Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects

Abstract

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...

Leveraging the bfloat16 Artificial Intelligence Datatype For Higher-Precision Computations

“…People have been proposing using mixed precision to refine other problems like eigenvalue problems for years such as in [13]. More recently, there has been success with FP32 Eigenvalue solvers which are compute intensive and are the bottleneck in quantum chemistry problems [14]. These applications consume a huge fraction of large supercomputers.…”

ESSEX: Equipping Sparse Solvers For Exascale

“…7333 s 265 s mixed (9) 316 s 257 s mixed 11 Therefore, mixed precision has been implemented in all BEAST schemes mentioned above, allowing an adaptive strategy to automatically switch from single to double precision after a given residual tolerance is reached. A comprehensive description and results are presented in [4]. These results and our initial investigations also suggest that increased precision beyond double precision (i.e., quad precision) will have no benefit for the convergence rate until a certain double precision specific threshold is reached; convergence beyond this point would require all operations to be carried out with increased precision.…”

“…Scalabilty, performance, and portability have been tested on three top-10 supercomputers covering the full range of architecures available during the ESSEX project time frame: Piz Daint 2 (heterogeneous CPU-GPU), OakForest-PACS 3 (many-core), and SuperMUC-NG 4 (standard multi-core).…”

“…https://www.cscs.ch/computers/piz-daint/ 3. https://www.cc.u-tokyo.ac.jp/en/supercomputer/ofp/service/ 4. https://doku.lrz.de/display/PUBLIC/SuperMUC-NG.…”

An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations

“…Third, the emergence of machine learning has enhanced the design of computer architecture for the acceleration of low-precision (single-or half-precision) calculation. The efficient use of low-precision calculation, typically in mixed-precision calculation, will be important in any high-performance computational science field [8,9]. A posteriori verification methods guarantee satisfactory numerical reliability when low-precision calculation is used.…”