Is the Ideal Approximation Operator Always "Ideal" for a Particular C/F Splitting?

By Erin Molloy

University of Illinois at Urbana-Champaign

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Given a coarse grid, the ideal prolongation operator is defined by $ %5Cmathbf{P}_%5Cstar = %5Cbegin{bmatrix}%5Cmathbf{W} & %5Cmathbf{I} %5Cend{bmatrix}^T$, where the weight matrix, $ %5Cmathbf{W} = %5Cmathbf{A}_{FF}^{-1} %5Cmathbf{A}_{FC}$ , interpolates a set of fine grid variable ($ F$-points) from a set of coarse grid variable ($ C$-points), and the identity matrix, $ %5Cmathbf{I}$, represents the injection of $ C$-points to and from the coarse grid (Falgout and Vassilevski, 2004). In this talk, we consider $ %5Cmathbf{P}_%5Cstar$, 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 $ %5Cmathbf{P}_%5Cstar$. Finally, we argue that $ %5Cmathbf{P}_%5Cstar$ may be misleading in demonstrating the ``ideal'' nature of interpolation of a given $ C/F$ splitting by providing numerical evidence that hierarchies built using $ %5Cmathbf{P}_%5Cstar$ 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 $ %5Cmathbf{P}_%5Cstar$ may have poor convergence.

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Researchers should cite this work as follows:

  • Erin Molloy (2016), "Is the Ideal Approximation Operator Always "Ideal" for a Particular C/F Splitting?,"

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NanoBio Node, Aly Taha

University of Illinois at Urbana-Champaign