In lattice QCD computations a substantial amount of work is spent in solving linear systems arising in Wilson's discretization of the Dirac equations. We show first numerical results of the extension of the two-level DD-αAMG method to a true multilevel method based on our parallel MPI-C implementation. Using additional levels pays off, allowing to cut down the core minutes spent on one system solve by a factor of approximately 700 compared to standard Krylov subspace methods and yielding another speed-up of a factor of 1.7 over the two-level approach.

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The Adaptive Aggregation-based Domain Decomposition Multigrid method [1] is extended for two degenerate flavors of twisted mass fermions. By fine-tuning the parameters we achieve a speed-up of the order of hundred times compared to the conjugate gradient algorithm for the physical value of the pion mass. A thorough analysis of the aggregation parameters is presented, which provides a novel insight into multigrid methods for lattice QCD independently of the fermion discretization.

In this paper, we consider a classical form of optimal algebraic multigrid (AMG) interpolation that directly minimizes the two-grid convergence rate and compare it with the so-called ideal form that minimizes a certain weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap algebraic multigrid setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the BAMG method with sparse interpolation matrix (and spanning multiple levels) outperforms the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.

This work concerns the development of an Algebraic Multilevel method for computing stationary vectors of Markov chains. We present an efficient Bootstrap Algebraic Multilevel method for this task. In our proposed approach, we employ a multilevel eigensolver, with interpolation built using ideas based on compatible relaxation, algebraic distances, and least squares fitting of test vectors. Our adaptive variational strategy for computation of the state vector of a given Markov chain is then a combination of this multilevel eigensolver and associated multilevel preconditioned GMRES iterations. We show that the Bootstrap AMG eigensolver by itself can efficiently compute accurate approximations to the state vector. An additional benefit of the Bootstrap approach is that it yields an accurate interpolation operator for many other eigenmodes. This in turn allows for the use of the resulting AMG hierarchy to accelerate the MLE steps using standard multigrid correction steps. Further, we mention that our method, unlike other existing multilevel methods for Markov Chains, does not employ any special processing of the coarse-level systems to ensure that stochastic properties of the fine-level system are maintained there. The proposed approach is applied to a range of test problems, involving non-symmetric stochastic M-matrices, showing promising results for all problems considered.

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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.

Abstract. This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.