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Modifications of the conventional muti-grid algorithm are explored to
avoid the use of smoothing iterations. In the full multi-grid algorithm,
classical smoothing iterations (e.g. Gauss-Seidel) reduce high-frequency
components of the error and a coarse-grid approach reduces the
low-frequency components of the error. The problem here is that two
methods with different structures are being combined, which introduces
additional complexity in the convergence analysis of multi-grid methods.
Then, the idea is to avoid the use of smoothing iterations by using
different inter-grid configurations. In an analogy with perfect
reconstruction filters, a configuration of inter-grid operators is found
such that the approximation error is completely cancelled after one
iteration of the algorithm. The main assumption for this configuration
to work is a particular aliasing pattern in the eigen-vectors of the
linear system. Examples and simulations are presented to evaluate the
performance of the new algorithm.
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