The inverse method provides tsunami forecasts through regression of near-field water-level data against a set of mareograms computed for unit slip of predetermined subfaults. The present paper describes an improvement to this inverse method by resolving the timing of subfault movement in the inversion. This provides additional degrees of freedom in the regression analysis and accounts for rupture propagation and rise time of tsunamigenic earthquakes. The method is implemented for the Peru-Chile subduction zone, in which the seismic source parameters are deduced from previous studies of seismotectonics and historical earthquakes. A hindcast analysis of the 1995 Antofagasta-Chile tsunami demonstrates the capability of the adaptive inverse algorithm in improving the near-field inversion results and consequently the far-field forecast.