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

TL;DR

This paper introduces direct multi-grid methods for solving linear systems efficiently by leveraging harmonic aliasing patterns, perfect reconstruction filters, and domain decomposition concepts, eliminating the need for smoothing iterations.

Contribution

It proposes novel direct multi-grid algorithms that avoid smoothing, using harmonic aliasing patterns and mirror filters, connecting multi-grid methods with filter bank theory.

Findings

Derived conditions for direct solutions in multi-grid methods

Developed configurations of direct multi-grid solvers using mirror filters

Established a strong analogy between multi-grid solutions and perfect reconstruction in filter banks

Abstract

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.