Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices

We propose a Preconditioned Locally Harmonic Residual (PLHR) method for computing several interior eigenpairs of a generalized Hermitian eigenvalue problem, without traditional spectral transformations, matrix factorizations, or inversions. PLHR is based on a short-term recurrence, easily extended to a block form, computing eigenpairs simultaneously. PLHR can take advantage of Hermitian positive definite preconditioning, e.g., based on an approximate inverse of an absolute value of a shifted matrix, introduced in [SISC, 35 (2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is efficient and robust for certain classes of large-scale interior eigenvalue problems, involving Laplacian and Hamiltonian operators, especially if memory requirements are tight

Equipping Sparse Solvers for Exascale (ESSEX / ESSEX II)

The ESSEX project is funded by the German DFG priority programme 1648 "Software for Exascale Computing" (SPPEXA). In 2016 it has entered its second funding phase, ESSEX-II. ESSEX investigated programming concepts and numerical algorithms for scalable, efficient and robust iterative sparse matrix applications on exascale systems. Starting with successful blueprints and prototype solutions identified in ESSEX-I, the second phase project ESSEX-II developed a collection of broadly usable and scalable sparse eigenvalue solvers with high hardware efficiency for the computer architectures to come. Project activities were organized along the traditional software layers of low-level parallel building blocks (kernels), algorithm implementations, and applications. The classic abstraction boundaries separating these layers were broken in ESSEX by strongly integrating objectives: scalability, numerical reliability, fault tolerance, and holistic performance and power engineering. The basic building block library supports an elaborate MPI+X approach that is able to fully exploit hardware heterogeneity while exposing functional parallelism and data parallelism to all other software layers in a flexible way. In addition, facilities for fully asynchronous checkpointing, silent data corruption detection and correction, performance assessment, performance model validation, and energy measurements are provided transparently. The advanced building blocks were defined and employed by the developments at the algorithms layer. Here, ESSEX-II provides state-of-the-art library implementations of classic linear sparse eigenvalue solvers including block Jacobi-Davidson, Kernel Polynomial Method (KPM), and Chebyshev filter diagonalization (ChebFD) that are ready to use for production on modern heterogeneous compute nodes with best performance and numerical accuracy. Research in this direction included the development of appropriate parallel adaptive AMG software for the block Jacobi-Davidson method. Contour integral-based approaches were also covered in ESSEX-II and were extended in two directions: The FEAST method was further developed for improved scalability, and the Sakurai-Sugiura method (SSM) method was extended to nonlinear sparse eigenvalue problems. These developments were strongly supported by additional Japanese project partners from University of Tokyo, Computer Science, and University of Tsukuba, Applied Mathematics. The applications layer delivers scalable solutions for conservative (Hermitian) and dissipative (non- Hermitian) quantum systems with strong links to optics and biology and to novel materials such as graphene and topological insulators. This talk gives a survey on latest results of the ESSEX-II project

Albany: Using Component-based Design to Develop a Flexible, Generic Multiphysics Analysis Code

Abstract: Albany is a multiphysics code constructed by assembling a set of reusable, general components. It is an implicit, unstructured grid finite element code that hosts a set of advanced features that are readily combined within a single analysis run. Albany uses template-based generic programming methods to provide extensibility and flexibility; it employs a generic residual evaluation interface to support the easy addition and modification of physics. This interface is coupled to powerful automatic differentiation utilities that are used to implement efficient nonlinear solvers and preconditioners, and also to enable sensitivity analysis and embedded uncertainty quantification capabilities as part of the forward solve. The flexible application programming interfaces in Albany couple to two different adaptive mesh libraries; it internally employs generic integration machinery that supports tetrahedral, hexahedral, and hybrid meshes of user specified order. We present the overall design of Albany, and focus on the specifics of the integration of many of its advanced features. As Albany and the components that form it are openly available on the internet, it is our goal that the reader might find some of the design concepts useful in their own work. Albany results in a code that enables the rapid development of parallel, numerically efficient multiphysics software tools. In discussing the features and details of the integration of many of the components involved, we show the reader the wide variety of solution components that are available and what is possible when they are combined within a simulation capability. Key Words: partial differential equations, finite element analysis, template-based generic programmin

Holographic entanglement entropy of local quenches in AdS4/CFT3: a finite-element approach

Understanding quantum entanglement in interacting higher-dimensional conformal field theories is a challenging task, as direct analytical calculations are often impossible to perform. With holographic entanglement entropy, calculations of entanglement entropy turn into a problem of finding extremal surfaces in a curved spacetime, which we tackle with a numerical finiteelement approach. In this paper, we compute the entanglement entropy between two half-spaces resulting from a local quench, triggered by a local operator insertion in a CFT3. We find that the growth of entanglement entropy at early time agrees with the prediction from the first law, as long as the conformal dimension Δ of the local operator is small. Within the limited time region that we can probe numerically, we observe deviations from the first law and a transition to sub-linear growth at later time. In particular, the time dependence at large Δ shows qualitative differences to the simple logarithmic time dependence familiar from the CFT2 case. We hope that our work will motivate further studies, both numerical and analytical, on entanglement entropy in higher dimensions

Real-Space Mesh Techniques in Density Functional Theory

This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.Comment: 70 pages, 11 figures. To be published in Reviews of Modern Physic