  The solution of eigenproblems is often a key computational bottleneck that
limits the tractable system size of numerical algorithms, among them electronic
structure theory in chemistry and in condensed matter physics. Large
eigenproblems can easily exceed the capacity of a single compute node, thus
must be solved on distributed-memory parallel computers. We here present
GPU-oriented optimizations of the ELPA two-stage tridiagonalization eigensolver
(ELPA2). On top of cuBLAS-based GPU offloading, we add a CUDA kernel to speed
up the back-transformation of eigenvectors, which can be the computationally
most expensive part of the two-stage tridiagonalization algorithm. We benchmark
the performance of this GPU-accelerated eigensolver on two hybrid CPU-GPU
architectures, namely a compute cluster based on Intel Xeon Gold CPUs and
NVIDIA Volta GPUs, and the Summit supercomputer based on IBM POWER9 CPUs and
NVIDIA Volta GPUs. Consistent with previous benchmarks on CPU-only
architectures, the GPU-accelerated two-stage solver exhibits a parallel
performance superior to the one-stage counterpart. Finally, we demonstrate the
performance of the GPU-accelerated eigensolver developed in this work for
routine semi-local KS-DFT calculations comprising thousands of atoms.
