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Reports (Research Report) Year : 2020

Comparative study of harmonic and Rayleigh-Ritz procedures with applications to deflated conjugate gradients

Abstract

Harmonic Rayleigh-Ritz and Raleigh-Ritz projection techniques are compared in the context of iterative procedures to solve for small numbers of least dominant eigenvectors of large symmetric positive definite matrices. The procedures considered are (i) locally optimal conjugate gradient (CG) methods, i.e., LOBCG, (ii) thick-restart Lanczos methods, and (iii) recycled linear CG solvers, e.g., eigCG. Approaches based on principles of local optimality are adapted to enable the use of harmonic projection techniques. Upon investigating the search spaces generated by these methods, it is found that LOBCG and thick-restart Lanczos methods can be adapted, which is not the case of eigCG. Explanations are also given as to why eigCG works so well in comparison to other recycling strategies. Numerical experiments show that, while approaches based on harmonic projections consistently result in a faster convergence of eigen-residuals, they generally do not yield better convergence of the forward error of eigenvectors, until the Rayleigh quotients have converged. Then, the effect of recycling strategies is investigated on deflation for the resolution of sequences of linear systems. While non-locally optimal recycling strategies need to solve more linear systems in order to fully develop their effect on convergence, they eventually reach similar behaviors to those of locally optimal recycling procedures. While implementations based on Init-CG are robust for systems with multiple right-hand sides, this is not the case for multiple operators.
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Dates and versions

hal-02434043 , version 1 (09-01-2020)

Identifiers

  • HAL Id : hal-02434043 , version 1

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Nicolas Venkovic, Paul Mycek, Luc Giraud, Olivier Le Maitre. Comparative study of harmonic and Rayleigh-Ritz procedures with applications to deflated conjugate gradients. [Research Report] Cerfacs. 2020. ⟨hal-02434043⟩
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