AMG preconditioners

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

where A = (a_ij) ∈ ℝ^n×n is a nonsingular sparse matrix; for ease of presentation we assume A has a symmetric sparsity pattern.

Let us consider as finest index space the set of row (column) indices of A, i.e., Ω = {1, 2,…,n}. 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 A, without assuming any knowledge of the geometry of the problem from which A originates. A vector space ℝ^n_k is associated with Ω^k, where n_k is the size of Ω^k. For all k < nlev, a restriction operator and a prolongation one are built, which connect two levels k and k + 1:

the matrix A^k+1 is computed by using the previous operators according to the Galerkin approach, i.e.,

In the current implementation of MLD2P4 we have R^k = (P^k)^T A smoother with iteration matrix M^k is set up at each level k < nlev, 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 LU factorization means computing and storing the L and U factors). The construction of the hierarchy of AMG components described so far corresponds to the so-called build phase of the preconditioner.

procedure V-cycle ( k k k)
k,A ,b ,u

then

u^k = u^k + M^k ( )
bk - Akuk

b^k+1 = R^k+1 (bk - Akuk )

u^k+1 = V-cycle ( )
k + 1,Ak+1,bk+1,0

u^k = u^k + P^k+1u^k+1

u^k = u^k + M^k ( )
bk - Akuk

else

u^k =

^-1b^k

endif

return u^k

end

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 w = B^-1v, where B denotes the preconditioner, usually within an iteration of a Krylov solver [23]. 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 [22].

4.1 AMG preconditioners