  In this paper, we introduce a new nonlinear evolution partial differential
equation for sparse deconvolution problems. The proposed PDE has the form of
continuity equation that arises in various research areas, e.g. fluid dynamics
and optimal transportation, and thus has some interesting physical and
geometric interpretations. The underlying optimization model that we consider
is the standard $\ell_1$ minimization with linear equality constraints, i.e.
$\min_u\{\|u\|_1 : Au=f\}$ with $A$ being an under-sampled convolution
operator. We show that our PDE preserves the $\ell_1$ norm while lowering the
residual $\|Au-f\|_2$. More importantly the solution of the PDE becomes sparser
asymptotically, which is illustrated numerically. Therefore, it can be treated
as a natural and helpful plug-in to some algorithms for $\ell_1$ minimization
problems, e.g. Bregman iterative methods introduced for sparse reconstruction
problems in [W. Yin, S. Osher, D. Goldfarb, and J. Darbon,SIAM J. Imaging Sci.,
1 (2008), pp. 143-168]. Numerical experiments show great improvements in terms
of both convergence speed and reconstruction quality.
