Andreas Frommer

Sebastian Birk, Andreas Frommer

We consider the task of computing solutions of linear systems that only differ by a shift with the identity matrix as well as linear systems with several different right hand sides. In the past Krylov subspace methods have been developed which exploit either the need for solutions to multiple right hand sides (e.g. deflation type methods and block methods) or multiple shifts (e.g. shifted CG) with some success. In this paper we present a block Krylov subspace method which, based on a block Lanczos process, exploits both features - shifts and multiple right hand sides - at once. Such situations arise, for example, in lattice QCD simulations within the Rational Hybrid Monte Carlo algorithm. We give numerical evidence that our method is superior to applying other iterative methods to each of the systems individually as well as, in some cases, to shifted or block Krylov subspace methods.

Somaiyeh Rashedi, Sebastian Birk, Andreas Frommer et al.

Block and global Krylov subspace methods have been proposed as methods adapted to the situation where one iteratively solves systems with the same matrix and several right hand sides. These methods are advantageous, since they allow to cast the major part of the arithmetic in terms of matrix-block vector products, and since, in the block case, they take their iterates from a potentially richer subspace. In this paper we consider the most established Krylov subspace methods which rely on short recurrencies, i.e. BiCG, QMR and BiCGStab. We propose modifications of their block variants which increase numerical stability, thus at least partly curing a problem previously observed by several authors. Moreover, we develop modifications of the "global" variants which almost halve the number of matrix-vector multiplications. We present a discussion as well as numerical evidence which both indicate that the additional work present in the block methods can be substantial, and that the new "economic" versions of the "global" BiCG and QMR method can be considered as good alternatives to the BiCGStab variants.

James Brannick, Andreas Frommer, Karsten Kahl et al.

The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics, the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations it preserves the important physical property of chiral symmetry, at the expense of requiring much more effort when solving systems with this operator. We present a preconditioning technique based on another lattice discretization, the Wilson-Dirac operator. The mathematical analysis precisely describes the effect of this preconditioning in the case that the Wilson-Dirac operator is normal. Although this is not exactly the case in realistic settings, we show that current smearing techniques indeed drive the Wilson-Dirac operator towards normality, thus providing a motivation why our preconditioner works well in computational practice. Results of numerical experiments in physically relevant settings show that our preconditioning yields accelerations of up to one order of magnitude.