Three methods based on the three-dimensional shallow ice approximation of glacier flow are devised that infer a glacier's subglacial topography from the observation of its time-evolving surface and mass balance. The quasi-stationary inverse method relying on the apparent surface mass-balance description of the glacier's evolution is first exposed. Second, the transient inverse method that iteratively updates the bedrock topography with the surface topography discrepancy is formulated. Third, a shape optimization algorithm is presented. The aim of the paper is to collect these methods, analyze their differences, and identify what brings the sophistication of shape optimization for reconstructing subglacial topographies. The three methods are compared to the ice thickness estimation method (ITEM) on direct measurements on Gries glacier, Swiss Alps. The paper concludes with a detailed discussion on the sensitivity of the shape optimization method to the model parameters.