Numerical simulation of physical processes is often constrained by our ability to solve the complex linear systems at the core of the computation. Multiscale methods, such as multigrid, provide optimal or near-optimal order solution techniques for many of these systems, relying on the use of complementary problems to reduce errors that simple iterative methods, such as Jacobi or Gauss-Seidel, are slow to resolve. Thus, classical geometric and algebraic multigrid methods rely on (implicit) assumptions about the character of these matrices in order to develop appropriately complementary coarse-grid correction processes for a given relaxation scheme.

The aim of the adaptive multigrid framework is to reduce the restrictions imposed by such assumptions, thus allowing for efficient black-box multigrid solution of a wider class of problems. There are, however, many challenges in altogether removing the reliance on assumptions about the errors left after relaxation, particularly in the choice of coarse-grid points. In this talk, we introduce the adaptive AMG framework and discuss its application to problems in heterogeneous media. Recent research on purely algebraic criteria for coarse grid selection will also be discussed.