Iterative methods for solving linear systems on massively parallel architectures

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 (UMR_7598) - 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 thesis, we first study the use of so-called Enlarged Krylov subspaces for reducing the number of iterations, and therefore the overall communication, of Krylov methods. We consider a reformulation of the Conjugate Gradient (CG) method using these enlarged Krylov subspaces: the Enlarged Conjugate Gradient (ECG) method. This method is first studied from a theoretical point of view. In particular, we show that its convergence speed is close to that of the so-called Deflated Conjugate Gradient method. In order to mitigate the effect of the extra arithmetic operations induced by the method, we explain how to dynamically reduce the number of search directions during the iterations. We then present the parallel design of two variants of the ECG method as well as their corresponding dynamic versions. Using a block Jacobi preconditioner, we show that our implementation scales up to several thousands of cores, and it can be significantly faster than the PETSc implementation of the CG method. We then focus on the Cosmic Microwave Background (CMB) analysis. We investigate the usage of so--called recycling strategies in this context. As a result of the multiplicity of the smallest eigenvalue, these techniques may not improve the convergence in some cases. Hence, we propose a cheap procedure to adapt the initial guess that permits to reduce the overall number of iterations in such situations.
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Olivier Tissot. Iterative methods for solving linear systems on massively parallel architectures. Numerical Analysis [math.NA]. Sorbonne Université, 2019. English. ⟨tel-02428348⟩

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