JuliaCon 2018

Abstract
We approximate a function rigorously with Julia: find a Taylor series that is close, using automatic differentiation, and bound the resulting error using intervals, giving a “Taylor model”. We use this to get guaranteed approximate solutions of ordinary differential equations in the form of tubes.
Description
We will show how to use Julia to approximate functions in a rigorous way: find a truncated Taylor series that is close to the function, using automatic differentiation techniques provided by the TaylorSeries.jl package. Next, bound the error from truncating the series, using intervals from the IntervalArithmetic.jl package. The resulting object is a Taylor model, as provided by the new TaylorModels.jl package.
We can now perform rigorous calculations on functions, e.g. multiplying two functions or finding sin(f) for a function f, by manipulating the corresponding Taylor models. This has many applications, for example global optimization: we can find rigorous optima of arbitrary functions by optimizing a polynomial instead, which is easier (although not “easy”).
We will focus on showing how Taylor models can be used as a building block to solve ordinary differential equations (ODEs): find a Taylor series expansion of the solution to the ODE and bound the error. This gives approximations to the solution, in the form of tubes that are guaranteed to enclose the true solution of the ODE.