DOUG 0.2

Smoothed Coarse Spaces

This is done after fine aggregates are found (see Aggregation).

Input
Set of ( $ n_s=0 $ non-overlapping) aggregates $ W=\{ W_j : j=1,...,n_a \} $
Output
Restriction matrix $ R_0:\mathbb{R}^{|\mathcal{N}|}\rightarrow \mathbb{R}^{n_a} $, Interpolation matrix $ R_0^T $ and coarse matrix $ A_0 $
  1. Form the aggregate projector operator $ P:\mathbb{R}^{|\mathcal{N}|}\rightarrow\mathbb{R}^{n_a} $, where $ P_{jk}=\{1 \mbox{ if } x_k\in W_j \mbox{ or } 0, \mbox{ otherwise}\} $
  2. Form the restriction operator $ R_0=PS^{n_s} $, with $ S=(I-\omega A) $ (applying $ n_s $ times a damped Jacobi smoother); aggregates grow by $ n_s $ layers as well, forming overlaps
  3. Form the coarse problem matrix $ A_0 $ through sparse matrix multiplication $ A_0=R_0AR_0^T $