Two New Block Krylov Methods for Linear Systems with Multiple Right-hand Sides

Abstract : Solving a sequence of large linear systems with several right-hand sides given simultaneously or in sequence, is at the core of many problems in the computational sciences, such as in radar cross section calculation in electromagnetism, wave scattering and wave propagation in acoustics, uncertainty quantification, quantum chromodynamics, data handling, time-dependent problems, and various source locations in seismic and parametric studies in general. In that framework, block Krylov approaches appear as good candidates for the solution as the Krylov subspaces associated with each right-hand side are shared to enlarge the search space. They are attractive not only because of this numerical feature (larger search subspace), but also from a computational view point as they enable higher data reusability consequently locality (BLAS3-like implementation). These nice data features comply with the memory constraint of modern multicore architectures. In this talk, we introduce two newly-developed block Krylov methods for linear systems with multiple right-hand sides available in two situations. For “simultaneous” right-hand sides, a block GMRES method is presented, which can address the problems related to spectral augmentation at restart and partial convergence of some linear combinations of the right-hand sides. For right- hand sides available one after each other, including the case where there are massive number (like tens of thousands) of right-hand sides associated with a single matrix so that all of them cannot be solved at once but rather need to be split into chunks of possible variable sizes, a recycling block flexible GMRES variant is developed by combining sub-space augmentation techniques in the generalized minimum residual norm framework to recycle spectral information between each restart and each block of right-hand sides, inexact breakdown detection in such a sub-space augmentation, and flexible preconditioner to cope with constraints on some applications while also enabling mixed-precision calculation. We demonstrate the efficiency of the two new algorithms on a set of numerical experiments including mixed arithmetic calculations.
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https://hal.inria.fr/hal-01334648
Contributor : Luc Giraud <>
Submitted on : Tuesday, June 21, 2016 - 10:25:19 AM
Last modification on : Tuesday, May 28, 2019 - 3:00:49 PM

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Yan-Fei Jing, Emmanuel Agullo, Bruno Carpentieri, Luc Giraud, Ting-Zhu Huang. Two New Block Krylov Methods for Linear Systems with Multiple Right-hand Sides. IMA Conference on Numerical Linear Algebra and Optimization, Sep 2016, Birmingham, United Kingdom. ⟨hal-01334648⟩

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