The process of ultrafast laser regenerative amplification has complex dynamic behavior that requires to build a model for solution and iterative calculation. Based on the improved Frantz-Nodvik equation, the output properties of the Yb:KGW regenerative amplifier are calculated, the influences of pumping intensity, crystal length and pumping duration on the output performance of the amplifier are investigated, and the output behaviors of the regenerative amplifier with different repetition rates are also analyzed. Based on the theoretical analysis, a regenerative amplifier is constructed and the laser amplification results with pulse energy of 1 mJ at repetition rate of 1 kHz are realized, which are in good agreement with the theoretical analysis results. The improved Frantz-Nodvik equation gives an insight into the design of laser amplification which is scalable to high pulse energy with high stability at high repetition rate.