Interval methods with Julia: Finding one million roots in one second

Abstract: The Julia language provides a remarkably productive environment for scientific computing, with a unique combination of interactivity and speed, and is particularly suitable for defining operations on new mathematical objects, such as intervals. I will present our free / open-source packages for interval arithmetic and interval methods (juliaintervals.github.io), written in pure Julia and comparable to state-of-the-art libraries. They use the composability coming from Julia's "multiple-dispatch"-based design and generic programming to integrate with other packages in the "ecosystem", including linear algebra, automatic differentiation, and plotting. The foundation is IntervalArithmetic.jl , which is almost compliant with the IEEE-1788 standard. Applications currently implemented include root finding, global optimization, constraint programming, Taylor models, and validated integration of ODEs. I will also show how Julia's facilities for parallel computing allow us to create user-defined objects on GPUs and manipulate them using the same Julia code. As an example benchmark, we find and verify one million stationary points of the two-dimensional transcendental Griewank function in under one second. Joint work with Luis Benet (ICF-UNAM).

analysis of PDEsclassical analysis and ODEsdynamical systemsfunctional analysisnumerical analysis

Audience: researchers in the discipline

CRM CAMP (Computer-Assisted Mathematical Proofs) in Nonlinear Analysis

Series comments: To have access to the zoom details of the talks, please register at www.crm.math.ca/camp-nonlinear