Adaptive or self-correcting algebraic multigrid methods employ the idea of using the method to improve the method by automatically identifying the so-called algebraically smooth components that are crucial for obtaining optimal multigrid convergence. PCG is a non-multilevel form of this basic idea. Achi Brandt's bootstrap multigrid method is a example of an adaptive multigrid algorithm.

In this talk, we will introduce two new adaptive multigrid algorithms: one based on element-free AMGe ideas and the other based on smoothed aggregation. Comments will be made on convergence theory for these methods, and numerical results will be given.