Generalization of partitioned Runge--Kutta methods for adjoint systems

2020/03/22 by Takeru Matsuda, Matsuda, Takeru, Yuto Miyatake +1
Computer Science · Mathematics · #FOS: Mathematics #Fractional Differential Equations Solutions #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2003.09789

openalex publication_date 2020/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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