The inverse problem of interest is the reconstruction of an astronomical object and wavefront aberrations due to atmospheric turbulence from a sequence of short time exposure (speckle) phase diversity images obtained from a groundbased telescope. A regularized least squares approach is taken, and two numerical techniques are applied to the resulting unconstrained optimization problem. The first technique is the limited memory BFGS method with line search globalization. The second is a Newton/trust region iteration in which the trust region subproblem is solved using a truncated conjugate gradient method. A numerical comparison based on real data is presented, and the roles of preconditioning and cost functional reduction (elimination of the object) are examined.