Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

In this report, we use the complex-step derivative approximation technique to compute sensitivities for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history functions. We compare the results with exact derivatives and with those computed using the classical sensitivity equations whenever possible. Our results demonstrate that the implementation of the complex-step method using the method of steps and the Matlab solver dde23 provides a very good approximation of sensitivities as long as discontinuities in the initial data do not cause loss of smoothness in the solution: that is, even when the underlying smoothness with respect to the initial data for the Cauchy-Riemann derivation of the method does not hold. We conclude with remarks on our findings regarding the complex-step method for computing sensitivities for simpler ordinary differential equation systems in the event of lack of smoothness with respect to parameters.