PARAREAL ALGORITHM IMPLEMENTATION AND SIMULATION IN JULIA
Tyler M. Masthay and Saverio Perugini
Institute for Computational and Engineering Sciences, University of Texas Austin, tyler@ices.utexas.edu
Department of Computer Science, University of Dayton, 300 College Park Dayton, Ohio 45469–2160, saverio@udayton.edu
We present an implementation of the parareal algorithm—an integration technique to solve differential equations in parallel—first proposed in 2001 by Lions, Maday, and Turinici [@Lions:2001]—in the Julia programming language [@Julia:2014] for a fully general, first-order, initial-value problem. We also provide a graphical simulation of the parareal algorithm intended to be used in a numerical analysis course to both introduce the parareal algorithm to students and aid them in an investigation of the types of curves for which the parareal algorithm might be practical. Our implementation of the parareal algorithm accepts both coarse and fine integrators as functional arguments. We provide implementations of Euler’s method and another Runge-Kutta integration technique as the integrators. The final two functions in the source code file \texttt{parareal.jl} are functions implementing Euler’s method and another Runge-Kutta integration technique that can be used as examples to be passed as first-class functions as coarse or fine integration techniques to the \texttt{parareal} or \texttt{simulate} functions. A Git repository of both the implementation and graphical simulation is available in GitHub at https://github.com/sperugin/Parareal-Implementation-and-Simulation-in-Julia.git. All of the graphical plots are generated with the Julia Plots package available at https://juliaplots.github.io/. A video describing this application of Julia is available on YouTube at https://www.youtube.com/watch?v=MtgbeLO6ZM4. The purpose of this software is pedagogical: as a simulation to introduce students to the parareal algorithm and the concept of concurrency, and as a tool for (graphically) investigating the performance of the algorithm.
Language: Julia Version: 1.3.1 Compiler: Julia Built-in Interpreter: Julia Built-in Test Platform: Windows (64-bit) Download page: https://julialang.org/downloads/
parareal.jl: accepts any fine and coarse integrator (e.g., it can accept any Runge-Kutta integration technique).
This is a short tutorial on the parareal algorithm as presented by Lions, Maday, and Turinici. It provides the implementation of the parareal algorithm for a 1D, first-order ODE. In addition, there is a simulator that provides a simulation of how the algorithm executes. More details are outlined in the preprint https://arxiv.org/abs/1706.08569.
Provides an implementation of a basic parareal algorithm for a 1D, first-order ODE. It also includes a simulation that details how the algorithm works.
Provides two test cases. The first converges quickly (in the eyeball norm) and is a case where the algorithm may provide speedup. The other converges very slowly, indicating a test case where parareal version will likely be slower than sequential algorithm.