Reordering strategy for blocking optimization in sparse linear solvers

Gregoire Pichon, Mathieu Faverge, Pierre Ramet, Jean Roman
HiePACS (High-End Parallel Algorithms for Challenging Numerical Simulations), LaBRI (Laboratoire Bordelais de Recherche en Informatique), Inria Bordeaux – Sud-Ouest
hal-01276746, (13 October 2016)

   title={Reordering strategy for blocking optimization in sparse linear solvers},

   author={Pichon, Gr{‘e}goire and Faverge, Mathieu and Ramet, Pierre and Roman, Jean},


   school={Inria Bordeaux Sud-Ouest; LaBRI-Laboratoire Bordelais de Recherche en Informatique; Bordeaux INP; Universit{‘e} de Bordeaux}


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Solving sparse linear systems is a problem that arises in many scientific applications, and sparse direct solvers are a time consuming and key kernel for those applications and for more advanced solvers such as hybrid direct-iterative solvers. For this reason, optimizing their performance on modern architectures is critical. The preprocessing steps of sparse direct solvers, ordering and block-symbolic factorization, are two major steps that lead to a reduced amount of computation and memory and to a better task granularity to reach a good level of performance when using BLAS kernels. With the advent of GPUs, the granularity of the block computation became more important than ever. In this paper, we present a reordering strategy that increases this block granularity. This strategy relies on the block-symbolic factorization to refine the ordering produced by tools such as Metis or Scotch, but it does not impact the number of operations required to solve the problem. We integrate this algorithm in the PaStiX solver and show an important reduction of the number of off-diagonal blocks on a large spectrum of matrices. This improvement leads to an increase in efficiency of up to 20% on GPUs.
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