Direct Multi-grid Methods for Linear Systems with Harmonic Aliasing Patterns

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.