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

摘要

Multi-level numerical methods that obtain the exact solution of a linearsystem are presented. The methods are devised by combining ideas from the fullmulti-grid algorithm and perfect reconstruction filters. The problem is statedas whether a direct solver is possible in a full multi-grid scheme by avoidingsmoothing iterations and using different coarse grids at each step. The coarsegrids must form a partition of the fine grid and thus establishes a strongconnection with domain decomposition methods. An important analogy isestablished between the conditions for direct solution in multi-grid solversand perfect reconstruction in filter banks. Furthermore, simple solutions ofthese conditions for direct multi-grid solvers are found by using mirrorfilters. As a result, different configurations of direct multi-grid solvers areobtained and studied.