Erin Molloy
Is the ideal approximation operator always ``ideal'' for a particular C/F splitting?

University of Illinois at Urbana-Champaign
Department of Computer Science
emolloy2@illinois.edu
Luke Olson
Jacob Schroder

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.