4. Algebraic Multigrid

4.1. Introduction. The focus in the application of standard multigrid methods is on the continuous problem to be solved. With the geometry of the problem known, the user discretizes the corresponding operators on a sequence of increasingly finer grids, each grid generally being a uniform refinement of the previous one, with transfer operators between the grids. The coarsest grid is sufficiently coarse to make the cost of solving the (residual) problem there negligible, while the finest is chosen to provide some desired degree of accuracy. The solution process, which involves relaxation, transfer of residuals from fine to coarse grids, and interpolation of corrections from coarse to fine levels, is a very efficient solver for the problem on the finest grid, provided the above “multigrid components” are properly chosen.

Roughly, the efficiency of proper multigrid methods is due to the fact that error only slightly affected by relaxation ( smooth error) can be easily approximated on a coarser grid by solving the residual equation there, where it is cheaper to compute. This error approximation is interpolated to the fine grid and used to correct the solution. Generally, uniform coarsening and linear interpolation are used, so the key to constructing an efficient multigrid algorithm is to pick the relaxation process that quickly reduces error not in the range of interpolation.

The algebraic multigrid (AMG) approach is developed to solve matrix equations using the principles of usual multigrid methods. In contrast to “geometric” multigrid methods, the relaxation used in AMG is fixed. The coarsening process (picking the coarse “grid” and defining interpolation) is performed automatically in a way that ensures the range of interpolation approximates those errors not efficiently reduced by relaxation. From a theoretical point of view, the process is best understood in the context of symmetric M-matrices, although, in practice, its use is not restricted to such cases. The underlying idea of the coarsening process is to exploit the fact that the form of the error after relaxation can be approximately expressed using the equations themselves, so that the coarse grid can be chosen and interpolation defined if the equations are used directly. This makes AMG attractive as a “black box” solver. In addition, AMG can be used for many kinds of problems, described below, where the application of standard multigrid methods is difficult or impossible.