Residual scaling techniques in multigrid, I: Equivalence proof

This paper addresses the residual scaling techniques (coarse-grid-correction optimization techniques) in multigrid methods. We surveyed recent developments in this area and prove the equivalence of the overweighted residual technique and the overcorrection technique. This leads to the proof of mathematical equivalence of the prescaling and postscaling acceleration techniques. Two theorems have been proved to unify the concept of the residual scaling techniques. These theoretical results clear the way for developing efficient prescaling acceleration techniques for practical applications. Those practical prescaling acceleration techniques are discussed in a companion paper: Residual scaling techniques, II: practical applications.