  Recently, significant connections between compressed sensing problems and
optimization of a particular class of functions relating to solutions of
Hamilton-Jacobi equation was discovered. In this paper we introduce a fast
approximate algorithm to optimize this particular class of functions and
subsequently find the solution to the compressed sensing problem. Although we
emphasize that the methodology of our algorithm finds an approximate solution,
numerical experiments show that our algorithm perfectly recovers the solution
when the solution is relatively sparse with respect to the number of
measurements. In these scenarios, the recovery is extremely fast compare to
other methods available. Numerical experiments also demonstrate that the
algorithm exhibits a sharp phase transition in success rate of recovery of the
solution to compressed sensing problems as sparsity of solution varies. The
algorithm proposed here is parameter free (except a tolerance parameter due to
numerical machine precision), and very easy to implement.
