The two-grid algorithm confronts a shifted unitary orthogonal method

Abstract

In this paper I describe a new optimal Krylov subspace solver for shifted unitary matrices called the Shifted Unitary Orthogonal Method (SUOM). This algorithm is used as a benchmark against any improvement like the two-grid algorithm. I use the latter to show that the overlap operator can be inverted by successive inversions of the truncated overlap operator. This strategy results in large gains compared to SUOM.It is well-known that overlap fermions [1] lead to much more expensive computations than standard fermions, i.e. Wilson or Kogut-Sussking fermions. This is obvious since for every application of the overlap operator an extra linear system solving is needed. For the time being, it seems that to get chiral symmetry at finite lattice spacing one should wait for a Petaflops computer being built.However, algorithmic research is far from exhausted. In this paper I give an example that this is the case if one uses the two-grid algorithm [2]. Before I do this, I introduce briefly an optimal Krylov subspace solver for shifted unitary matrices. SUOM: A NEW OPTIMAL KRYLOV SOLVERConsider the task of solving the linear system:where V = γ 5 sign(H W ) is a unitary matrix, 1l the identity matrix, H W the Hermitian Wilson operator, c 1 = (1 + m q )/2, c 2 = (1 − m q )/2 and m q the bare fermion mass. The overlap operator D is non-Hermitian. For such operators GMRES (Generalised Minimal Residual) and FOM (Full Orthogonalisation Method) are known to be the fastest. It is shown that when the norm-minimising process of GMRES is converging rapidly, the residual norms in the corresponding Galerkin process of FOM exhibit similar behaviour [3]. But they are based on long recurrences and thus require to store a large number of vectors of the size of matrix columns. However, exploiting the fact that the overlap operator is a shifted unitary matrix one can construct a GM-RES type algorithm with short recurrences [4]. Similarly, a short recurrences algorithm can be obtained from FOM. The method is based on an observation of Rutishauser [5] that for upper Hessenberg unitary matrices one can write H = LU −1 , where L and U are lower and upper bidiagonal matrices. Applying this decomposition for the Arnoldi iteration:one obtains an algorithm which constructs Arnoldi vectors Q k by short recurrences [6]:(1.3)Projecting the linear system (1.1) onto the Krylov subspace one gets:k )y k = e 1 (1.4) which can be equivalently written as:Note that the matrix on the left hand side is tridiagonal. It can be shown that one can solve 1

Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory

“…This idea is implemented in SHUMR (Shifted Unitary Minimal Residual) optimal algorithm which has been part of QCDLAB 1.0., and it is shown in the Appendix C. This algorithm is tested before also in case of SU(3) gauge theory and gives good results [17]. Based on the idea of a two level algorithm as proposed in [11,12], in this work we develop a faster algorithm, the preconditioned GMRESR [18] (Generalized Minimal Residual Method -Recursive) algorithm developed it as part of QCDLAB 1.0 package, in U(1) gauge field background. So in this way we have added in QCDLAB 1.0 new efficient routines.…”

“…In practice computing quark propagator in lattice is an inversion problem of the Dirac operator matrix representing these quarks. Motivated by the results of the two grid algorithm that we proposed [11,12] as the fastest overlap solver in SU(3) gauge theory, which didn't converges for all quark masses, we thought to test this idea in less dimensions such as U(1) gauge theory. In our case we want to calculate the domain wall fermion propagator, but in order to develop fast algorithms, we use the truncated overlap variant of domain wall fermions in 2+1 dimensions with the extra finite dimension N3.…”