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**Example text**

The coarse-level set of variables is first selected geometrically (taking every fourth fine-level ghost variable, in a certain fixed 2D pattern) . Then this set is enhanced using the compatible relaxation tool (see Sec. 17), thereby adding another 10%-20% of the ghost variables to the coarse level. The coarse-level equations have been derived using either a 3 x 3 or 5 x 5 coarse-grid stencil, each including also all those extra coarse variables added (following the compatible relaxation test) at the corresponding region .

7 abo ve should be used for coarsening. A new approach (point ed out in [30]) is being developed for calculating N eigenfunctions of a differenti al opera tor discretized on N g gridpoints in O(Ng log N log:) computer operations, € being t he accuracy. This approach is based on the observation t hat "neighbor ing" eigenfunct ions are distinguishable from each oth er only at lar ge enough scales, and hence, in suitable repr esentations, one can use a common description of their details at finer scales, and pr ogressively separate t hem out only on increasingly coarse r grids.

The extension to higher dimensions (discussed in [120, §6]) is far from t rivial, Multiscale Scientific Computation 31 and intimately related to the extension of the wave/ray multigrid methods to variable coefficients (see Sec. 1) and to general matrices (see Sec. 2). A work related to the computation of the eigenvalue of a matrix and the fast updating of the singular-value decomposition of matrices is reported in Sec. 2 below. 1) and for the solution of corresponding integral and integro-differential equations.