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{\large Artem Napov \\
{\bf Algebraic analysis of aggregation-based multigrid }}

Université Libre de Bruxelles \\ Service de Métrologie Nucléaire (CP 165/84) \\ Avenue F D Roosevelt 50 \\ B-1050 Bruxelles - Belgique
\\
{\tt anapov@ulb.ac.be}\\
Yvan Notay\end{center}

Algebraic multigrid techniques based on coarsening by aggregation
\cite{bul93,br85} have gained in popularity over the last years
\cite{no09,muno08}. The growing interest for such approaches is motivated
by their low setup cost and memory requirements, combined with a (close
to) optimal convergence when used with K-cycle \cite{nova07}.

In this talk, we present a convergence analysis of two-grids methods
based on aggregation \cite{agganl}. For diagonally dominant symmetric
(M-)matrices, we show that the analysis can be conducted locally; that
is, the convergence factor can be bounded above by computing separately
for each aggregate a parameter which in some sense measures its quality.
The procedure is purely algebraic and can be used to control the quality
of automatic coarsening algorithms. Assuming the aggregation pattern
sufficiently regular, we show that the resulting bound is asymptotically
sharp for a large class of elliptic boundary value problems, including
problems with variable and discontinuous coefficients. In particular, the
analysis of typical examples reveals that the convergence rate is
insensitive to discontinuities under some reasonable assumptions on the
aggregation scheme.

\begin{thebibliography}{6}
\bibitem{bul93}\textsc{V. E. Bulgakov}, \textit{Multi-level iterative
technique and aggregation concept with semi-analytical preconditioning
for solving boundary-value problems}, Comm. Numer. Methods Engrng.,
9(1993), {pp. 649--657}.
\bibitem{br85}\textsc{D. Braess}, \textit{Towards algebraic multigrid for
elliptic problems of second order}, {Computing}, {55}(1995), {pp.
379--393}.
\bibitem{no09}\textsc{Y. Notay}, \textit{An aggregation-based algebraic
multigrid method}, Electronic Transactions on Numerical Analysis, to
appear, 2009.
\bibitem{muno08} \textsc{A. C. Muresan} and \textsc{Y. Notay},
\textit{Analysis of aggregation-based multigrid}, SIAM J. Sci. Comp.,
30(2008), pp. 1082-1103.
\bibitem{nova07} \textsc{Y. Notay} and \textsc{P.S. Vassilevski},
\textit{Recursive Krylov-based multigrid cycles}, Numer. Lin. Alg. Appl.,
15(2008), pp. 473-487.
\bibitem{agganl} \textsc{A. Napov} and\textsc{ Y. Notay},
\textit{Algebraic analysis of aggregation-based multigrid}, Tech. Report
GANMN 09-04, Universit\'e Libre de Bruxelles, Brussels, Belgium, 2009.
\url{http://homepages.ulb.ac.be/~anapov}.
\end{thebibliography}


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