Hybrid Numerical Solvers for Massively Parallel Eigenvalue Computation and Their Benchmark with Electronic Structure Calculations

TL;DR

This paper develops and benchmarks hybrid numerical solvers for large-scale eigenvalue problems on supercomputers, demonstrating improved performance with newer libraries and analyzing bottlenecks relevant for exa-scale computing.

Contribution

It introduces a hybrid solver framework combining multiple libraries for massively parallel eigenvalue problems and provides benchmark results on supercomputers.

Findings

ELPA and EigenExa outperform ScaLAPACK in benchmarks.

The reducer can be a bottleneck in exa-scale supercomputers.

Benchmark results guide future research directions.

Abstract

Optimally hybrid numerical solvers were constructed for massively parallel generalized eigenvalue problem (GEP).The strong scaling benchmark was carried out on the K computer and other supercomputers for electronic structure calculation problems in the matrix sizes of M = 10^4-10^6 with upto 105 cores. The procedure of GEP is decomposed into the two subprocedures of the reducer to the standard eigenvalue problem (SEP) and the solver of SEP. A hybrid solver is constructed, when a routine is chosen for each subprocedure from the three parallel solver libraries of ScaLAPACK, ELPA and EigenExa. The hybrid solvers with the two newer libraries, ELPA and EigenExa, give better benchmark results than the conventional ScaLAPACK library. The detailed analysis on the results implies that the reducer can be a bottleneck in next-generation (exa-scale) supercomputers, which indicates the guidance for future research. The code was developed as a middleware and a mini-application and will appear online.