BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nicholas J. Higham (University of Manchester\, UK) DTSTART:20200429T140000Z DTEND:20200429T150000Z DTSTAMP:20260423T022008Z UID:E-NLA/2 DESCRIPTION:Title: Ar e Numerical Linear Algebra Algorithms Accurate at Extreme Scale and at Low Precisions?\nby Nicholas J. Higham (University of Manchester\, UK) as part of E-NLA - Online seminar series on numerical linear algebra\n\n\nAb stract\nThe advent of exascale computing will bring the capability to solv e dense linear systems of order $10^8$. At the same time\, computer hardwa re is increasingly supporting low precision floating-point arithmetics\, s uch as the IEEE half precision and bfloat16 arithmetics. The standard rou nding error bound for the inner product of two $n$-vectors $x$ and $y$ is $|fl(x^Ty) - x^Ty| \\le n u |x|^T|y|$\, where $u$ is the unit roundoff\, and the bound is approximately attainable. This bound provides useful in formation only if $nu < 1$. Yet $nu > 1$ for exascale-size problems solve d in single precision and also for problems of order $n > 2048$ solved in half precision. Standard error bounds for matrix multiplication\, LU facto rization\, and so on\, are equally uninformative in these situations. Yet the supercomputers in the TOP500 are there by virtue of having successfull y solved linear systems of orders up to $10^7$\, and deep learning impleme ntations routinely use half precision with apparent success.\n\nHave we re ached the point where our techniques for analyzing rounding errors\, honed over 70 years of digital computation\, are unable to predict the accurac y of numerical linear algebra computations that are now routine? I will sh ow that the answer is "no": we can understand the behaviour of extreme-sca le and low accuracy computations. The explanation lies in algorithmic desi gn techniques (both new and old) that help to reduce error growth along wi th a new probabilistic approach to rounding error analysis.\n LOCATION:https://researchseminars.org/talk/E-NLA/2/ END:VEVENT END:VCALENDAR