ABSTRACTWith the conclusion of the Last Glacial Maximum (LGM), about 20 000 years ago, ended the most recent long-lasting cold phase in Earth history. This last glacial advance left a strong observable imprint on the landscape, such as moraines, trimlines and other glacial geomorphic features. These features reflect the extent of former glaciers and ice caps, which in turn provides information on past temperature and precipitation conditions. Here we present an inverse approach to reconstruct the equilibrium line altitudes (E) from observed ice extents. The ice-flow model is developed solving the mass conservation equation using the shallow ice approximation and implemented using Graphical Processing Units (GPUs). We present the theoretical basis of the inversion method, which relies on a Tikhonov regularization, and demonstrate its ability to constrain spatial variations in mass balance with idealized and real glaciers.