Non-linear eigensolver-based alternative to traditional SCF methods
15 May 2013-Journal of Chemical Physics (American Institute of Physics)-Vol. 138, Iss: 19, pp 194101-194101
TL;DR: In this article, the authors revisited the selfconsistent procedure in electronic structure calculations using a highly efficient and robust algorithm for solving the non-linear eigenvector problem, i.e., H({ψ})ψ = Eψ.
Abstract: The self-consistent procedure in electronic structure calculations is revisited using a highly efficient and robust algorithm for solving the non-linear eigenvector problem, i.e., H({ψ})ψ = Eψ. This new scheme is derived from a generalization of the FEAST eigenvalue algorithm to account for the non-linearity of the Hamiltonian with the occupied eigenvectors. Using a series of numerical examples and the density functional theory-Kohn/Sham model, it will be shown that our approach can outperform the traditional SCF mixing-scheme techniques by providing a higher converge rate, convergence to the correct solution regardless of the choice of the initial guess, and a significant reduction of the eigenvalue solve time in simulations.
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TL;DR: This work proposes improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximation based on the work of Zolotarev, and improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.
Abstract: The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.
100 citations
TL;DR: The distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals are emphasized and it is shown how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension.
Abstract: Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. Finally we highlight the power of rational in comparison with polynomial approximations for some of these problems.
74 citations
13 Nov 2016
TL;DR: This paper highlights a recent development within the software package that allows the dominant computational task, solving a set of complex linear systems, to be performed with a distributed memory solver.
Abstract: The FEAST algorithm and eigensolver for interior eigenvalue problems naturally possesses three distinct levels of parallelism. The solver is then suited to exploit modern computer architectures containing many interconnected processors. This paper highlights a recent development within the software package that allows the dominant computational task, solving a set of complex linear systems, to be performed with a distributed memory solver. The software, written with a reverse-communication-interface, can now be interfaced with any generic MPI linear-system solver using a customized data distribution for the eigenvector solutions. This work utilizes two common "black-box" distributed memory linear-systems solvers (Cluster-MKL-Pardiso and MUMPS), as well as our own application-specific domain-decomposition MPI solver, for a collection of 3-dimensional finite-element systems. We discuss and analyze how parallel resources can be placed at all three levels simultaneously in order to achieve good scalability and optimal use of the computing platform.
35 citations
TL;DR: A rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions based on which a robust eigen-solver, called RSRR, is developed for solving general NEPs.
Abstract: Summary
Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen-solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, called RSRR, is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples. This article is protected by copyright. All rights reserved.
19 citations
TL;DR: In this article, the plasmon resonances in a number of representative 1-D finite carbon-based nanostructures using first-principle computational electronic spectroscopy studies were identified.
Abstract: Tomonaga-Luttinger (T-L) theory predicts collective plasmon resonances in 1-D nanostructure conductors of finite length, that vary roughly in inverse proportion to the length of the structure. In-depth quantitative understanding of such resonances which have not been clearly identified in experiments so far, would be invaluable for future generations of nano-photonic and nano-electronic devices that employ 1-D conductors. Here we provide evidence of the plasmon resonances in a number of representative 1-D finite carbon-based nanostructures using first-principle computational electronic spectroscopy studies. Our special purpose real-space/real-time all-electron Time-Dependent Density-Functional Theory (TDDFT) simulator can perform excited-states calculations to obtain correct frequencies for known optical transitions, and capture various nanoscopic effects including collective plasmon excitations. The presence of 1-D plasmons is universally predicted by the various numerical experiments, which also demonstrate a phenomenon of resonance splitting. For the metallic carbon nanotubes under study, the plasmons are expected to be related to the T-L plasmons of infinitely long 1-D structures.
9 citations
References
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TL;DR: An overview of NWChem is provided focusing primarily on the core theoretical modules provided by the code and their parallel performance, as well as Scalable parallel implementations and modular software design enable efficient utilization of current computational architectures.
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TL;DR: In this article, a procedure is given for accelerating the convergence of slowly converging quasi-Newton-Raphson type algorithms for large systems of linear equations, where the number of parameters is so large that the calculation and storage of the hessian is no longer practical.
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TL;DR: A new numerical algorithm for solving the symmetric eigenvalue problem is presented, which takes its inspiration from the contour integration and density matrix representation in quantum mechanics.
Abstract: A fast and stable numerical algorithm for solving the symmetric eigenvalue problem is presented. The technique deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques and takes its inspiration from the contour integration and density-matrix representation in quantum mechanics. It will be shown that this algorithm---named FEAST---exhibits high efficiency, robustness, accuracy, and scalability on parallel architectures. Examples from electronic structure calculations of carbon nanotubes are presented, and numerical performances and capabilities are discussed.
379 citations
TL;DR: In this article, the Broyden update of the inverse Jacobian is shown to be superior to updating the Jacobian itself, and the Anderson mixing and Broyden updating are shown to cancel out each other.
352 citations
TL;DR: The present algorithm is benchmarked against the direct inversion iterative subspace method based on the commutator of the density and Fock matrices developed by Pulay (DIIS), and it is shown how EDIIS can detect the presence and determine the value of fractional occupations in KS-DFT.
Abstract: A direct inversion iterative subspace version of the relaxed constrained algorithm is found to be a very powerful convergence acceleration technique for the solution of the self-consistent field equations found in the Hartree–Fock method and Kohn–Sham-based density functional theory (KS-DFT). The present algorithm, abbreviated EDIIS, is benchmarked against the direct inversion iterative subspace method based on the commutator of the density and Fock matrices developed by Pulay (DIIS). Our findings indicate that while EDIIS is able to rapidly bring the density matrix from any initial guess to a solution region, the DIIS method is faster when the density matrix is close to convergence. Consequently, we propose a combination of EDIIS and DIIS methods, which is both very robust and highly efficient. We also show how EDIIS can detect the presence and determine the value of fractional occupations in KS-DFT.
232 citations