  Given a multigrid procedure for linear systems with coefficient matrices
$A_n$, we discuss the optimality of a related multigrid procedure with the same
smoother and the same projector, when applied to properly related algebraic
problems with coefficient matrices $B_n$: we assume that both $A_n$ and $B_n$
are positive definite with $A_n\le \vartheta B_n$, for some positive
$\vartheta$ independent of $n$. In this context we prove the Two-Grid method
optimality. We apply this elementary strategy for designing a multigrid
solution for modifications of multilevel structured (Toeplitz, circulants,
Hartley, sine ($\tau$ class) and cosine algebras) linear systems, in which the
coefficient matrix is banded in a multilevel sense and Hermitian positive
definite. In such a way, several linear systems arising from the approximation
of integro-differential equations with various boundary conditions can be
efficiently solved in linear time (with respect to the size of the algebraic
problem). Some numerical experiments are presented and discussed, both with
respect to Two-Grid and multigrid procedures.
