Given a coarse grid, the ideal prolongation operator is defined by , where the weight matrix, , interpolates a set of fine grid variable (-points) from a set of coarse grid variable (-points), and the identity matrix, , represents the injection of -points to and from the coarse grid (Falgout and Vassilevski, 2004). In this talk, we consider , constructed from both traditional splittings and splittings corresponding to aggregates, for several challenging problems. We demonstrate the effects of the splitting on the convergence and complexity of . Finally, we argue that may be misleading in demonstrating the ``ideal'' nature of interpolation of a given splitting by providing numerical evidence that hierarchies built using converge more slowly than hierarchies built from alternative prolongation operators with the same 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 may have poor convergence.
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