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AMG preconditioners

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

\begin{displaymath}
Ax=b,
\end{displaymath} (2)

where $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ is a nonsingular sparse matrix; for ease of presentation we assume $A$ 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 $A$, i.e., $\Omega = \{1, 2, \ldots, n\}$. Any algebraic multilevel preconditioners implemented in MLD2P4 generates a hierarchy of index spaces and a corresponding hierarchy of matrices,

\begin{displaymath}\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
\quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \end{displaymath}

by using the information contained in $A$, without assuming any knowledge of the geometry of the problem from which $A$ originates. A vector space $\mathbb{R}^{n_{k}}$ is associated with $\Omega^k$, where $n_k$ is the size of $\Omega^k$. For all $k < nlev$, a restriction operator and a prolongation one are built, which connect two levels $k$ and $k+1$:

\begin{displaymath}
 P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad 
 R^k \in \mathbb{R}^{n_{k+1}\times n_k};
\end{displaymath}

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

\begin{displaymath}
 A^{k+1}=R^kA^kP^k.
\end{displaymath}

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.

Figure 1: Application phase of a V-cycle preconditioner.
\framebox{
\begin{minipage}{.85\textwidth}
\begin{tabbing}
\quad \=\quad \=\quad...
...mm]
\>endif  [1mm]
\>return $u^k$  [1mm]
end
\end{tabbing}
\end{minipage}
}

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 $w=B^{-1}v$, where $B$ 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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Next: Smoothed Aggregation Up: Multigrid Background Previous: Multigrid Background   Contents