Algorithmic patterns for H\mathcal{H}-matrices on many-core processors

In this work, we consider the reformulation of hierarchical ($\mathcal{H}$) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). $\mathcal{H}$ matrices approximate specific dense matrices, e.g., from discretized integral equations or kernel ridge regression, leading to log-linear time complexity in dense matrix-vector products. The parallelization of $\mathcal{H}$ matrix operations on many-core processors is difficult due to the complex nature of the underlying algorithms. While previous algorithmic advances for many-core hardware focused on accelerating existing $\mathcal{H}$ matrix CPU implementations by many-core processors, we here aim at totally relying on that processor type. As main contribution, we introduce the necessary parallel algorithmic patterns allowing to map the full $\mathcal{H}$ matrix construction and the fast matrix-vector product to many-core hardware. Here, crucial ingredients are space filling curves, parallel tree traversal and batching of linear algebra operations. The resulting model GPU implementation hmglib is the, to the best of the authors knowledge, first entirely GPU-based Open Source $\mathcal{H}$ matrix library of this kind. We conclude this work by an in-depth performance analysis and a comparative performance study against a standard $\mathcal{H}$ matrix library, highlighting profound speedups of our many-core parallel approach

A Linear Algebra Approach to Fast DNA Mixture Analysis Using GPUs

Analysis of DNA samples is an important step in forensics, and the speed of analysis can impact investigations. Comparison of DNA sequences is based on the analysis of short tandem repeats (STRs), which are short DNA sequences of 2-5 base pairs. Current forensics approaches use 20 STR loci for analysis. The use of single nucleotide polymorphisms (SNPs) has utility for analysis of complex DNA mixtures. The use of tens of thousands of SNPs loci for analysis poses significant computational challenges because the forensic analysis scales by the product of the loci count and number of DNA samples to be analyzed. In this paper, we discuss the implementation of a DNA sequence comparison algorithm by re-casting the algorithm in terms of linear algebra primitives. By developing an overloaded matrix multiplication approach to DNA comparisons, we can leverage advances in GPU hardware and algoithms for Dense Generalized Matrix-Multiply (DGEMM) to speed up DNA sample comparisons. We show that it is possible to compare 2048 unknown DNA samples with 20 million known samples in under 6 seconds using a NVIDIA K80 GPU.Comment: Accepted for publication at the 2017 IEEE High Performance Extreme Computing conferenc

NVIDIA Tensor Core Programmability, Performance & Precision

The NVIDIA Volta GPU microarchitecture introduces a specialized unit, called "Tensor Core" that performs one matrix-multiply-and-accumulate on 4x4 matrices per clock cycle. The NVIDIA Tesla V100 accelerator, featuring the Volta microarchitecture, provides 640 Tensor Cores with a theoretical peak performance of 125 Tflops/s in mixed precision. In this paper, we investigate current approaches to program NVIDIA Tensor Cores, their performances and the precision loss due to computation in mixed precision. Currently, NVIDIA provides three different ways of programming matrix-multiply-and-accumulate on Tensor Cores: the CUDA Warp Matrix Multiply Accumulate (WMMA) API, CUTLASS, a templated library based on WMMA, and cuBLAS GEMM. After experimenting with different approaches, we found that NVIDIA Tensor Cores can deliver up to 83 Tflops/s in mixed precision on a Tesla V100 GPU, seven and three times the performance in single and half precision respectively. A WMMA implementation of batched GEMM reaches a performance of 4 Tflops/s. While precision loss due to matrix multiplication with half precision input might be critical in many HPC applications, it can be considerably reduced at the cost of increased computation. Our results indicate that HPC applications using matrix multiplications can strongly benefit from using of NVIDIA Tensor Cores.Comment: This paper has been accepted by the Eighth International Workshop on Accelerators and Hybrid Exascale Systems (AsHES) 201

Computational Physics on Graphics Processing Units

The use of graphics processing units for scientific computations is an emerging strategy that can significantly speed up various different algorithms. In this review, we discuss advances made in the field of computational physics, focusing on classical molecular dynamics, and on quantum simulations for electronic structure calculations using the density functional theory, wave function techniques, and quantum field theory.Comment: Proceedings of the 11th International Conference, PARA 2012, Helsinki, Finland, June 10-13, 201

A scalable H-matrix approach for the solution of boundary integral equations on multi-GPU clusters

In this work, we consider the solution of boundary integral equations by means of a scalable hierarchical matrix approach on clusters equipped with graphics hardware, i.e. graphics processing units (GPUs). To this end, we extend our existing single-GPU hierarchical matrix library hmglib such that it is able to scale on many GPUs and such that it can be coupled to arbitrary application codes. Using a model GPU implementation of a boundary element method (BEM) solver, we are able to achieve more than 67 percent relative parallel speed-up going from 128 to 1024 GPUs for a model geometry test case with 1.5 million unknowns and a real-world geometry test case with almost 1.2 million unknowns. On 1024 GPUs of the cluster Titan, it takes less than 6 minutes to solve the 1.5 million unknowns problem, with 5.7 minutes for the setup phase and 20 seconds for the iterative solver. To the best of the authors' knowledge, we here discuss the first fully GPU-based distributed-memory parallel hierarchical matrix Open Source library using the traditional H-matrix format and adaptive cross approximation with an application to BEM problems