Deconvolution is the problem of estimating the distribution of a random variable from a sample with additive measurement error. Most penalised MLE methods restrict the space of possible solutions to a fixed a priori finite basis. However, we find a finite-dimensional subspace that contains the penalised MLE and use standard optimisation methods to obtain the infinite-dimensional MLE. This method is consistent and performs well in simulations. We also present a novel application of deconvolution - bootstrap sampling. Our motivating example is a likelihood ratio test to determine the rank of Nonnegative Matrix Factorisation (NMF). We estimate the null distribution via a bootstrap. Optimisation for NMF often finds a local optimum. A computationally more efficient approach than taking multiple starting points, is to bootstrap with optimisation error, then use deconvolution to remove it. This approach performs as well using multiple starting points, with much shorter computation time.