How AD can help solve differential-algebraic equations. (2nd November 2018)

• Record Type: Journal Article
• Authors: Pryce, John D.; Nedialkov, Nedialko S.; Tan, Guangning; Li, Xiao
• Is Part Of: Optimization methods and software. Volume 33:Number 4/6(2018)
• Page Start: 729
• Page End: 749
• Publication Date: 2018-11-02
• Subjects: algorithmic differentiation -- differential-algebraic equations -- dummy derivatives -- Lagrangians; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2018.1428605
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of the so-called index reduction or regularization, to prepare them for numerical solution. This is often done with the help of a computer algebra system. We show in two significant cases that it can be done efficiently by pure algorithmic differentiation. The first is the Dummy Derivatives method; here we give a mainly theoretical description, with tutorial examples. The second is the solution of a mechanical system directly from its Lagrangian formulation. Here, we outline the theory and show several non-trivial examples of using the 'Lagrangian facility' of the Nedialkov–Pryce initial-value solver DAETS, namely a spring-mass-multi-pendulum system; a prescribed-trajectory control problem; and long-time integration of a model of the outer planets of the solar system, taken from the DETEST testing package for ODE solvers.