Exploiting variable precision in GMRES

Serge Gratton (1, 2) , Ehouarn Simon (1, 2) , David Titley-Peloquin (3) , Philippe Toint (4)

1 IRIT-APO - Algorithmes Parallèles et Optimisation
2 Toulouse INP - Institut National Polytechnique (Toulouse)
3 McGill University = Université McGill [Montréal, Canada]
4 UNamur - Université de Namur [Namur]

We describe how variable precision floating-point arithmetic can be used to compute inner products in the iterative solver GMRES. We show how the precision of the inner products carried out in the algorithm can be reduced as the iterations proceed, without affecting the convergence rate or final accuracy achieved by the iterates. Our analysis explicitly takes into account the resulting loss of orthogonality in the Arnoldi vectors. We also show how inexact matrix-vector products can be incorporated into this setting.

Mots clés: GMRES algorithm AMS Subject Codes-15A06; 65F10; 65F25; 97N20; variable precision arithmetic; inexact inner products; inexact matrix-vector products; Arnoldi algorithm

Domaines: Analyse numérique [math.NA]

Serge Gratton, Ehouarn Simon, David Titley-Peloquin, Philippe Toint. Exploiting variable precision in GMRES. 2020. ⟨hal-02943241⟩