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Let

,

and

be the normalized work counts for the multiscale
smoothing, residual restriction and correction steps within the equivalent multigrid
implementation of our standard multiscale cycles. If we replace

by

for

we will obtain the
corresponding work counts

,

for the improved multiscale cycles.
Then, in the case of geometric multiscale decompositions (

)
and coarsest scale

, we have the
following standard cost estimates for multigrid-like implementations of our proposed
cycles [

21]:

- Standard multiscale V(1,1)-cycle:
,
- Improved multiscale V(1,1)-cycle:
,
- Standard multiscale W(1,1)-cycle:
,

where

and

.
From the representation

one sees that the application of

within the improved cycles is at least
by a factor of nine more expensive than the application of

alone.
Here, we use 9-point stencils and consider only the costs for the application of

and

.
Hence, the improved versions are more expensive than a standard multiscale
W-cycle.

Regarding the costs for our AMG-based multiscale cycles there are unfortunately no
predictive cost analyses available.
Therefore, we define as * grid* and * non-standard operator complexities* (adapted
to an AMG-based non-standard form) the two numbers

where nnz

denotes the number of nonzero entries of a matrix

.
Grid complexity

gives an accurate measure of the total storage required for the residual
and correction vectors with respect to the different scales.
Non-standard operator complexity

indicates the total storage space necessary for the
blocks of the non-standard form as well as for the fine-grid operator that are used within
the multiscale cycles.
It generalizes the well known AMG operator complexity [

10].
Furthermore,

gives a measure of the
work involved in the solve phase in the case of multiscale V(1,1)-cycles.
Here, we assume that the costs for
the detail correction (essentially ILU(0)(

)) and for the computation of the residuals
are proportional to the number of nonzero entries of the corresponding blocks of the
non-standard form.

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*Frank Kiefer*

*2001-10-25*