The eruption of the Anak Krakatau volcano, Indonesia, on 22 December 2018 induced a destructive tsunami (the Sunda Strait tsunami), which was recorded by four nearby tidal gauges. In this study we invert the tsunami records and recover the tsunami generation process. Two tsunami sources are obtained, a static one of instant initial water elevation and a time-dependent one accounting for the continuous evolution of water height. The time-dependent results are found to reproduce the tsunami recordings more satisfactorily. The complete tsunami generation process lasts approximately 9 min and features a two-stage evolution with similar intensity. Each stage lasts about 3.5 min and elevates a water volume of about 0.13 km3. The time, duration and volume of the volcano eruption in general agree with seismic records and geomorphological interpretations. We also test different sizes of the potential source region, which lead to different maximum wave height in the source area, but all the results of time-dependent tsunami sources show the robust feature of two stages of wave generation. Our results imply a time-dependent and complex process of tsunami generation during the volcano eruption.