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 of multigrid hierarchies constructed with $\mathbf{P}_\star$ . 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 from $\mathbf{P}_\star$ 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 coarsening aggressively to yield a small set of C points for which $\mathbf{P}_\star$ may have relatively poor convergence.