@conference {1168,
	title = {The Design and Performance of Batched BLAS on Modern High-Performance Computing Systems},
	booktitle = {International Conference on Computational Science (ICCS 2017)},
	year = {2017},
	month = {2017-06},
	publisher = {Elsevier},
	organization = {Elsevier},
	address = {Z{\"u}rich, Switzerland},
	abstract = {A current trend in high-performance computing is to decompose a large linear algebra problem into batches containing thousands of smaller problems, that can be solved independently, before collating the results. To standardize the interface to these routines, the community is developing an extension to the BLAS standard (the batched BLAS), enabling users to perform thousands of small BLAS operations in parallel whilst making efficient use of their hardware. We discuss the benefits and drawbacks of the current batched BLAS proposals and perform a number of experiments, focusing on a general matrix-matrix multiplication (GEMM), to explore their affect on the performance. In particular we analyze the effect of novel data layouts which, for example, interleave the matrices in memory to aid vectorization and prefetching of data. Utilizing these modifications our code outperforms both MKL1 CuBLAS2 by up to 6 times on the self-hosted Intel KNL (codenamed Knights Landing) and Kepler GPU architectures, for large numbers of double precision GEMM operations using matrices of size 2 {\texttimes} 2 to 20 {\texttimes} 20.},
	keywords = {Batched BLAS, BLAS, High-performance computing, Memory management, Parallel processing, Scientific computing},
	doi = {DOI:10.1016/j.procs.2017.05.138},
	author = {Jack Dongarra and Sven Hammarling and Nicholas J. Higham and Samuel Relton and Pedro Valero-Lara and Mawussi Zounon}
}
@conference {1170,
	title = {Optimized Batched Linear Algebra for Modern Architectures},
	booktitle = {Euro-Par 2017},
	year = {2017},
	month = {2017-08},
	publisher = {Springer},
	organization = {Springer},
	address = {Santiago de Compostela, Spain},
	abstract = {Solving large numbers of small linear algebra problems simultaneously is becoming increasingly important in many application areas. Whilst many researchers have investigated the design of efficient batch linear algebra kernels for GPU architectures, the common approach for many/multi-core CPUs is to use one core per subproblem in the batch. When solving batches of very small matrices, 2 {\texttimes} 2 for example, this design exhibits two main issues: it fails to fully utilize the vector units and the cache of modern architectures, since the matrices are too small. Our approach to resolve this is as follows: given a batch of small matrices spread throughout the primary memory, we first reorganize the elements of the matrices into a contiguous array, using a block interleaved memory format, which allows us to process the small independent problems as a single large matrix problem and enables cross-matrix vectorization. The large problem is solved using blocking strategies that attempt to optimize the use of the cache. The solution is then converted back to the original storage format. To explain our approach we focus on two BLAS routines: general matrix-matrix multiplication (GEMM) and the triangular solve (TRSM). We extend this idea to LAPACK routines using the Cholesky factorization and solve (POSV). Our focus is primarily on very small matrices ranging in size from 2 {\texttimes} 2 to 32 {\texttimes} 32. Compared to both MKL and OpenMP implementations, our approach can be up to 4 times faster for GEMM, up to 14 times faster for TRSM, and up to 40 times faster for POSV on the new Intel Xeon Phi processor, code-named Knights Landing (KNL). Furthermore, we discuss strategies to avoid data movement between sockets when using our interleaved approach on a NUMA node.},
	doi = {https://doi.org/10.1007/978-3-319-64203-1_37},
	author = {Jack Dongarra and Sven Hammarling and Nicholas J. Higham and Samuel Relton and Mawussi Zounon}
}
@techreport {1173,
	title = {PLASMA 17 Performance Report},
	journal = {Innovative Computing Laboratory Technical Report},
	number = {ICL-UT-17-11},
	year = {2017},
	month = {2017-06},
	publisher = {University of Tennessee},
	abstract = {PLASMA (Parallel Linear Algebra for Multicore Architectures) is a dense linear algebra package at the forefront of multicore computing. PLASMA is designed to deliver the highest possible performance from a system with multiple sockets of multicore processors. PLASMA achieves this objective by combining state of the art solutions in parallel algorithms, scheduling, and software engineering. PLASMA currently offers a collection of routines for solving linear systems of equations and least square problems.},
	author = {Maksims Abalenkovs and Negin Bagherpour and Jack Dongarra and Mark Gates and Azzam Haidar and Jakub Kurzak and Piotr Luszczek and Samuel Relton and Jakub Sistek and David Stevens and Panruo Wu and Ichitaro Yamazaki and Asim YarKhan and Mawussi Zounon}
}
@techreport {1172,
	title = {PLASMA 17.1 Functionality Report},
	journal = {Innovative Computing Laboratory Technical Report},
	number = {ICL-UT-17-10},
	year = {2017},
	month = {2017-06},
	publisher = {University of Tennessee},
	abstract = {PLASMA (Parallel Linear Algebra for Multicore Architectures) is a dense linear algebra package at the forefront of multicore computing. PLASMA is designed to deliver the highest possible performance from a system with multiple sockets of multicore processors. PLASMA achieves this objective by combining state of the art solutions in parallel algorithms, scheduling, and software engineering. PLASMA currently offers a collection of routines for solving linear systems of equations and least square problems.},
	author = {Maksims Abalenkovs and Negin Bagherpour and Jack Dongarra and Mark Gates and Azzam Haidar and Jakub Kurzak and Piotr Luszczek and Samuel Relton and Jakub Sistek and David Stevens and Panruo Wu and Ichitaro Yamazaki and Asim YarKhan and Mawussi Zounon}
}
