Scalable Linear Solvers based on Enlarged Krylov subspaces with Dynamic Reduction of Search Directions

Laura Grigori 1 Olivier Tissot 1
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
INSMI - Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris, LJLL - Laboratoire Jacques-Louis Lions
Abstract : Krylov methods are widely used for solving large sparse linear systems of equations. On distributed architectures, their performance is limited by the communication needed at each iteration of the algorithm. In this paper, we study the use of so-called enlarged Krylov subspaces for reducing the number of iterations, and therefore the overall communication, of Krylov methods. In particular, we consider a reformulation of the Conjugate Gradient method using these enlarged Krylov subspaces: the enlarged Conjugate Gradient method. We present the parallel design of two variants of the enlarged Conjugate Gradient method as well as their corresponding dynamic versions where the number of search directions is dynamically reduced during the iterations. For a linear elasticity problem with heterogeneous coecients, using a block Jacobi preconditioner, we show that this implementation scales up to 16; 384 cores, and is up to 5.7 times faster than the PETSc implementation of PCG.
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Laura Grigori, Olivier Tissot. Scalable Linear Solvers based on Enlarged Krylov subspaces with Dynamic Reduction of Search Directions. [Research Report] RR-9190, Inria Paris; Laboratoire Jacques-Louis Lions, UPMC, Paris. 2018, pp.1-30. ⟨hal-01828521⟩

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