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LOCATION:Room 5.0A52
DTSTART;TZID=Europe/Stockholm:20250616T160000
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UID:submissions.pasc-conference.org_PASC25_sess138_msa125@linklings.com
SUMMARY:Probabilistic Error Analysis of Limited-Precision Stochastic Round
 ing
DESCRIPTION:El-Mehdi El arar (University of Rennes, INRIA)\n\nClassical pr
 obabilistic rounding error analysis is well suited to stochastic rounding 
 (SR), yielding strong results for floating-point algorithms relying on sum
 mation. For many numerical linear algebra algorithms, one can prove probab
 ilistic error bounds that grow as $\mathcal{O}(\sqrt{n}u)$, where $n$ is t
 he problem size and $u$ is the unit roundoff. These bounds are asymptotica
 lly tighter than the worst-case ones, which grow as $\mathcal{O}(nu)$. For
  certain algorithms, SR is unbiased. However, all these results were deriv
 ed under the assumption that SR is implemented exactly, which requires a n
 umber of random bits that is too large to be suitable for practical implem
 entations. We investigate the effect of using $r$ random bits in probabili
 stic SR error analysis. To this end, we introduce a new rounding mode, lim
 ited-precision SR. Considering the $r$ used, this new rounding mode accura
 tely matches hardware implementations, unlike the ideal SR generally used 
 in the literature. We show that this new rounding mode is biased and that 
 the bias is a function of $r$. As $r$ approaches infinity, however, the bi
 as disappears, and limited-precision SR converges to the ideal SR. We deve
 lop a novel model for probabilistic error analysis of algorithms employing
  SR. Several numerical examples corroborate our theoretical findings.\n\nD
 omain: Climate, Weather, and Earth Sciences, Engineering, Computational Me
 thods and Applied Mathematics\n\nSession Chair: Roman Iakymchuk (Uppsala U
 niversity, Umeå University)\n\n
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