Numerical methods for computing sensitivities for ODEs and DDEs.

*(English)*Zbl 1362.65070Summary: We investigate the performance of the adjoint approach and the variational approach for computing the sensitivities of the least squares objective function commonly used when fitting models to observations. We note that the discrete nature of the objective function makes the cost of the adjoint approach for computing the sensitivities dependent on the number of observations. In the case of ordinary differential equations (ODEs), this dependence is due to having to interrupt the computation at each observation point during numerical solution of the adjoint equations. Each observation introduces a jump discontinuity in the solution of the adjoint differential equations. These discontinuities are propagated in the case of delay differential equations (DDEs), making the performance of the adjoint approach even more sensitive to the number of observations for DDEs. We quantify this cost and suggest ways to make the adjoint approach scale better with the number of observations. In numerical experiments, we compare the adjoint approach with the variational approach for computing the sensitivities.

##### MSC:

65L05 | Numerical methods for initial value problems |

65L03 | Numerical methods for functional-differential equations |

34A34 | Nonlinear ordinary differential equations and systems, general theory |

34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |

##### Keywords:

ordinary differential equations; delay differential equations; adjoint method; variational equations; sensitivities; least squares; fitting; numerical experiment##### Software:

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\textit{J. Calver} and \textit{W. Enright}, Numer. Algorithms 74, No. 4, 1101--1117 (2017; Zbl 1362.65070)

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