29 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
Energy-adaptive Riemannian optimization on the Stiefel manifold
This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energyadaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes
Energy-adaptive Riemannian optimization on the Stiefel manifold
This paper addresses the numerical solution of nonlinear eigenvector problems
such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational
physics and chemistry. These problems characterize critical points of energy
minimization problems on the infinite-dimensional Stiefel manifold. To
efficiently compute minimizers, we propose a novel Riemannian gradient descent
method induced by an energy-adaptive metric. Quantified convergence of the
methods is established under suitable assumptions on the underlying problem. A
non-monotone line search and the inexact evaluation of Riemannian gradients
substantially improve the overall efficiency of the method. Numerical
experiments illustrate the performance of the method and demonstrates its
competitiveness with well-established schemes.Comment: accepted for publication in M2A
ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers
Solving the electronic structure from a generalized or standard eigenproblem
is often the bottleneck in large scale calculations based on Kohn-Sham
density-functional theory. This problem must be addressed by essentially all
current electronic structure codes, based on similar matrix expressions, and by
high-performance computation. We here present a unified software interface,
ELSI, to access different strategies that address the Kohn-Sham eigenvalue
problem. Currently supported algorithms include the dense generalized
eigensolver library ELPA, the orbital minimization method implemented in
libOMM, and the pole expansion and selected inversion (PEXSI) approach with
lower computational complexity for semilocal density functionals. The ELSI
interface aims to simplify the implementation and optimal use of the different
strategies, by offering (a) a unified software framework designed for the
electronic structure solvers in Kohn-Sham density-functional theory; (b)
reasonable default parameters for a chosen solver; (c) automatic conversion
between input and internal working matrix formats, and in the future (d)
recommendation of the optimal solver depending on the specific problem.
Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800
basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table
ELSI -- An open infrastructure for electronic structure solvers
Routine applications of electronic structure theory to molecules and periodic systems need to compute the electron density from given Hamiltonian and, in case of non-orthogonal basis sets, overlap matrices. System sizes can range from few to thousands or, in some examples, millions of atoms. Different discretization schemes (basis sets) and different system geometries (finite non-periodic vs. infinite periodic boundary conditions) yield matrices with different structures. The ELectronic Structure Infrastructure (ELSI) project provides an open-source software interface to facilitate the implementation and optimal use of high-performance solver libraries covering cubic scaling eigensolvers, linear scaling density-matrix-based algorithms, and other reduced scaling methods in between. In this paper, we present recent improvements and developments inside ELSI, mainly covering (1) new solvers connected to the interface, (2) matrix layout and communication adapted for parallel calculations of periodic and/or spin-polarized systems, (3) routines for density matrix extrapolation in geometry optimization and molecular dynamics calculations, and (4) general utilities such as parallel matrix I/O and JSON output. The ELSI interface has been integrated into four electronic structure code projects (DFTB+, DGDFT, FHI-aims, SIESTA), allowing us to rigorously benchmark the performance of the solvers on an equal footing. Based on results of a systematic set of large-scale benchmarks performed with Kohn–Sham density-functional theory and density-functional tight-binding theory, we identify factors that strongly affect the efficiency of the solvers, and propose a decision layer that assists with the solver selection process. Finally, we describe a reverse communication interface encoding matrix-free iterative solver strategies that are amenable, e.g., for use with planewave basis sets. Program summary: Program title: ELSI Interface CPC Library link to program files: http://dx.doi.org/10.17632/473mbbznrs.1 Licensing provisions: BSD 3-clause Programming language: Fortran 2003, with interface to C/C++ External routines/libraries: BLACS, BLAS, BSEPACK (optional), EigenExa (optional), ELPA, FortJSON, LAPACK, libOMM, MPI, MAGMA (optional), MUMPS (optional), NTPoly, ParMETIS (optional), PETSc (optional), PEXSI, PT-SCOTCH (optional), ScaLAPACK, SLEPc (optional), SuperLU_DIST Nature of problem: Solving the electronic structure from given Hamiltonian and overlap matrices in electronic structure calculations. Solution method: ELSI provides a unified software interface to facilitate the use of various electronic structure solvers including cubic scaling dense eigensolvers, linear scaling density matrix methods, and other approaches