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Jacob Sanders deleted file Molecular Vibrations 21.tex
over 9 years ago
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Since the molecular mechanics Hessians illustrate the best-case scenario in which the sparsity \(S\) scales linearly with the number of atoms \(N\), we attempted to recover these Hessians directly in the basis of atomic coordinates via the compressed sensing procedure we have outlined by sampling columns in the Fourier basis. Fig.~\ref{fig:MMNColumnsVsNRings} illustrates the number of columns which must be sampled to recover the Hessians to within a relative error of \(10^{-3}\) as a function of the size of the polyacene. Far fewer than the total number of columns in the entire matrix need to be sampled. Even more attractive is the fact that the number of columns grows quite slowly with the size of the polyacene, consistent with the best-case \(O\left(\log N \right)\times OE\) scaling result obtained above. This result indicates that our compressed sensing approach is especially promising for the calculation of Hessian matrices for large systems.
(For comparison, we also recovered the Hessians in their sparsest possible basis, which is their eigenbasis. This procedure is not practical since knowing the eigenbasis requires knowing the entire Hessian beforehand, but it illustrates how the compressed sensing procedure can be improved further if an appropriate sparsifying transformation is known.)
