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“Adaptive Algebraic Multigrid Preconditioners in Quantum Chromodynamics”

James Brannick
Department of Applied Mathematics
University of Colorado at Boulder
April 15, 2005
Standard algebraic multigrid methods assume explicit knowledge of so-called algebraically-smooth or near-kernel components, which loosely speaking are errors that give relatively small residuals. Typically, these methods automatically generate a sequence of coarse problems under the assumption that the near-kernel is locally constant. The difficulty in applying algebraic multigrid to lattice QCD is that the near-kernel components can be far from constant, often exhibiting little or no apparent smoothness. In fact, the local character of these components appears to be random, depending on the randomness of the so-called "gauge" group. Hence, no apriori knowledge of the local character of the near-kernel is readily available.
This talk develops an adaptive algebraic multigrid (AMG) preconditioner suitable for the linear systems arising in lattice QCD.The method is a recently developed extension of smoothed aggregation, aSA, in which good convergence properties are achieved in situations where explicit knowledge of the near-kernel components may not be available. This extension is accomplished using the method itself to determine near-kernel components automatically, by applying it carefully to the homogeneous matrix equation, Ax=0. The coarsening process is modified to use and improve the computed components. Preliminary results with model 2D QCD problems suggest that this approach may yield optimal multigrid-like performance that is uniform in matrix dimension and gauge-group randomness.

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