@article {1262,
	title = {Algorithms and Optimization Techniques for High-Performance Matrix-Matrix Multiplications of Very Small Matrices},
	journal = {Parallel Computing},
	volume = {81},
	year = {2019},
	month = {2019-01},
	pages = {1{\textendash}21},
	abstract = {Expressing scientific computations in terms of BLAS, and in particular the general dense matrix-matrix multiplication (GEMM), is of fundamental importance for obtaining high performance portability across architectures. However, GEMMs for small matrices of sizes smaller than 32 are not sufficiently optimized in existing libraries. We consider the computation of many small GEMMs and its performance portability for a wide range of computer architectures, including Intel CPUs, ARM, IBM, Intel Xeon Phi, and GPUs. These computations often occur in applications like big data analytics, machine learning, high-order finite element methods (FEM), and others. The GEMMs are grouped together in a single batched routine. For these cases, we present algorithms and their optimization techniques that are specialized for the matrix sizes and architectures of interest. We derive a performance model and show that the new developments can be tuned to obtain performance that is within 90\% of the optimal for any of the architectures of interest. For example, on a V100 GPU for square matrices of size 32, we achieve an execution rate of about 1600 gigaFLOP/s in double-precision arithmetic, which is 95\% of the theoretically derived peak for this computation on a V100 GPU. We also show that these results outperform currently available state-of-the-art implementations such as vendor-tuned math libraries, including Intel MKL and NVIDIA CUBLAS, as well as open-source libraries like OpenBLAS and Eigen.},
	keywords = {Autotuning, Batched GEMM, HPC, Matrix-matrix product, optimization, Small matrices},
	doi = {https://doi.org/10.1016/j.parco.2018.10.003},
	author = {Ian Masliah and Ahmad Abdelfattah and Azzam Haidar and Stanimire Tomov and Marc Baboulin and Jo{\"e}l Falcou and Jack Dongarra}
}
@techreport {1229,
	title = {Algorithms and Optimization Techniques for High-Performance Matrix-Matrix Multiplications of Very Small Matrices},
	journal = {Innovative Computing Laboratory Technical Report},
	number = {ICL-UT-18-09},
	year = {2018},
	month = {2018-09},
	publisher = {Innovative Computing Laboratory, University of Tennessee},
	abstract = {Expressing scientific computations in terms of BLAS, and in particular the general dense matrix-matrix multiplication (GEMM), is of fundamental importance for obtaining high performance portability across architectures. However, GEMMs for small matrices of sizes smaller than 32 are not sufficiently optimized in existing libraries. We consider the computation of many small GEMMs and its performance portability for a wide range of computer architectures, including Intel CPUs, ARM, IBM, Intel Xeon Phi, and GPUs. These computations often occur in applications like big data analytics, machine learning, high-order finite element methods (FEM), and others. The GEMMs are grouped together in a single batched routine. For these cases, we present algorithms and their optimization techniques that are specialized for the matrix sizes and architectures of interest. We derive a performance model and show that the new developments can be tuned to obtain performance that is within 90\% of the optimal for any of the architectures of interest. For example, on a V100 GPU for square matrices of size 32, we achieve an execution rate of about 1; 600 gigaFLOP/s in double-precision arithmetic, which is 95\% of the theoretically derived peak for this computation on a V100 GPU. We also show that these results outperform currently available state-of-the-art implementations such as vendor-tuned math libraries, including Intel MKL and NVIDIA CUBLAS, as well as open-source libraries like OpenBLAS and Eigen.},
	author = {Ian Masliah and Ahmad Abdelfattah and Azzam Haidar and Stanimire Tomov and Marc Baboulin and Jo{\"e}l Falcou and Jack Dongarra}
}
@article {1341,
	title = {Accelerating Tensor Contractions in High-Order FEM with MAGMA Batched},
	year = {2017},
	month = {2017-03},
	publisher = {SIAM Conference on Computer Science and Engineering (SIAM CSE17), Presentation},
	address = {Atlanta, GA},
	author = {Ahmad Abdelfattah and Marc Baboulin and Veselin Dobrev and Jack Dongarra and Christopher Earl and Jo{\"e}l Falcou and Azzam Haidar and Ian Karlin and Tzanio Kolev and Ian Masliah and Stanimire Tomov}
}
@conference {964,
	title = {High-performance Matrix-matrix Multiplications of Very Small Matrices},
	booktitle = {22nd International European Conference on Parallel and Distributed Computing (Euro-Par{\textquoteright}16)},
