Many problems in physics and engineering rely on complex simulations to make predictions and learn inputs to a system.  Using these simulations efficiently and comparing them with data has become an important research area for statistical modeling.  I will present work on an example in cosmology that includes many issues common to these applications. The matter power spectrum describes the density of matter in the Universe as a function of scale.  Accurate predictions of this function could be combined with observations to constrain unknown cosmological parameters, in particular those describing dark energy. Unfortunately, accurate predictions can only be obtained from computationally expensive simulations which are impractical to produce in large numbers. This talk covers a number of statistical modeling techniques that were combined to produce a prediction scheme, called the Cosmic Emu, for the matter power spectrum. The scheme used a training set of simulations from a carefully selected set of cosmological parameters. These simulations produce noisy realizations of the spectrum, so we developed an adaptive process convolution model to estimate a smooth spectrum for each input setting. These smooth estimates were used to train a well-developed dimension reduction and Gaussian process-based method that can give fast predictions of the smooth spectrum at new settings of the cosmological inputs for which simulations are not available. The resulting emulator was released to the astro community who, of course, wanted more. To satisfy them, we developed modifications of the methods that allowed us to extend the Emu to cover more of the spectrum and include a new input parameter without needing to redo the costly simulation results.