High-Performance Solvers for Dense Hermitian Eigenproblems

We introduce a new collection of solvers - subsequently called EleMRRR - for large-scale dense Hermitian eigenproblems. EleMRRR solves various types of problems: generalized, standard, and tridiagonal eigenproblems. Among these, the last is of particular importance as it is a solver on its own right, as well as the computational kernel for the first two; we present a fast and scalable tridiagonal solver based on the Algorithm of Multiple Relatively Robust Representations - referred to as PMRRR. Like the other EleMRRR solvers, PMRRR is part of the freely available Elemental library, and is designed to fully support both message-passing (MPI) and multithreading parallelism (SMP). As a result, the solvers can equally be used in pure MPI or in hybrid MPI-SMP fashion. We conducted a thorough performance study of EleMRRR and ScaLAPACK's solvers on two supercomputers. Such a study, performed with up to 8,192 cores, provides precise guidelines to assemble the fastest solver within the ScaLAPACK framework; it also indicates that EleMRRR outperforms even the fastest solvers built from ScaLAPACK's components

Parallel eigensolvers in plane-wave Density Functional Theory

We consider the problem of parallelizing electronic structure computations in plane-wave Density Functional Theory. Because of the limited scalability of Fourier transforms, parallelism has to be found at the eigensolver level. We show how a recently proposed algorithm based on Chebyshev polynomials can scale into the tens of thousands of processors, outperforming block conjugate gradient algorithms for large computations

Studies in Rheology: Molecular Simulation and Theory

With an enormous advance in the capability of computers during the last fewdecades, the computer simulation has become an important tool for scientific researches in many areas such as physics, chemistry, biology, and so on. In particular, moleculardynamics (MD) simulations have been proven to be of a great help in understanding the rheology of complex fluids from the fundamental microscopic viewpoint. There are two important standard flows in rheology: shear flow and elongational flow. While there exist suitable nonequilibrium MD (NEMD) algorithms of shear flows, such as the Lees-Edwards purely boundary-driven algorithm and the so-called SLLOD algorithm as a field-driven algorithm, a proper NEMD algorithm for elongational flow has been lacking. The main difficulty of simulating elongational flow lies in the limited simulation time available due to the contraction of one or two dimensions dictated by itskinematics. This problem, however, has been partially resolved by Kraynik and Reinelt’s ingenious discovery of the temporal and spatial periodicity of lattice vectors in planar elongational flow (PEF). Although there have been a few NEMD simulations of PEF using their idea, another serious defect has recently been reported when using the SLLOD algorithm in PEF: for adiabatic systems, the total linear momentum of the system in the contracting direction grows exponentially with time, which eventually leads to an aphysical phase transition.This problem has been completely resolved by using the so-called ‘proper-SLLOD’ or ‘p-SLLOD’ algorithm, whose development has been one of the mainaccomplishments of this study. The fundamental correctness of the p-SLLOD algorithm has been demonstrated quite thoroughly in this work through detailed theoretical analyses together with direct simulation results. Both theoretical and simulation works achieved in this research are expected to play a significant role in advancing the knowledge of rheology, as well as that of NEMD simulation itself for other types of flow in general. Another important achievement in this work is the demonstration of the possibility of predicting a liquid structure in nonequilibrium states by employing a concept of ‘hypothetical’ nonequilibrium potentials. The methodology developed in this work has been shown to have good potential for further developments in this field

Evaluation of Directive-Based GPU Programming Models on a Block Eigensolver with Consideration of Large Sparse Matrices

Achieving high performance and performance portability for large-scale scientific applications is a major challenge on heterogeneous computing systems such as many-core CPUs and accelerators like GPUs. In this work, we implement a widely used block eigensolver, Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG), using two popular directive based programming models (OpenMP and OpenACC) for GPU-accelerated systems. Our work differs from existing work in that it adopts a holistic approach that optimizes the full solver performance rather than narrowing the problem into small kernels (e.g., SpMM, SpMV). Our LOPBCG GPU implementation achieves a 2.8$${\times }$$–4.3$${\times }$$ speedup over an optimized CPU implementation when tested with four different input matrices. The evaluated configuration compared one Skylake CPU to one Skylake CPU and one NVIDIA V100 GPU. Our OpenMP and OpenACC LOBPCG GPU implementations gave nearly identical performance. We also consider how to create an efficient LOBPCG solver that can solve problems larger than GPU memory capacity. To this end, we create microbenchmarks representing the two dominant kernels (inner product and SpMM kernel) in LOBPCG and then evaluate performance when using two different programming approaches: tiling the kernels, and using Unified Memory with the original kernels. Our tiled SpMM implementation achieves a 2.9$${\times }$$ and 48.2$${\times }$$ speedup over the Unified Memory implementation on supercomputers with PCIe Gen3 and NVLink 2.0 CPU to GPU interconnects, respectively