Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid

Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver. Algebraic multigrid (AMG) and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error. For smoothed aggregation (SA) multigrid methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the near-kernel or near-nullspace of the weak form. This paper introduces an extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the near-kernel components is unavailable. This extension is accomplished in an adaptive process that uses the method itself to determine near-kernel components and adjusts the coarsening processes accordingly.