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The use of compressed sensing and sparse sampling methods for scientific development has been dominated by experimental applications~\cite{Kazimierczuk_2011,Holland_2011,Gross_2010,Zhu_2012,Sanders2012,Doneva_2010}. However compressed sensing is also becoming a tool for theoretical applications~\cite{Schaeffer_2013,Almeida_2012,Nelson_2013}. In particular, in previous work we have shown that compressed sensing can also be used to reduce the amount of computation in numerical simulations~\cite{Andrade2012b}.  In this article, we extend apply  compressed sensing to the problem of reconstructing scientific computing  matrices. This method has two key properties. First, the cost of the procedure is proportional to the size of the number of non-zero elements in the matrix, without the need to know \emph{a priori} the location of the non-zero elements. Second, the rescontruction is exact. Furthermore, such a method is useful not only for sparse matrices. It makes it practical to devise methods to find bases where the matrix is known or suspected to be sparse based on the characteristics and previous knowledge of each problem. To demonstrate the power of our approach, we apply these ideas to the determination of the vibrational modes of molecules from electronic structure methods. These methods require the calculation of the matrix of the second derivatives of the energy with respect to the nuclear displacements, known as the force-constant or Hessian matrix. This matrix is routinely obtained in numerical simulations by chemists and physicists, but it is relatively expensive to compute when accurate quantum mechanical methods are used. Our application shows that how we can exploit the sparsity of the matrix to make important improvements in the efficiency of this calculation, and at the same time makes it practical to bootstrap this calculations using lower accuracy models, something that previously was not worthwile to do.