https://inria.hal.science/hal-02947991v1Beaupère, MatthiasMatthiasBeaupèreALPINES - Algorithms and parallel tools for integrated numerical simulations - INSMI - Institut National des Sciences Mathématiques et de leurs Interactions - Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique - LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris CitéGrigori, LauraLauraGrigoriALPINES - Algorithms and parallel tools for integrated numerical simulations - INSMI - Institut National des Sciences Mathématiques et de leurs Interactions - Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique - LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris CitéCommunication avoiding low rank approximation based on QR with tournament pivotingHAL CCSD2020QR decompositionCommunication avoiding algorithmSubset selectionLow rank approximation[INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation[INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA]Beaupère, MatthiasExtreme-Scale Mathematically-based Computational Chemistry - EMC2 - 2019-09-01 - 2026-02-26 - 810367 - VALID - 2020-09-24 13:15:372023-03-24 14:53:182020-09-24 13:39:18enPreprints, Working Papers, ...https://inria.hal.science/hal-02947991v1application/pdf1We introduce a parallel algorithm for computing the low rank approximation $A_k$ of a large matrix $A$ which minimizes the number of messages exchanged between processors (modulo polylogarithmic factors) and has guarantees for the approximations of the singular values of $A$ provided by $A_k$. This operation is essential in many applications in scientific computing and data analysis when dealing with large data sets. Our algorithm is based on QR factorization that consists in selecting a subset of columns from the matrix $A$ that allow to approximate the range of $A$, and then projecting the columns of $A$ on a basis of the subspace spanned by those columns. The selection of columns is performed by using tournament pivoting, a strategy introduced previously for matrices partitioned into blocks of columns. This strategy is extended here to matrices partitioned along both dimensions that are distributed on a two-dimensional grid of processors, and also to tournaments with more general reduction trees. Performance results show that the algorithm scales well on up to $1024$ cores of $16$ nodes.