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This article presents a new method to compute matrices on numerical simulations based on the ideas of sparse sampling and compressed sensing. It The methods  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.   Our calculations, where our  results show that in some cases our approach can improve the  overall scaling of the procedure.  Moreover, it procedure  can significantly be improved in some cases. Moreover, we see a siginificant  speed-up in  actual calculations, in particular since especially when we use  our approachallows us  to bootstrap the calculations usingresult from  lower-accuracy methods. or approximated results.