This article has two parts. In the first part of this article, an inverse FEL autoaccelerator experiment which uses the Columbia FEL accelerator system (V = 800 kV, I = 400 A) is described. High-power 1.5 mm radiation is developed in an optical resonator located along the first section of the undulator (which is 40 cm long and has a period of 1.43 cm and a magnetic field < 1 kG). The electron beam then moves into the second section of the undulator (which is 50 cm long and has its period tapered from 1.8 to 2.4 cm). A fraction of the electrons in the beam is accelerated in the second section by stimulated absorption of the FEL wave. A multimode two-dimensional simulation shows that approximately 50% of the FEL carried power should accelerate a group of trapped electrons to approximately 1200 kV. Measurements of the spectrum of FEL power shows that the stimulated absorption does in fact occur. In the second part of this article, we describe the theory of an inverse FEL beat-wave accelerator (IFELBWA). The essential idea is to use the inverse FEL mechanism to bunch a dense, low-energy electron beam, and then use the electric field generated between the bunches to accelerate a higher-energy electron beam. The phase velocity of the beat wave responsible for acceleration is enhanced by propagating the wave in a waveguide of smaller radius. Accelerating electric fields of the order of 100 MeV/m can be achieved with an electron beam of current density 20 kA/cm². The parameters of a proof-of-principle experiment are presented.