Satellite and in situ observations of ice sheet outlet glaciers around the turn of the 21st century showed that rapid changes in ice dynamics are possible and important for the evolution of ice sheets. When attempting to model these dynamic changes the conditions at the ice-bed interface are crucial. Inverse methods can be used to infer basal properties, such as the basal yield stress, from abundant surface velocity observations by using a physical model of ice flow. Inverse methods are very powerful, but they need to be applied with care, otherwise errors can dominate the solution. In this study we investigate the potentials and caveats of inverse methods. Synthetic experiments can be designed where basal conditions are assumed and an ice flow model is used to produce a set of 'synthetic' surface velocities. These can then be used to examine and evaluate inverse methods. We find that in iterative inverse methods it is essential to use a stopping criterion that will prevent overfitting the data. We introduce a new and rapidly-converging iterative inverse method called Incomplete Gauss Newton method, where the linearized problem is partly minimized in each step.In a practical application of inverse methods to the terminus region of Jakobshavn Isbræ, Greenland we investigate changes in basal conditions over time by performing inversions for different years of available surface velocity data. We find a decrease in basal yield stress in the lower areas of the glacier that agrees with effective pressure changes due to the changes in ice geometry. This supports an ocean and terminus driven system. The difference between the modeled and observed velocity fields, called residual, contains information about the ability to reproduce the velocities when only adjustment of the basal condition is allowed. With a properly regularized inversion the residual patterns can be used to investigate sources of error in the system. We find that the ice geometry and the model simplifications influence the ability to reproduce observed velocity fields more than the error in observed velocity does. This indicates that further progress must come from model improvements and improved capabilities to measure bedrock geometry.