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The solution of (generalized) eigenvalue problems for symmetric or Hermitian matrices is a common subtask of many numerical calculations in electronic structure theory or materials science. Depending on the scientific problem, solving the eigenvalue problem can easily amount to a sizeable fraction of the whole numerical calculation, and quite often is even the dominant part by far. For researchers in the field of computational materials science, an efficient and scalable solution of the eigenvalue problem is thus of major importance. The ELPA-library (Eigenvalue SoLvers for Petaflop-Applications) is a well-established dense direct eigenvalue solver library, which has proven to be very efficient and scalable up to very large core counts. It is in a wide-spread production use on a large variety of HPC systems worldwide, and is applied by many codes in the field of materials science. In this paper, we describe the latest optimizations of the ELPAlibrary for new HPC architectures of the Intel Skylake processor family with an AVX-512 SIMD instruction set, or for HPC systems accelerated with recent GPUs. Apart from those direct hardware-related optimizations, we also describe a complete redesign of the API in a modern modular way, which, apart from a much simpler and more flexible usability, leads to a new path to access system-specific performance optimizations. In order to ensure optimal performance for a particular scientific setting or a specific HPC system, the new API allows the user to influence in straightforward way the internal details of the algorithms and of performance-critical parameters used in the ELPA-library. On top of that, we introduced an autotuning functionality, which allows for finding the best settings in a self-contained automated way, without the need of much user effort. In situations where many eigenvalue problems with similar settings have to be solved consecutively, the autotuning process of the ELPA-library can be done "on-the-fly", without the need of preceding the simulation with an "artificial" autotuning step. Practical applications from materials science which rely on reaching a numerical convergence limit by so-called self-consistency iterations can profit from the on-the-fly autotuning. On some examples of scientific interest, simulated with the FHI-aims [17] application, the advantages of the latest optimizations of the ELPA-library are demonstrated.

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We first briefly report on the status and recent achievements of the ELPA-AEO (Eigenvalue Solvers for Petaflop Applications -Algorithmic Extensions and Optimizations) and ESSEX II (Equipping Sparse Solvers for Exascale) projects. In both collaboratory efforts, scientists from the application areas, mathematicians, and computer scientists work together to develop and make available efficient highly parallel methods for the solution of eigenvalue problems. Then we focus on a topic addressed in both projects, the use of mixed precision computations to enhance efficiency. We give a more detailed description of our approaches for benefiting from either lower or higher precision in three selected contexts and of the results thus obtained.Keywords ELPA-AEO · ESSEX · eigensolver · parallel · mixed precision IntroductionEigenvalue computations are at the core of simulations in various application areas, including quantum physics and electronic structure computations. Being able to best utilize the capabilities of current and emerging high-end computing systems is essential for further improving such simulations with respect to space/time resolution or by including additional effects in the models. Given these needs, the ELPA-AEO and ESSEX-II projects contribute to the development and implementation of efficient highly parallel methods for eigenvalue problems, in different contexts.Both projects are aimed at adding new features (concerning, e.g., performance and resilience) to previously developed methods and at providing additional functionality with new methods. Building on the results of the first ESSEX funding phase [14,34], ESSEX-II again focuses on iterative methods for very large eigenproblems arising, e.g., in quantum physics. ELPA-AEO's main application area is electronic structure computation, and for these moderately sized eigenproblems direct methods are often superior. Such methods are available in the widely used ELPA library [19], which had originated in an earlier project [2] and is being improved further and extended with ELPA-AEO.In Sections 2 and 3 we briefly report on the current state and on recent achievement in the two projects, with a focus on aspects that may be of particular interest to prospective users of the software or the underlying methods.Mixed precision in the ELPA-AEO and ESSEX-II projects 3In Section 4 we turn to computations involving different precisions. Looking at three examples from the two projects we describe how lower or higher precision is used to reduce the computing time. The ELPA-AEO projectIn the ELPA-AEO project, chemists, mathematicians and computer scientists from the Max Planck Computing and Data Facility in Garching, the Fritz Haber Institute of the Max Planck Society in Berlin, the Technical University of Munich, and the University of Wuppertal collaborate to provide highly scalable methods for solving moderately-sized (n 10 6 ) Hermitian eigenvalue problems. Such problems arise, e.g., in electronic structure computations, and during the earlier ELPA project, efficient...