## High-Performance Tensor Contractions for GPUs

Innovative Computing Laboratory, University of Tennessee, Knoxville, TN, USA

University of Tennessee Computer Science Technical Report, no. UT-EECS-16-738, 2016

@article{abdelfattah2016high,

title={High-Performance Tensor Contractions for GPUs},

author={Abdelfattah, A and Baboulin, M and Dobrev, V and Dongarra, J and Earl, C and Falcou, J and Haidar, A and Karlin, I and Kolev, Tz and Masliah, I and others},

year={2016}

}

We present a computational framework for high-performance tensor contractions on GPUs. High-performance is difficult to obtain using existing libraries, especially for many independent contractions where each contraction is very small, e.g., sub-vector/warp in size. However, using our framework to batch contractions plus application-specifics, we demonstrate close to peak performance results. In particular, to accelerate large scale tensor-formulated high-order finite element method (FEM) simulations, which is the main focus and motivation for this work, we represent contractions as tensor index reordering plus matrix-matrix multiplications (GEMMs). This is a key factor to achieve algorithmically many-fold acceleration (vs. not using it) due to possible reuse of data loaded in fast memory. In addition to using this context knowledge, we design tensor data-structures, tensor algebra interfaces, and new tensor contraction algorithms and implementations to achieve 90+% of a theoretically derived peak on GPUs. On a K40c GPU for contractions resulting in GEMMs on square matrices of size 8 for example, we are 2.8x faster than CUBLAS, and 8.5x faster than MKL on 16 cores of Intel Xeon ES-2670 (Sandy Bridge) 2.60GHz CPUs. Finally, we apply autotuning and code generation techniques to simplify tuning and provide an architecture-aware, user-friendly interface.

January 29, 2016 by hgpu