Sensitivity Approximation by the Peano-Baker Series

In this paper we develop a new method for numerically approximating sensitivitiesin parameter-dependent ordinary differential equations (ODEs). Our approach,intended for situations where the standard forward and adjoint sensitivity analysisbecome too computationally costly for practical purposes, is based on the PeanoBaker series from control theory. We give a representation, using this series, for thesensitivity matrix S of an ODE system and use the representation to construct anumerical method for approximating S. We prove that, under standard regularityassumptions, the error of our method scales as O(∆t2max), where ∆tmax is the largesttime step used when numerically solving the ODE. We illustrate the performanceof the method in several numerical experiments, taken from both the systemsbiology setting and more classical dynamical systems. The experiments show thesought-after improvement in running time of our method compared to the forwardsensitivity approach. For example, in experiments involving a random linear system,the forward approach requires roughly √n longer computational time, where n isthe dimension of the parameter space, than our proposed method.QC 20230824