The Special Math of Translating Theory to Software In Differential Eqs | Chris Rackauckas | ASE60 скачать в хорошем качестве

The Special Math of Translating Theory to Software In Differential Eqs | Chris Rackauckas | ASE60

Numerical analysis describes a pristine mathematical theory about optimal numerical algorithms under assumptions which do not necessarily hold in practice. For example, in theory good ODE approximations have fast convergence as the step size approaches zero. But, in practice, a good differential equation solver takes as big steps as possible. How must one change their mathematical reasoning to match the software world? This talk will describe the nitty gritty reasoning that comes into play when building mathematical software. We will describe how one of the most important aspects of optimizing a Runge-Kutta method happens to be alternative floating point power approximations, and how "suboptimal methods" can be more optimal by taking advantage of certain assembly instructions in modern hardware. This demonstrates how the creation of mathematical software is a discipline unto itself. Contents 00:00 Welcome and introduction 01:40 First problem: Small ODEs in pharmacometrics 02:49 Euler’s method and Runge-Kutta methods 07:28 Why not just use arbitrarily high order methods? 11:06 Dormand-Prince as default solver (e.g., ode45) 13:59 Can we drop the Dormand-Prince simplifying assumption? 14:56 Yes – this is why Julia defaults to Tsit5 16:35 Origins of Vern solvers 22:08 Building in adaptivity for solvers 25:20 Going beyond explicit Runge-Kutta methods 26:14 When to choose a non-BDF approach for stiff ODE solvers 29:54 Final comments and questions S/O to https://github.com/agchesebro for the video timestamps! Want to help add timestamps to our YouTube videos to help with discoverability? Find out more here: https://github.com/JuliaCommunity/You... Interested in improving the auto generated captions? Get involved here: https://github.com/JuliaCommunity/You...