Consider a homogeneous medium so that wave propagation is fully coherent. Narrow band, active array imaging of localized scatterers is described mathematically as seeking sparse solutions of underdetermined linear system Ax=b. I will describe an ordinary differential equations approach to the analysis of numerical algorithms for constructing l₁ minimizing solutions to these underdetermined linear systems. 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. I will also mention what could be done when only the intensities are recorded and measured at the array of receivers. The problem of recovering an image from intensity-only measurements, known as the phase retrieval problem, arises in many situations in which it is difficult, or impossible, to measure and record the phases of the signals received at the detectors. Mathematically it means we only know moduli of b in the right-hand side of the system Ax=b.