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I've been reading about the algebraic multigrid algorithm and came across polynomial smoothers in this paper. It's my understanding that usually the coarse-level matrices

$A_H = I_h^HA_hI_H^h$

are formed explicitly. However, there's still the issue of how to define the coarse space corresponding to a given smoother. The paper suggests using polynomial smoothing as an adjunct to smoothed aggregation, in which case the prolongation operator is

$I_H^h = p(A)\cdot\mathscr{I}_H^h$

where $p$ is some shifted Chebyshev polynomial and $\mathscr{I}$ is the aggregation operator.

Is explicitly forming $A_H$ with this approach prohibitively expensive? Moreover, am I even correct in assuming you need to explicitly compute the coarse level matrices?

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Do I have to form $A_H$ explicitly?

It is not strictly necessary, but if you define it implicitly, then the complexity grows from $O(n)$ to $O(n \log n)$, making the algorithm significantly less efficient. It also becomes inconvenient to define properly-scaled pointwise smoothers, which costs another factor.

Is forming $I_H^h = p(A) \mathscr I_H^h$ and ultimately $A_H$ prohibitively expensive?

Forming $I_H^h$ is usually inexpensive because $p(A)$ is almost always chosen to be damped Jacobi (a degree 1 polynomial). Even so, construction of $A_H$ is usually the most expensvie part of multigrid setup. If $I_H^h$ were chosen to have more overlap, for example by smoothing the aggregates multiple times, grid complexity would be much higher.

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