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9 changes: 4 additions & 5 deletions EuroPar/2-amg-prec.tex
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\section{Algebraic MultiGrid (AMG) Preconditioners}~\label{AMG}
%
MultiGrid methods are widely used as preconditioners of iterative
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Jacobi or Gauss-Seidel, to reduce highly oscillatory error components,
while the coarse-grid correction corresponds to the solution of the
resulting residual equation in an appropriately chosen coarse space
aimed at reducing the leftover error components. In the classical
aimed at reducing the leftover error components. In the original
multigrid approach, the coarser grid and the interpolation operator
for transfer the coarse-grid solution to the original (fine) grid are
predefined by the geometry of the problem. The recursive application
Expand All @@ -39,13 +38,13 @@ \section{Algebraic MultiGrid (AMG) Preconditioners}~\label{AMG}
problem and relying only on system matrix entries. In this work we
refer to an algebraic coarsening process based on aggregation of
unknowns, where coarse-grid unknowns are agglomerates of the original
unknowns. In particular, AMG preconditioners available in MLD2P4 rely
ones. In particular, AMG preconditioners available in MLD2P4 rely
on a decoupled version of \emph{the smoothed aggregation} algorithm
described in~\cite{BrezinaVanek96,BrezinaVanek99}. This procedure is
currently implemented on the host CPUs, therefore, for sake of space,
we refer the reader to~\cite{mld2p4-2-guide} for more details on the
algorithm and its parallel implementation.
Our main aim here is to describe all the main Linear Algebra
Our main aim here is to describe all the basic linear algebra
operations needed for the application of the AMG preconditioner within
an iterative linear solver, since their efficient implementation on
GPUs was the main focus of this work.
Expand Down Expand Up @@ -87,7 +86,7 @@ \section{Algebraic MultiGrid (AMG) Preconditioners}~\label{AMG}
level $k$ for relaxation.
Main computational kernels in the application of the preconditioner
are then sparse matrix-vector multiplications and inversion of the
matrix operator $M^k$. In the simple case of Jacobi method,
matrix operator $M^k$. In the simple case of Jacobi method is
$M^k=diag(A^k)$, therefore it corresponds to a highly parallel vector
update operation. On the other hand, more robust iterative methods,
such as the Gauss-Seidel method or incomplete factorizations, are
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