Architecture-Aware Algorithms for Scalable Performance and Resilience on Heterogeneous Architectures

The goal of the Extreme-scale Algorithms & Software Institute (EASI) is to close the �application-architecture performance gap� by exploring algorithms and runtime improvements that will enable key science applications to better exploit the architectural features of DOE extreme-scale systems. For the past year of the project, our efforts at the University of Tennessee have concentrated on, and made significant progress related to, the following high-level EASI goals: � Develop multi-precision and architecture-aware implementations of Krylov, Poisson, Helmholtz solvers, and dense factorizations for heterogeneous multi-core systems; � Explore new methods of algorithm resilience, and develop new algorithms with these capabilities; � Develop runtime support for adaptable algorithms that are dealing with resilience, scalability; � Distribute the new algorithms and runtime support through widely used software packages; � Establish a strong outreach program to disseminate results, interact with colleagues and train students and junior members of our community

A Novel Hybrid CPU-GPU Generalized Eigensolver for Electronic Structure Calculations Based on Fine Grained Memory Aware Tasks

The adoption of hybrid CPU–GPU nodes in traditional supercomputing platforms such as the Cray-XK6 opens acceleration opportunities for electronic structure calculations in materials science and chemistry applications, where medium-sized generalized eigenvalue problems must be solved many times. These eigenvalue problems are too small to effectively solve on distributed systems, but can benefit from the massive computing power concentrated on a single-node, hybrid CPU–GPU system. However, hybrid systems call for the development of new algorithms that efficiently exploit heterogeneity and massive parallelism of not just GPUs, but of multicore/manycore CPUs as well. Addressing these demands, we developed a generalized eigensolver featuring novel algorithms of increased computational intensity (compared with the standard algorithms), decomposition of the computation into fine-grained memory aware tasks, and their hybrid execution. The resulting eigensolvers are state-of-the-art in high-performance computing, significantly outperforming existing libraries. We describe the algorithm and analyze its performance impact on applications of interest when different fractions of eigenvectors are needed by the host electronic structure code. </jats:p

Mixed-Precision Numerical Linear Algebra Algorithms: Integer Arithmetic Based LU Factorization and Iterative Refinement for Hermitian Eigenvalue Problem

Mixed-precision algorithms are a class of algorithms that uses low precision in part of the algorithm in order to save time and energy with less accurate computation and communication. These algorithms usually utilize iterative refinement processes to improve the approximate solution obtained from low precision to the accuracy we desire from doing all the computation in high precision. Due to the demand of deep learning applications, there are hardware developments offering different low-precision formats including half precision (FP16), Bfloat16 and integer operations for quantized integers, which uses integers with a shared scalar to represent a set of equally spaced numbers. As new hardware architectures focus on bringing performance in these formats, the mixed-precision algorithms have more potential leverage on them and outmatch traditional fixed-precision algorithms. This dissertation consists of two articles. In the first article, we adapt one of the most fundamental algorithms in numerical linear algebra---LU factorization with partial pivoting--- to use integer arithmetic. With the goal of obtaining a low accuracy factorization as the preconditioner of generalized minimal residual (GMRES) to solve systems of linear equations, the LU factorization is adapted to use two different fixed-point formats for matrices L and U. A left-looking variant is also proposed for matrices with unbounded column growth. Finally, GMRES iterative refinement has shown that it can work on matrices with condition numbers up to 10000 with the algorithm that uses int16 as input and int32 accumulator for the update step. The second article targets symmetric and Hermitian eigenvalue problems. In this section we revisit the SICE algorithm from Dongarra et al. By applying the Sherman-Morrison formula on the diagonally-shifted tridiagonal systems, we propose an updated SICE-SM algorithm. By incorporating the latest two-stage algorithms from the PLASMA and MAGMA software libraries for numerical linear algebra, we achieved up to 3.6x speedup using the mixed-precision eigensolver with the blocked SICE-SM algorithm for iterative refinement when compared with full double complex precision solvers for the cases with a portion of eigenvalues and eigenvectors requested