	year = {2016},
	month = {2016-08},
	publisher = {Springer International Publishing},
	organization = {Springer International Publishing},
	address = {Grenoble, France},
	abstract = {The use of the general dense matrix-matrix multiplication (GEMM) is fundamental for obtaining high performance in many scientific computing applications. GEMMs for small matrices (of sizes less than 32) however, are not sufficiently optimized in existing libraries. In this paper we consider the case of many small GEMMs on either CPU or GPU architectures. This is a case that often occurs in applications like big data analytics, machine learning, high-order FEM, and others. The GEMMs are grouped together in a single batched routine. We present specialized for these cases algorithms and optimization techniques to obtain performance that is within 90\% of the optimal. We show that these results outperform currently available state-of-the-art implementations and vendor-tuned math libraries.},
	author = {Ian Masliah and Ahmad Abdelfattah and Azzam Haidar and Stanimire Tomov and Jo{\"e}l Falcou and Jack Dongarra}
}
@techreport {929,
	title = {High-Performance Tensor Contractions for GPUs},
	journal = {University of Tennessee Computer Science Technical Report},
	number = {UT-EECS-16-738},
	year = {2016},
	month = {2016-01},
	publisher = {University of Tennessee},
	abstract = {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 2.8{\texttimes} faster than CUBLAS, and 8.5{\texttimes} 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.},
	author = {Ahmad Abdelfattah and Marc Baboulin and Veselin Dobrev and Jack Dongarra and Christopher Earl and Jo{\"e}l Falcou and Azzam Haidar and Ian Karlin and Tzanio Kolev and Ian Masliah and Stanimire Tomov}
}
@conference {942,
	title = {High-Performance Tensor Contractions for GPUs},
	booktitle = {International Conference on Computational Science (ICCS{\textquoteright}16)},
	year = {2016},
	month = {2016-06},
	address = {San Diego, CA},
	abstract = {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 2.8{\texttimes} faster than CUBLAS, and 8.5{\texttimes} faster than MKL on 16 cores of Intel Xeon E5-2670 (Sandy Bridge) 2.60GHz CPUs. Finally, we apply autotuning and code generation techniques to simplify tuning and provide an architecture-aware, user-friendly interface.},
	keywords = {Applications, Batched linear algebra, FEM, gpu, Tensor contractions, Tensor HPC},
	author = {Ahmad Abdelfattah and Marc Baboulin and Veselin Dobrev and Jack Dongarra and Christopher Earl and Jo{\"e}l Falcou and Azzam Haidar and Ian Karlin and Tzanio Kolev and Ian Masliah and Stanimire Tomov}
}
@article {1346,
	title = {Towards a High-Performance Tensor Algebra Package for Accelerators},
	year = {2015},
	month = {2015-09},
	publisher = {moky Mountains Computational Sciences and Engineering Conference (SMC15)},
	address = {Gatlinburg, TN},
	author = {Marc Baboulin and Veselin Dobrev and Jack Dongarra and Christopher Earl and Jo{\"e}l Falcou and Azzam Haidar and Ian Karlin and Tzanio Kolev and Ian Masliah and Stanimire Tomov}
}
@conference {763,
	title = {A Parallel Solver for Incompressible Fluid Flows},
	booktitle = {International Conference on Computational Science (ICCS 2013)},
	year = {2013},
	month = {2013-06},
	publisher = {Elsevier B.V.},
	organization = {Elsevier B.V.},
	address = {Barcelona, Spain},
	abstract = {The Navier-Stokes equations describe a large class of fluid flows but are difficult to solve analytically because of their nonlin- earity. We present in this paper a parallel solver for the 3-D Navier-Stokes equations of incompressible unsteady flows with constant coefficients, discretized by the finite difference method. We apply the prediction-projection method which transforms the Navier-Stokes equations into three Helmholtz equations and one Poisson equation. For each Helmholtz system, we apply the Alternating Direction Implicit (ADI) method resulting in three tridiagonal systems. The Poisson equation is solved using partial diagonalization which transforms the Laplacian operator into a tridiagonal one. We describe an implementation based on MPI where the computations are performed on each subdomain and information is exchanged on the interfaces, and where the tridiagonal system solutions are accelerated using vectorization techniques. We present performance results on a current multicore system.},
	keywords = {ADI, Navier-Stokes equations, Parallel computing, Partial diagonalization, Prediction-projection, SIMD},
	doi = {DOI: 10.1016/j.procs.2013.05.207},
	author = {Yushan Wang and Marc Baboulin and Jo{\"e}l Falcou and Yann Fraigneau and Olivier Le Ma{\^\i}tre}
}
