Exploiting variable precision in GMRES

Serge Gratton, Ehouarn Simon, David Titley-Peloquin, Philippe Toint

Research output: Working paper

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Abstract

We describe how variable precision floating point arithmetic can be used 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.
Original languageEnglish
Number of pages19
Volume1907:10550
Publication statusPublished - Jul 2019

Fingerprint

GMRES
Digital arithmetic
Arnoldi
Iterative Solver
Floating-point Arithmetic
Cross product
Matrix Product
Iterate
Convergence Rate
Iteration

Keywords

  • numerical analysis
  • variable precision
  • Krylov methods
  • ilinear algebra

Cite this

Gratton, S., Simon, E., Titley-Peloquin, D., & Toint, P. (2019). Exploiting variable precision in GMRES.
Gratton, Serge ; Simon, Ehouarn ; Titley-Peloquin, David ; Toint, Philippe. / Exploiting variable precision in GMRES. 2019.
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Gratton, S, Simon, E, Titley-Peloquin, D & Toint, P 2019 'Exploiting variable precision in GMRES'.

Exploiting variable precision in GMRES. / Gratton, Serge; Simon, Ehouarn; Titley-Peloquin, David; Toint, Philippe.

2019.

Research output: Working paper

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AU - Titley-Peloquin, David

AU - Toint, Philippe

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N2 - We describe how variable precision floating point arithmetic can be used inthe iterative solver GMRES. We show how the precision of the inner productscarried out in the algorithm can be reduced as the iterations proceed, withoutaffecting the convergence rate or final accuracy achieved by the iterates.Our analysis explicitly takes into account the resulting loss of orthogonalityin the Arnoldi vectors. We also show how inexact matrix-vector products canbe incorporated into this setting.

AB - We describe how variable precision floating point arithmetic can be used inthe iterative solver GMRES. We show how the precision of the inner productscarried out in the algorithm can be reduced as the iterations proceed, withoutaffecting the convergence rate or final accuracy achieved by the iterates.Our analysis explicitly takes into account the resulting loss of orthogonalityin the Arnoldi vectors. We also show how inexact matrix-vector products canbe incorporated into this setting.

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KW - Krylov methods

KW - ilinear algebra

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Gratton S, Simon E, Titley-Peloquin D, Toint P. Exploiting variable precision in GMRES. 2019 Jul.

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    Research Output

    A note on solving nonlinear optimization problems in variable precision

    Research output: Working paper