On the Efficient Methods to Solve ODEs and BVPs Using Automatic Differentiation

Abstract

A large number of physical phenomena are modeled by a system of ODEs or a system of implicit ODEs. We demonstrate applicability of automatic differentiation (AD) for solving: (1) Boundary value problems in ODEs and implicit ODEs. (2) Initial state and parameter estimation problems. The impact of using AD is two fold. Firstly, efficient methods for computing the gradient vectors and Jacobian matrices have been developed using AD. Secondly the process of getting derivatives via AD is robust, more user friendly, and provides error free derivatives. Furthermore, techniques using AD have been developed which exploit structure in the user's computation, and particularly the structure we observe in boundary value problems or state/parameter estimation problems. We demonstrate by a few experiments the efficiency gained by the usage of AD in solving these problems.