Generalization of partitioned Runge-Kutta methods for adjoint systems

This study computes the gradient of a function of numerical solutions of ordinary differential equations (ODEs) with respect to the initial condition. The adjoint method computes the gradient approximately by solving the corresponding adjoint system numerically. In this context, Sanz-Serna [SIAM Rev., 58 (2016), pp. 3-33] showed that when the initial value problem is solved by a Runge-Kutta (RK) method, the gradient can be exactly computed by applying an appropriate RK method to the adjoint system. Focusing on the case where the initial value problem is solved by a partitioned RK (PRK) method, this paper presents a numerical method, which can be seen as a generalization of PRK methods, for the adjoint system that gives the exact gradient. (C) 2020 The Author(s). Published by Elsevier B.V.

Automated summary

This study calculates the gradient of a function of numerical solutions of ODEs with respect to the initial condition, approximating it using the adjoint method. It is shown that when solving the initial value problem with a Runge-Kutta method, the gradient can be exactly computed, and a generalized numerical method for the adjoint system is proposed for partitioned RK methods.