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Smoothed Aggregation

In order to define the prolongator , used to compute the coarse-level matrix , MLD2P4 uses the smoothed aggregation algorithm described in [2,25]. The basic idea of this algorithm is to build a coarse set of indices by suitably grouping the indices of into disjoint subsets (aggregates), and to define the coarse-to-fine space transfer operator by applying a suitable smoother to a simple piecewise constant prolongation operator, with the aim of improving the quality of the coarse-space correction.

Three main steps can be identified in the smoothed aggregation procedure:

1. aggregation of the indices of to obtain ;
2. construction of the prolongator ;
3. application of and to build .

In order to perform the coarsening step, the smoothed aggregation algorithm described in [25] is used. In this algorithm, each index corresponds to an aggregate of , consisting of a suitably chosen index and indices that are (usually) contained in a strongly-coupled neighborood of , i.e.,

| (3)

for a given threshold (see [25] for the details). Since this algorithm has a sequential nature, a decoupled version of it is applied, where each processor independently executes the algorithm on the set of indices assigned to it in the initial data distribution. This version is embarrassingly parallel, since it does not require any data communication. On the other hand, it may produce some nonuniform aggregates and is strongly dependent on the number of processors and on the initial partitioning of the matrix . Nevertheless, this parallel algorithm has been chosen for MLD2P4, since it has been shown to produce good results in practice [5,7,24].

The prolongator is built starting from a tentative prolongator , defined as

| (4)

where is the aggregate of corresponding to the index . is obtained by applying to a smoother :

in order to remove nonsmooth components from the range of the prolongator, and hence to improve the convergence properties of the multi-level method [2,23]. A simple choice for is the damped Jacobi smoother:

where is the diagonal matrix with the same diagonal entries as , is the filtered matrix defined as

| (5)

and is an approximation of , where is the spectral radius of [2]. In MLD2P4 this approximation is obtained by using as an estimate of . Note that for systems coming from uniformly elliptic problems, filtering the matrix has little or no effect, and can be used instead of . The latter choice is the default in MLD2P4.

Next: Smoothers and coarsest-level solvers Up: Multigrid Background Previous: AMG preconditioners Contents