@inbook {,
	title = {Batch QR Factorization on GPUs: Design, Optimization, and Tuning},
	booktitle = { Lecture Notes in Computer Science},
	volume = {13350},
	year = {2022},
	month = {2022-06},
	publisher = {Springer International Publishing},
	organization = {Springer International Publishing},
	address = {Cham},
	abstract = {QR factorization of dense matrices is a ubiquitous tool in high performance computing (HPC). From solving linear systems and least squares problems to eigenvalue problems, and singular value decompositions, the impact of a high performance QR factorization is fundamental to computer simulations and many applications. More importantly, the QR factorization on a batch of relatively small matrices has acquired a lot of attention in sparse direct solvers and low-rank approximations for Hierarchical matrices. To address this interest and demand, we developed and present a high performance batch QR factorization for Graphics Processing Units (GPUs). We present a multi-level blocking strategy that adjusts various algorithmic designs to the size of the input matrices. We also show that following the LAPACK QR design convention, while still useful, is significantly outperformed by unconventional code structures that increase data reuse. The performance results show multi-fold speedups against the state of the art libraries on the latest GPU architectures from both NVIDIA and AMD.},
	keywords = {Batch linear algebra, GPU computing, QR factorization},
	isbn = {978-3-031-08750-9},
	doi = {10.1007/978-3-031-08751-6_5},
	url = {https://link.springer.com/chapter/10.1007/978-3-031-08751-6_5},
	author = {Abdelfattah, Ahmad and Stanimire Tomov and Dongarra, Jack},
	editor = {Groen, Derek and de Mulatier, C{\'e}lia and Paszy{\'n}ski, Maciej and Krzhizhanovskaya, Valeria V. and Dongarra, Jack J. and Sloot, Peter M. A.}
}
@article {924,
	title = {Assessing the Cost of Redistribution followed by a Computational Kernel: Complexity and Performance Results},
	journal = {Parallel Computing},
	volume = {52},
	year = {2016},
	month = {2016-02},
	pages = {22-41},
	abstract = {The classical redistribution problem aims at optimally scheduling communications when reshuffling from an initial data distribution to a target data distribution. This target data distribution is usually chosen to optimize some objective for the algorithmic kernel under study (good computational balance or low communication volume or cost), and therefore to provide high efficiency for that kernel. However, the choice of a distribution minimizing the target objective is not unique. This leads to generalizing the redistribution problem as follows: find a re-mapping of data items onto processors such that the data redistribution cost is minimal, and the operation remains as efficient. This paper studies the complexity of this generalized problem. We compute optimal solutions and evaluate, through simulations, their gain over classical redistribution. We also show the NP-hardness of the problem to find the optimal data partition and processor permutation (defined by new subsets) that minimize the cost of redistribution followed by a simple computational kernel. Finally, experimental validation of the new redistribution algorithms are conducted on a multicore cluster, for both a 1D-stencil kernel and a more compute-intensive dense linear algebra routine.},
	keywords = {Data partition, linear algebra, parsec, QR factorization, Redistribution, Stencil},
	doi = {doi:10.1016/j.parco.2015.09.005},
	author = {Julien Herrmann and George Bosilca and Thomas Herault and Loris Marchal and Yves Robert and Jack Dongarra}
}
@article {917,
	title = {Mixing LU-QR Factorization Algorithms to Design High-Performance Dense Linear Algebra Solvers},
	journal = {Journal of Parallel and Distributed Computing},
	volume = {85},
	year = {2015},
	month = {2015-11},
	pages = {32-46},
	abstract = {This paper introduces hybrid LU{\textendash}QR algorithms for solving dense linear systems of the form Ax=b. Throughout a matrix factorization, these algorithms dynamically alternate LU with local pivoting and QR elimination steps based upon some robustness criterion. LU elimination steps can be very efficiently parallelized, and are twice as cheap in terms of floating-point operations, as QR steps. However, LU steps are not necessarily stable, while QR steps are always stable. The hybrid algorithms execute a QR step when a robustness criterion detects some risk for instability, and they execute an LU step otherwise. The choice between LU and QR steps must have a small computational overhead and must provide a satisfactory level of stability with as few QR steps as possible. In this paper, we introduce several robustness criteria and we establish upper bounds on the growth factor of the norm of the updated matrix incurred by each of these criteria. In addition, we describe the implementation of the hybrid algorithms through an extension of the PaRSEC software to allow for dynamic choices during execution. Finally, we analyze both stability and performance results compared to state-of-the-art linear solvers on parallel distributed multicore platforms. A comprehensive set of experiments shows that hybrid LU{\textendash}QR algorithms provide a continuous range of trade-offs between stability and performances.},
	keywords = {lu factorization, Numerical algorithms, QR factorization, Stability; Performance},
	doi = {doi:10.1016/j.jpdc.2015.06.007},
	author = {Mathieu Faverge and Julien Herrmann and Julien Langou and Bradley Lowery and Yves Robert and Jack Dongarra}
}
@article {752,
	title = {Hierarchical QR Factorization Algorithms for Multi-core Cluster Systems},
	journal = {Parallel Computing},
	volume = {39},
	year = {2013},
	month = {2013-05},
	pages = {212-232},
	abstract = {This paper describes a new QR factorization algorithm which is especially designed for massively parallel platforms combining parallel distributed nodes, where a node is a multi-core processor. These platforms represent the present and the foreseeable future of high-performance computing. Our new QR factorization algorithm falls in the category of the tile algorithms which naturally enables good data locality for the sequential kernels executed by the cores (high sequential performance), low number of messages in a parallel distributed setting (small latency term), and fine granularity (high parallelism). Each tile algorithm is uniquely characterized by its sequence of reduction trees. In the context of a cluster of nodes, in order to minimize the number of inter-processor communications (aka, {\textquoteleft}{\textquoteleft}communication-avoiding{\textquoteright}{\textquoteright}), it is natural to consider hierarchical trees composed of an {\textquoteleft}{\textquoteleft}inter-node{\textquoteright}{\textquoteright} tree which acts on top of {\textquoteleft}{\textquoteleft}intra-node{\textquoteright}{\textquoteright} trees. At the intra-node level, we propose a hierarchical tree made of three levels: (0) {\textquoteleft}{\textquoteleft}TS level{\textquoteright}{\textquoteright} for cache-friendliness, (1) {\textquoteleft}{\textquoteleft}low-level{\textquoteright}{\textquoteright} for decoupled highly parallel inter-node reductions, (2) {\textquoteleft}{\textquoteleft}domino level{\textquoteright}{\textquoteright} to efficiently resolve interactions between local reductions and global reductions. Our hierarchical algorithm and its implementation are flexible and modular, and can accommodate several kernel types, different distribution layouts, and a variety of reduction trees at all levels, both inter-node and intra-node. Numerical experiments on a cluster of multi-core nodes (i) confirm that each of the four levels of our hierarchical tree contributes to build up performance and (ii) build insights on how these levels influence performance and interact within each other. Our implementation of the new algorithm with the DAGUE scheduling tool significantly outperforms currently available QR factorization software for all matrix shapes, thereby bringing a new advance in numerical linear algebra for petascale and exascale platforms.},
	keywords = {Cluster, Distributed memory, Hierarchical architecture, multi-core, numerical linear algebra, QR factorization},
	author = {Jack Dongarra and Mathieu Faverge and Thomas Herault and Mathias Jacquelin and Julien Langou and Yves Robert}
}
