next up previous contents
Next: Getting Started Up: Multigrid Background Previous: Multigrid Background   Contents


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}


next up previous contents
Next: Getting Started Up: Multigrid Background Previous: Multigrid Background   Contents