2,239 research outputs found
Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
The iterative diagonalization of a sequence of large ill-conditioned
generalized eigenvalue problems is a computational bottleneck in quantum
mechanical methods employing a nonorthogonal basis for {\em ab initio}
electronic structure calculations. We propose a hybrid preconditioning scheme
to effectively combine global and locally accelerated preconditioners for rapid
iterative diagonalization of such eigenvalue problems. In partition-of-unity
finite-element (PUFE) pseudopotential density-functional calculations,
employing a nonorthogonal basis, we show that the hybrid preconditioned block
steepest descent method is a cost-effective eigensolver, outperforming current
state-of-the-art global preconditioning schemes, and comparably efficient for
the ill-conditioned generalized eigenvalue problems produced by PUFE as the
locally optimal block preconditioned conjugate-gradient method for the
well-conditioned standard eigenvalue problems produced by planewave methods
High-Performance Solvers for Dense Hermitian Eigenproblems
We introduce a new collection of solvers - subsequently called EleMRRR - for
large-scale dense Hermitian eigenproblems. EleMRRR solves various types of
problems: generalized, standard, and tridiagonal eigenproblems. Among these,
the last is of particular importance as it is a solver on its own right, as
well as the computational kernel for the first two; we present a fast and
scalable tridiagonal solver based on the Algorithm of Multiple Relatively
Robust Representations - referred to as PMRRR. Like the other EleMRRR solvers,
PMRRR is part of the freely available Elemental library, and is designed to
fully support both message-passing (MPI) and multithreading parallelism (SMP).
As a result, the solvers can equally be used in pure MPI or in hybrid MPI-SMP
fashion. We conducted a thorough performance study of EleMRRR and ScaLAPACK's
solvers on two supercomputers. Such a study, performed with up to 8,192 cores,
provides precise guidelines to assemble the fastest solver within the ScaLAPACK
framework; it also indicates that EleMRRR outperforms even the fastest solvers
built from ScaLAPACK's components
Numerical Methods for the QCD Overlap Operator:III. Nested Iterations
The numerical and computational aspects of chiral fermions in lattice quantum
chromodynamics are extremely demanding. In the overlap framework, the
computation of the fermion propagator leads to a nested iteration where the
matrix vector multiplications in each step of an outer iteration have to be
accomplished by an inner iteration; the latter approximates the product of the
sign function of the hermitian Wilson fermion matrix with a vector. In this
paper we investigate aspects of this nested paradigm. We examine several Krylov
subspace methods to be used as an outer iteration for both propagator
computations and the Hybrid Monte-Carlo scheme. We establish criteria on the
accuracy of the inner iteration which allow to preserve an a priori given
precision for the overall computation. It will turn out that the accuracy of
the sign function can be relaxed as the outer iteration proceeds. Furthermore,
we consider preconditioning strategies, where the preconditioner is built upon
an inaccurate approximation to the sign function. Relaxation combined with
preconditioning allows for considerable savings in computational efforts up to
a factor of 4 as our numerical experiments illustrate. We also discuss the
possibility of projecting the squared overlap operator into one chiral sector.Comment: 33 Pages; citations adde
Taylor expansion and the Cauchy Residue Theorem for finite-density QCD
We present an update on our efforts to determine the Taylor coefficients of
the expansion of the pressure for finite-density QCD. Here, we explore
alternatives based on the Cauchy Residue Theorem, which allows us to use a
discretized contour to determine the desired spectral moments occurring in the
Taylor expansion of QCD at zero chemical potential.Comment: 6 pages, 4 figures, talk presented at the 36th Annual International
Symposium on Lattice Field Theory, July 22-28, 2018, East Lansing, MI, US
Multigrid accelerated simulations for Twisted Mass fermions
Simulations at physical quark masses are affected by the critical slowing
down of the solvers. Multigrid preconditioning has proved to deal effectively
with this problem. Multigrid accelerated simulations at the physical value of
the pion mass are being performed to generate and
gauge ensembles using twisted mass fermions. The adaptive aggregation-based
domain decomposition multigrid solver, referred to as DD-AMG method, is
employed for these simulations. Our simulation strategy consists of an hybrid
approach of different solvers, involving the Conjugate Gradient (CG),
multi-mass-shift CG and DD-AMG solvers. We present an analysis of the
multigrid performance during the simulations discussing the stability of the
method. This significant speeds up the Hybrid Monte Carlo simulation by more
than a factor at physical pion mass compared to the usage of the CG solver.Comment: 8 pages, 5 figures, proceedings for LATTICE 201
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