We discuss various numerical approaches to construct complex geometric optics solutions to $\overline{\partial}$ systems appearing in the context of inverse scattering for the Davey-Stewartson II equations. Potentials in the Schwartz class and with compact support are discussed. Of particular interest is the behavior for large values of the spectral parameter $|k|$. This is work in collaboration with K.McLaughlin, J. Sjöstrand and N. Stoilov.