deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 3c35faf..ee1725f 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
...
For example, if we know \emph{a priori} the eigenvectors of a \(N\times N\) diagonalizable matrix, then we can obtain the full matrix by only calculating the \(N\) diagonal elements. Similarly, a sparse matrix, which contains many zero elements, can be evaluated by calculating only the non-zero elements, if we know in advance where such elements are located. In this article, we present a general approach that can produce a considerable reduction in the cost of constructing a matrix in many scientific applications by substantially reducing the number of elements that need to be calculated.   The key numerical procedure of our approach is a method to cheaply recover sparse matrices with a cost that is essentially  proportional to the number of non-zero elements. The matrix reconstruction procedure is based on the increasingly popular compressed sensing approach~\cite{Candes2006,Donoho2006,Candes2008}, a state-of-the-art signal processing technique developed for minimizing the amount of data that needs to be measured to reconstruct a sparse signal.  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 apply compressed sensing to the problem of computing matrices. This method has two key properties. First, the cost of the procedure is proportional to quasi-linear with  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.  We begin by introducing discussing how this compressed sensing makes it practical to take a new approach for the calculation of matrices based on the idea of finding strategies to make the matrix sparse.   Next, we introduce  the mathematical foundations of the method of compressed sensing and applying it to the problem of sparse matrix reconstruction. This is the numerical tool that forms the foundation of our approach.Next, we discuss how this tool makes it practical to take a new approach for the calculation of matrices based on the idea of finding strategies to make the matrix sparse.  Finally, we illustrate these new ideas by applying them to the problem of obtaining molecular vibrations from quantum mechanical simulations.