Wednesday, March 9, 2022
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All day |
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12pm |
03/09/2022 - 12:00pm Abstract: Motivated by the advent of machine learning, the last few years saw the return of hardware-supported reduced-precision computing. Computations with fewer digits are faster and more memory and energy efficient, but a careful implementation and rounding error analysis are required to ensure that sensible results can still be obtained. This talk is divided in two parts in which we focus on reduced- and mixed-precision algorithms respectively. Reduced-precision algorithms obtain an as accurate solution as possible given the precision while avoiding catastrophic rounding error accumulation. Mixed-precision algorithms, on the other hand, combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision while retaining good accuracy. In the first part of the talk we study the accumulation of rounding errors in the solution of the heat equation, a proxy for parabolic PDEs, in reduced precision using round-to-nearest (RtN) and stochastic rounding (SR). We demonstrate how to implement the numerical scheme to reduce rounding errors and we present \emph{a priori} estimates for local and global rounding errors. While the RtN solution leads to rapid rounding error accumulation and stagnation, SR leads to much more robust implementations for which the error remains at roughly the same level of the working precision. In the second part of the talk we focus on mixed-precision explicit stabilised Runge-Kutta methods. We show that a naive mixed-precision implementation harms convergence and leads to error stagnation, and we present a more accurate alternative. We introduce new Runge-Kutta-Chebyshev schemes that only use $q\in\{1,2\}$ high-precision function evaluations to achieve a limiting convergence order of $O(\Delta t^{q})$, leaving the remaining evaluations in low precision. These methods are essentially as cheap as their fully low-precision equivalent and they are as accurate and (almost) as stable as their high-precision counterpart. Short bio: Matteo obtained is DPhil in Mathematics at the University of Oxford in early 2020 under the supervision of Patrick E. Farrell, Michael B. Giles, and Marie E. Rognes. From the end of 2019 Matteo worked as a postdoc at the University of Oxford under the supervision of Michael B. Giles until the end of 2021 when he joined the Oden Institute as a postdoctoral fellow working with Robert Moser and Karen E. Willcox. Matteo won the Charles Broyden prize for the best paper in the Optimization Methods and Software journal in 2020. Matteo’s research is in the field of computational stochastics, uncertainty quantification and computational and industrial mathematics, with a focus on multilevel (quasi-) Monte Carlo methods, PDEs with random coefficients, reduced- and mixed-precision numerical algorithms, and biomedical computing. Location:
https://yale.zoom.us/j/97458245891
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4pm |
03/09/2022 - 4:15pm Abstract: The random geometry on simply connected surfaces is a well established subject in probability. The key aspects of this theory include the scaling limit of random planar maps, Liouville quantum gravity, Schramm-Loewner evolution (SLE), and Liouville conformal field theory. The first half of my talk is an overview of these aspects. The second half of the talk is on the random annulus. Although the geometry locally looks the same as on simply connected surfaces, the conformal structure of a random annulus is now a random variable, since annuli with different moduli are not conformally equivalent. The law of the modulus for a uniformly sampled random annulus was predicted in string theory and quantum gravity. I will report the recent verification of this conjecture joint with Morris Ang and Guillaume Remy. Our method also yields several exact formulae on SLE that were predicted by Cardy (2006) via the non-rigorous Coulomb gas method, including the generating function of the number of non-contracting loops for a conformal loop ensemble on the annulus, and the annulus partition function for Werner’s self avoiding loop. Location: |