arXiv subjects

Matthias Rottmann

Publications and source records attributed to Matthias Rottmann.

80 records · Page 5Linked to original sources

arXiv2018

Least Angle Regression Coarsening in Bootstrap Algebraic Multigrid

The bootstrap algebraic multigrid framework allows for the adaptive construction of algebraic multigrid methods in situations where geometric multigrid methods are not known or not available at all. While there has been some work on adaptive coarsening in this framework in terms of algebraic distances, coarsening is the part of the adaptive bootstrap setup that is least developed. In this paper we try to close this gap by introducing an adaptive coarsening scheme that views interpolation as a local regression problem. In fact the bootstrap algebraic multigrid setup can be understood as a machine learning ansatz that learns the nature of smooth error by local regression. In order to turn this idea into a practical method we modify least squares interpolation to both avoid overfitting of the data and to recover a sparse response that can be used to extract information about the coupling strength amongst variables like in classical algebraic multigrid. In order to improve the so-found coarse grid we propose a post-processing to ensure stability of the resulting least squares interpolation operator. We conclude with numerical experiments that show the viability of the chosen approach.

math.NA↗

arXiv2016

Adaptive Aggregation-based Domain Decomposition Multigrid for Twisted Mass Fermions

The Adaptive Aggregation-based Domain Decomposition Multigrid method (arXiv:1303.1377) 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.

hep-lat↗

arXiv2016

DDalphaAMG for Twisted Mass Fermions

We present the Adaptive Aggregation-based Domain Decomposition Multigrid method extended to the twisted mass fermion discretization action. We show comparisons of results as a function of tuning the parameters that enter the twisted mass version of the DDalphaAMG library (https://github.com/sbacchio/DDalphaAMG). Moreover, we linked the DDalphaAMG library to the tmLQCD software package and give details on the performance of the multigrid solver during HMC simulations at the physical point.

hep-lat↗

arXiv2015

Adaptive algebraic multigrid on SIMD architectures

We present details of our implementation of the Wuppertal adaptive algebraic multigrid code DD-$α$AMG on SIMD architectures, with particular emphasis on the Intel Xeon Phi processor (KNC) used in QPACE 2. As a smoother, the algorithm uses a domain-decomposition-based solver code previously developed for the KNC in Regensburg. We optimized the remaining parts of the multigrid code and conclude that it is a very good target for SIMD architectures. Some of the remaining bottlenecks can be eliminated by vectorizing over multiple test vectors in the setup, which is discussed in the contribution of Daniel Richtmann.

physics.comp-ph↗

arXiv2015

(Approximate) Low-Mode Averaging with a new Multigrid Eigensolver

We present a multigrid based eigensolver for computing low-modes of the Hermitian Wilson Dirac operator. For the non-Hermitian case multigrid methods have already replaced conventional Krylov subspace solvers in many lattice QCD computations. Since the $γ_5$-preserving aggregation based interpolation used in our multigrid method is valid for both, the Hermitian and the non-Hermitian case, inversions of very ill-conditioned shifted systems with the Hermitian operator become feasible. This enables the use of multigrid within shift-and-invert type eigensolvers. We show numerical results from our MPI-C implementation of a Rayleigh quotient iteration with multigrid. For state-of-the-art lattice sizes and moderate numbers of desired low-modes we achieve speed-ups of an order of magnitude and more over PARPACK. We show results and develop strategies how to make use of our eigensolver for calculating disconnected contributions to hadronic quantities that are noisy and still computationally challenging. Here, we explore the possible benefits, using our eigensolver for low-mode averaging and related methods with high and low accuracy eigenvectors. We develop a low-mode averaging type method using only a few of the smallest eigenvectors with low accuracy. This allows us to avoid expensive exact eigensolves, still benefitting from reduced statistical errors.

hep-lat↗

arXiv2014

Multigrid Preconditioning for the Overlap Operator in Lattice QCD

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.

hep-lat↗

arXiv2014

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.

hep-lat↗

arXiv2012

Aggregation-based Multilevel Methods for Lattice QCD

In Lattice QCD computations a substantial amount of work is spent in solving the Dirac equation. In the recent past it has been observed that conventional Krylov solvers tend to critically slow down for large lattices and small quark masses. We present a Schwarz alternating procedure (SAP) multilevel method as a solver for the Clover improved Wilson discretization of the Dirac equation. This approach combines two components (SAP and algebraic multigrid) that have separately been used in lattice QCD before. In combination with a bootstrap setup procedure we show that considerable speed-up over conventional Krylov subspace methods for realistic configurations can be achieved.

hep-lat↗