Accelerating linear system solutions using randomization techniques

Marc Baboulin, Jack Dongarra, Julien Herrmann, Stanimire Tomov
Laboratoire de Recherche en Informatique – Universite Paris-Sud 11 (LRI), Universite Paris XI – Paris Sud
inria-00593306, 2011




   title={Accelerating linear system solutions using randomization techniques},

   author={Baboulin, Marc and Dongarra, Jack and Herrmann, Julien and Tomov, Stanimire},

   keywords={dense linear algebra; linear systems; LU factorization; randomization; multiplicative preconditioning; Graphics Processing Units},


   affiliation={Laboratoire de Recherche en Informatique – Universit{‘e} Paris-Sud 11 – LRI , Innovative Computing Laboratory – ICL},

   type={Rapport de recherche},







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We illustrate how linear algebra calculations can be enhanced by statistical techniques in the case of a square linear system Ax = b. We study a random transformation of A that enables us to avoid pivoting and then to reduce the amount of communication. Numerical experiments show that this randomization can be performed at a very affordable computational price while providing us with a satisfying accuracy when compared to partial pivoting. This random transformation called Partial Random Butter y Transformation (PRBT) is optimized in terms of data storage and flops count. We propose a solver where PRBT and the LU factorization with no pivoting take advantage of the latest generation of hybrid multicore/GPU machines and we compare its Gfl op/s performance with a solver implemented in a current parallel library.
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