Structure-function relationships occur throughout the sciences. Motivated by optimisation of such systems, we develop a framework for estimating the input modes from the singular value decomposition from the action of the forward operator alone. These can then be used to determine the input (structure) changes which induce the largest output (function) changes. The accuracy of the estimate is determined by reference to the method of snapshots. The proposed method is demonstrated on several example problems, and finally used to approximate the optimal airway treatment set for a problem in respiratory physiology.