices from numerical
simulations based on the ideas of sparse sampling and compressed sensing. The
method is useful for problems where the determination of the entries of a
matrix constitutes the computational bottleneck. We apply this new method to an
important problem in computational chemistry: the determination of molecular
vibrations from electronic structure calculations, where our results show that
the overall scaling of the procedure can be improved in some cases. Moreover,
our method provides a general framework for bootstrapping cheap low-accuracy
calculations in order to reduce the required number of expensive high-accuracy
calculations, resulting in a significant 3x speed-up in actual calculations.
