Generalization of partitioned Runge--Kutta methods for adjoint systems

TL;DR

This paper extends partitioned Runge--Kutta methods to exactly compute gradients of solutions to ODEs via adjoint systems, improving the accuracy of gradient calculations in numerical simulations.

Contribution

It introduces a generalized numerical method for adjoint systems that ensures exact gradient computation when using partitioned Runge--Kutta methods.

Findings

The method provides exact gradients for partitioned RK solutions.

It generalizes previous approaches to a broader class of partitioned methods.

Numerical experiments confirm the method's accuracy.

Abstract

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.