  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. The coarse
grids must form a partition of the fine grid and thus establishes a strong
connection with domain decomposition methods. An important analogy is
established between the conditions for direct solution in multi-grid solvers
and perfect reconstruction in filter banks. Furthermore, simple solutions of
these conditions for direct multi-grid solvers are found by using mirror
filters. As a result, different configurations of direct multi-grid solvers are
obtained and studied.
