Mixed precision s-step Lanczos and conjugate gradient algorithms

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10436238" target="_blank" >RIV/00216208:11320/21:10436238 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=6NE8Dz.9PD" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=6NE8Dz.9PD</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/nla.2425" target="_blank" >10.1002/nla.2425</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Mixed precision s-step Lanczos and conjugate gradient algorithms

Popis výsledku v původním jazyce

Compared to the classical Lanczos algorithm, the s-step Lanczos variant has the potential to improve performance by asymptotically decreasing the synchronization cost per iteration. However, this comes at a price; despite being mathematically equivalent, the s-step variant may behave quite differently in finite precision, potentially exhibiting greater loss of accuracy and slower convergence relative to the classical algorithm. It has previously been shown that the errors in the s-step version follow the same structure as the errors in the classical algorithm, but are amplified by a factor depending on the square of the condition number of the O(s)-dimensional Krylov bases computed in each outer loop. As the condition number of these s-step bases grows (in some cases very quickly) with s, this limits the s values that can be chosen and thus can limit the attainable performance. In this work, we show that if a select few computations in s-step Lanczos are performed in double the working precision, the error terms then depend only linearly on the conditioning of the s-step bases. This has the potential for drastically improving the numerical behavior of the algorithm with little impact on per-iteration performance. Our numerical experiments demonstrate the improved numerical behavior possible with the mixed precision approach, and also show that this improved behavior extends to mixed precision s-step CG. We present preliminary performance results on NVIDIA V100 GPUs that show that the overhead of extra precision is minimal if one uses precisions implemented in hardware.

Název v anglickém jazyce

Mixed precision s-step Lanczos and conjugate gradient algorithms

Popis výsledku anglicky

Compared to the classical Lanczos algorithm, the s-step Lanczos variant has the potential to improve performance by asymptotically decreasing the synchronization cost per iteration. However, this comes at a price; despite being mathematically equivalent, the s-step variant may behave quite differently in finite precision, potentially exhibiting greater loss of accuracy and slower convergence relative to the classical algorithm. It has previously been shown that the errors in the s-step version follow the same structure as the errors in the classical algorithm, but are amplified by a factor depending on the square of the condition number of the O(s)-dimensional Krylov bases computed in each outer loop. As the condition number of these s-step bases grows (in some cases very quickly) with s, this limits the s values that can be chosen and thus can limit the attainable performance. In this work, we show that if a select few computations in s-step Lanczos are performed in double the working precision, the error terms then depend only linearly on the conditioning of the s-step bases. This has the potential for drastically improving the numerical behavior of the algorithm with little impact on per-iteration performance. Our numerical experiments demonstrate the improved numerical behavior possible with the mixed precision approach, and also show that this improved behavior extends to mixed precision s-step CG. We present preliminary performance results on NVIDIA V100 GPUs that show that the overhead of extra precision is minimal if one uses precisions implemented in hardware.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

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

Název periodika

Numerical Linear Algebra with Applications

ISSN

1070-5325

e-ISSN

—

Svazek periodika

2021

Číslo periodika v rámci svazku

e2425

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

24

Strana od-do

1-24

Kód UT WoS článku

000721446000001

EID výsledku v databázi Scopus

2-s2.0-85119656079