Variable-resolution computational grids can substantially improve the benefit-to-cost ratio in many environmental modeling applications, but they can also introduce unwanted and unrealistic numerical anomalies if not properly utilized. For example, we showed in previous studies that resolved (non-parameterized) atmospheric convection develops more quickly as resolution increases. Furthermore, on variable grids that transition from resolved to parameterized convection, timing and intensity of the convection in both regimes is generally disparate unless special care is taken to tune the parameterization. In both cases, the convection that develops first (due to purely numerical reasons) tends to suppress convection elsewhere by inducing subsidence in the surrounding environment. This highly nonlinear competition, while desirable when induced by natural causes such as surface inhomogeneity, is highly undesirable when it is a numerical artifact of variable grid resolution and/or selective application of convective parameterization. Our current research is aimed at leveling the playing field for convection across a variable resolution grid so that the above problems are avoided. The underlying idea is to apply the same or very similar "convective machinery" to all areas of the grid. For convection-resolving regions of the grid, this machinery is simply the model grid itself along with explicit representation of dynamics and a bulk microphysics parameterization. For coarser regions of the grid, the local environment is sampled from a single grid column or small group of columns (depending on local resolution) and fed to a separate "convective processor", which determines the convective response to that environment and feeds the result back to the host grid. The convective processor can either (1) explicitly resolve convective activity for a subset of the sampled environments using a separate limited-area 3D high resolution simulation and apply a principal component analysis to estimate the specific responses in each of the remaining environments, or (2) estimate the essential elements of the convective response from lookup table entries that were previously generated for similar environments using method (1). Obviously, method (2) is extremely efficient while method (1) is computationally intensive, so the key is to construct clever algorithms that enable method (2) to be used as often as possible.