Hierarchically deflated conjugate gradient

Abstract

We present a multi-level algorithm for the solution of five dimensional chiral fermion formulations, including domain wall and Mobius Fermions. The algorithm operates on the red-black preconditioned Hermitian operator, and directly accelerates conjugate gradients on the normal equations. The coarse grid representation of this matrix is next-to-next-to-next-to-nearest neighbour and multiple algorithmic advances are introduced, which help minimise the overhead of the coarse grid. The treatment of the coarse grids is purely four dimensional, and the bulk of the coarse grid operations are nearest neighbour. The intrinsic cost of most of the coarse grid operations is therefore comparable to those for the Wilson case. We also document the implementation of this algorithm in the BAGEL/Bfm software package and report on the measured performance gains the algorithm brings to simulations at the physical point on IBM BlueGene/Q hardware.

37 pages, 6 figures

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