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AMG preconditioners

In order to describe the AMG preconditioners available in MLD2P4, we consider a linear system

| (2)

where is a nonsingular sparse matrix; for ease of presentation we assume is real, but the results are valid for the complex case as well.

Let us assume as finest index space the set of row (column) indices of , i.e., . Any algebraic multilevel preconditioners implemented in MLD2P4 generates a hierarchy of index spaces and a corresponding hierarchy of matrices,

by using the information contained in , without assuming any knowledge of the geometry of the problem from which originates. A vector space is associated with , where is the size of . For all , a restriction operator and a prolongation one are built, which connect two levels and :

the matrix is computed by using the previous operators according to the Galerkin approach, i.e.,

In the current implementation of MLD2P4 we have A smoother with iteration matrix is set up at each level , and a solver is set up at the coarsest level, so that they are ready for application (for example, setting up a solver based on the factorization means computing and storing the and factors). The construction of the hierarchy of AMG components described so far corresponds to the so-called build phase of the preconditioner.

Figure 1: Application phase of a V-cycle preconditioner.

The components produced in the build phase may be combined in several ways to obtain different multilevel preconditioners; this is done in the application phase, i.e., in the computation of a vector of type , where denotes the preconditioner, usually within an iteration of a Krylov solver [20]. An example of such a combination, known as V-cycle, is given in Figure 1. In this case, a single iteration of the same smoother is used before and after the the recursive call to the V-cycle (i.e., in the pre-smoothing and post-smoothing phases); however, different choices can be performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle and W-cycle [3], and a version of the K-cycle described in [19].

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