  We present an ordinary differential equations approach to the analysis of
algorithms for constructing $l_1$ minimizing solutions to underdetermined
linear systems of full rank. It involves a relaxed minimization problem whose
minimum is independent of the relaxation parameter. An advantage of using the
ordinary differential equations is that energy methods can be used to prove
convergence. The connection to the discrete algorithms is provided by the
Crandall-Liggett theory of monotone nonlinear semigroups. We illustrate the
effectiveness of the discrete optimization algorithm in some sparse array
imaging problems.
