Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator

In lattice QCD computations a substantial amount of work is spent in solving discretized versions of the Dirac equation. Conventional Krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions. We present a domain decomposition adaptive algebraic multigrid method used as a precondtioner to solve the "clover improved" Wilson discretization of the Dirac equation. This approach combines and improves two approaches, namely domain decomposition and adaptive algebraic multigrid, that have been used seperately in lattice QCD before. We show in extensive numerical test conducted with a parallel production code implementation that considerable speed-up over conventional Krylov subspace methods, domain decomposition methods and other hierarchical approaches for realistic system sizes can be achieved.Comment: Additional comparison to method of arXiv:1011.2775 and to mixed-precision odd-even preconditioned BiCGStab. Results of numerical experiments changed slightly due to more systematic use of odd-even preconditionin

Dynamical Simulations of Lattice QCD

Lattice calculations of Quantum Chromodynamics (QCD) are continuously becoming more realistic. Where Ukawa famously concluded only fourteen years ago that simulations including two physically light sea quarks are basically impossible even with today’s computers, algorithmic developments over the last years have changed this situation drastically. Nowadays up and down quark masses light enough to control the chiral extrapolation reliably are standard and also the sea quark effects of strange (and charm) quark are included.Modern lattice simulations are an intricate interplay between a large variety of numerical methods on one side and the computer hardware on the other side. The main areas of progress have been the solvers used for the Dirac equation, fermion determinant factorisations and better integrators for the molecular dynamics which is at the heart of most algorithms used for QCD simulations.In lattice QCD simulations the path integral is computed via a Markov Chain Monte Carlo method. In virtually all projects with dynamical fermions a variant of the Hybrid Monte Carlo algorithm is employed to generate the Markov chain, where the fields are updated using molecular dynamics. But there is considerable freedom in how to include the fermion determinant into the simulation. Factorisations of this determinant have been essential in the progress of recent years, being successful in particular together with improved integrators of the molecular dynamics.The solution of the Dirac equation constitutes the most computer time consuming element of simulations with fermions. The dramatic speedup for small fermion mass due to locally deflated solvers5, 6 has therefore had a significant impact on what is possible in the simulations. These algorithms have practically eliminated the increase in cost of the solution as the quark mass is lowered

Nonperturbative renormalization of nonlocal quark bilinears for quasi-PDFs on the lattice using an auxiliary field

Quasi-PDFs provide a path toward an ab initio calculation of parton distribution functions (PDFs) using lattice QCD. One of the problems faced in calculations of quasi-PDFs is the renormalization of a nonlocal operator. By introducing an auxiliary field, we can replace the nonlocal operator with a pair of local operators in an extended theory. On the lattice, this is closely related to the static quark theory. In this approach, we show how to understand the pattern of mixing that is allowed by chiral symmetry breaking, and obtain a master formula for renormalizing the nonlocal operator that depends on three parameters. We present an approach for nonperturbatively determining these parameters and use perturbation theory to convert to the MS-bar scheme. Renormalization parameters are obtained for two lattice spacings using Wilson twisted mass fermions and for different discretizations of the Wilson line in the nonlocal operator. Using these parameters we show the effect of renormalization on nucleon matrix elements with pion mass approximately 370 MeV, and compare renormalized results for the two lattice spacings. The renormalized matrix elements are consistent among the different Wilson line discretizations and lattice spacings.Comment: 6 pages, 6 figures. v2: added perturbative matching to MS-bar and additional reference

Quantum annealing for systems of polynomial equations

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of $10^{-8}$.Comment: 11 pages, 4 figures. Added example for a system of quadratic equations. Supporting code is available at https://github.com/cchang5/quantum_poly_solver . This is a post-peer-review, pre-copyedit version of an article published in Scientific Reports. The final authenticated version is available online at: https://www.nature.com/articles/s41598-019-46729-

Algebraic Multi-Grid solver for lattice QCD on Exascale hardware: Intel Xeon Phi

In this white paper we describe work done on the development of an efficient iterative solver for lattice QCD based on the Algebraic Multi-Grid approach (AMG) within the tmLQCD software suite. This development is aimed at modern computer architectures that will be relevant for the Exa-scale regime, namely multicore processors together with the Intel Xeon Phi coprocessor. Because of the complexity of this solver, implementation turned out to take a considerable effort. Fine tuning and optimization will require more work and will be the subject of further investigation. However, the work presented here provides a necessary initial step in this direction

pMR: A high-performance communication library

On many parallel machines, the time LQCD applications spent in communication is a significant contribution to the total wall-clock time, especially in the strong-scaling limit. We present a novel high-performance communication library that can be used as a de facto drop-in replacement for MPI in existing software. Its lightweight nature that avoids some of the unnecessary overhead introduced by MPI allows us to improve the communication performance of applications without any algorithmic or complicated implementation changes. As a first real-world benchmark, we make use of the pMR library in the coarse-grid solve of the Regensburg implementation of the DD-$\alpha$AMG algorithm. On realistic lattices, we see an improvement of a factor 2x in pure communication time and total execution time savings of up to 20%.Comment: 7 pages, 2 figures, Proceedings of Lattice 201

Second moment of the pion distribution amplitude with the momentum smearing technique

Using the second moment of the pion distribution amplitude as an example, we investigate whether lattice calculations of matrix elements of local operators involving covariant derivatives may benefit from the recently proposed momentum smearing technique for hadronic interpolators. Comparing the momentum smearing technique to the traditional Wuppertal smearing we find—at equal computational cost—a considerable reduction of the statistical errors. The present investigation was carried out using N_{f}=2+1 dynamical non-perturbatively order a improved Wilson fermions on lattices of different volumes and pion masses down to 220 MeV