Multi-level numerical methods that obtain the exact solution of a linear
system are presented. The methods are devised by combining ideas from the full
multi-grid algorithm and perfect reconstruction filters. The problem is stated
as whether a direct solver is possible in a full multi-grid scheme by avoiding
smoothing iterations and using different coarse grids at each step. These
coarse grids must form a partition of the fine grid and thus establishes a
strong connection with domain decomposition methods.