The eigSUMR inverter for overlap fermion

We discuss the usage and applicability of deflation methods for the overlap lattice Dirac operator, focussing on calculating the eigenvalues using a method similar to the eigCG algorithm used for other Dirac operators. The overlap operator, which contains several theoretical advantages over other formulations of lattice Quantum Chromodynamics, is more computationally expensive because it requires the computation of the matrix sign function. The principle change made compared to deflation methods for other formulations of lattice QCD is that it is necessary for best performance to tune or relax the accuracy of the matrix sign function as the computation proceeds. We adapt the eigCG algorithm for two inversion algorithms for overlap fermions, GMRESR(relCG) and GMRESR(relSUMR). Before deflation, the rate of convergence of these routines in terms of iterations is similar, but, since the Shifted Unitary Minimal Residual (SUMR) algorithm only requires one call to the matrix sign function compared to the two calls required for Conjugate Gradient (CG), SUMR is usually preferred for single inversions of the Dirac operator. We construct bounds for the required accuracy of the matrix sign function during the eigenvalue calculation. For the SUMR algorithm, we use a Galerkin projection to perform the deflation; while for the CG algorithm, we are able to use a considerably superior spectral pre-conditioner. The superior performance of the spectral preconditioner, and its need for less accurate eigenvalues, almost erodes SUMR's advantage over CG as an inversion algorithm. We see factor of three gains for the inversion algorithm from the deflation on our small test lattices. There is, however, a significant cost in the eigenvalue calculation because we cannot relax the accuracy of the matrix sign function as aggressively when calculating the eigenvalues as we do while performing the inversions.