===affil2:
University of Illinois at Urbana-Champaign
===firstname:
Erin 
===firstname4:

===firstname3:
Jacob
===lastname2:
Olson
===keyword1:
AMG
===lastname:
Molloy
===firstname5:

===affil6:

===lastname3:
Schroder
===email:
emolloy2@illinois.edu
===keyword2:
Multigrid
===keyword_other2:

===lastname6:

===affil5:

===otherauths:

===lastname4:

===affil4:

===lastname7:

===competition:
no
===affil7:

===firstname7:

===postal:
University of Illinois at Urbana-Champaign
Department of Computer Science, Rm 4332
201 N. Goodwin Ave.
Urbana, IL 61801
===firstname6:

===ABSTRACT:
Given a coarse grid, the ideal prolongation operator is defined by 
$\mathbf{P}_\star = \begin{bmatrix} \mathbf{W} & \mathbf{I} \end{bmatrix}^T$, 
where the weight matrix, $\mathbf{W} = \mathbf{A}_{FF}^{-1} \mathbf{A}_{FC}$ , 
interpolates a set of fine grid variable ($F$-points) from a set of coarse grid variable 
($C$-points), and the identity matrix, $\mathbf{I}$, represents the injection of $C$-points to and from the coarse grid (Falgout and Vassilevski, 2004). 
In this talk, we consider $\mathbf{P}_\star$, constructed from 
both traditional $C/F$ splittings and $C/F$ splittings corresponding to
aggregates, for several challenging problems. We demonstrate the effects of the
$C/F$ splitting on the convergence and complexity of $\mathbf{P}_\star$. 
Finally, we argue that $\mathbf{P}_\star$ may be misleading in demonstrating the 
``ideal'' nature of interpolation of a given $C/F$ splitting by providing numerical evidence that hierarchies built using $\mathbf{P}_\star$ converge more slowly 
than hierarchies built from alternative prolongation operators with the same $C/F$ splitting. This is important as we wish to minimize the number of levels in
a multigrid hierarchy by using a small set of C points for which $\mathbf{P}_\star$ 
may have poor convergence.
===affil3:
Lawrence Livermore National Laboratory
===keyword_other1:

===lastname5:

===affilother:

===title:
Is the ideal approximation operator always ``ideal'' for a particular C/F splitting?
===firstname2:
Luke