Fig. 1.

Illustration of the hierarchical optimization scheme using adjoint sensitivities. In the outer loop, θ is updated by the employed iterative gradient-based optimization method. When a new value of θ is proposed, an inner loop is entered, in which the optimal static parameters are computed for the given θ, and objective function value and gradient are returned before exiting the inner loop. Here, the solution of the inner problem is shown in detail. The red boxes involve the simulation of ODEs and are thus usually computationally more expensive. If the gradient is not required in some optimizer iteration, the adjoint and gradient steps can be omitted. Note that the dependence of s, b, σ, p, J and $\nabla J$ on θ is in the setting considered in this study only indirect via $\tilde{h}$, while in general also an explicit dependence is possible. (Color version of this figure is available at Bioinformatics online.)