No matter how large our computational resources are, we always strive to solve the next problem. In seismic imaging, we now have the computational resources to do acoustic full-waveform inversion in 2D, but we would like to do elastic 3D inversions and quantify the uncertainty in those results as well. Right now, this is not computationally feasible. At the same time we are typically interested in specific regions of the subsurface and not in the entire model, which should lead to significant reductions in computational cost. This is not typically the case however, because solutions need to be calculated throughout a large domain even to recover parameters only in a local region. Local wave equation solvers are one way to attack this problem, allowing us to do rapid inversions in a local region with a significantly reduced computational cost. In this presentation I will present computational developments that are allowing us to extend the local solver to 3D as well as examples of uncertainty quantification in 4D seismic, and the coupling of acoustic and elastic wave solvers to quickly recover elastic parameters.