Multistage mixed precision iterative refinement

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10446026" target="_blank" >RIV/00216208:11320/22:10446026 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=DFqPLcERLC" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=DFqPLcERLC</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/nla.2434" target="_blank" >10.1002/nla.2434</a>

Result language

angličtina

Original language name

Multistage mixed precision iterative refinement

Original language description

Low precision arithmetic, in particular half precision (16-bit) floating point arithmetic, is now available in commercial hardware. Using lower precision can offer significant savings in computation and communication costs with proportional savings in energy. Motivated by this, there has been a renewed interest in mixed precision iterative refinement schemes for solving linear systems Ax = b, and new variants of GMRES-based iterative refinement have been developed. Each particular variant with a given combination of precisions leads to different condition number-based constraints for convergence of the backward and forward errors, and each has different performance costs. The constraints for convergence given in the literature are, as an artifact of the analyses, often overly strict in practice, and thus could lead a user to select a more expensive variant when a less expensive one would have sufficed. In this work, we develop a multistage mixed precision iterative refinement solver which aims to combine existing mixed precision approaches to balance performance and accuracy and improve usability. For a user-specified initial combination of precisions, the algorithm begins with the least expensive approach and convergence is monitored via inexpensive computations with quantities produced during the iteration. If slow convergence or divergence is detected using particular stopping criteria, the algorithm switches to use a more expensive, but more reliable variant. A novel aspect of our approach is that, unlike existing implementations, our algorithm first attempts to use &quot;stronger&quot; GMRES-based solvers for the solution update before resorting to increasing the precision(s). In some scenarios, this can avoid the need to refactorize the matrix in higher precision. We perform extensive numerical experiments on a variety of random dense problems and problems from real applications which confirm the benefits of the multistage approach.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10102 - Applied mathematics

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2022

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Numerical Linear Algebra with Applications

ISSN

1070-5325

e-ISSN

—

Volume of the periodical

2022

Issue of the periodical within the volume

29(4)

Country of publishing house

GB - UNITED KINGDOM

Number of pages

24

Pages from-to

1-24

UT code for WoS article

000759325400001

EID of the result in the Scopus database

2-s2.0-85125105383