Parameter identification in ODE models with oscillatory dynamics: a Fourier regularization approach

In this paper we consider a parameter identification problem (PIP) for data oscillating in time, that can be described in terms of the dynamics of some ordinary differential equation (ODE) model, resulting in an optimization problem constrained by the ODEs. In problems with this type of data structure, simple application of the direct method of control theory (discretize-thenoptimize) yields a least-squares cost function exhibiting multiple 'low' minima. Since in this situation any optimization algorithm is liable to fail in the approximation of a good solution, here we propose a Fourier regularization approach that is able to identify an iso-frequency manifold S of codimension-one in the parameter space Omega subset of R-m, such that for all parameters in S the ODE solutions have the same frequency of the assigned data. Further to the identification of S, we propose to minimize on this manifold the least squares, the phase (or time lag) and infinity norm errors between data and simulations. Hence, the Fourier-PIP can be regarded as a new constrained optimization problem, where the iso-frequency sub-manifold represents a further constraint. First we describe our approach for simulated oscillatory data obtained with the two-parameter Schnakenberg model, in the Hopf regime. Finally, we apply Fourier-PIP regularization to follow original experimental data with the morphochemical model for electrodeposition (Lacitignola et al 2015 Eur. J. Appl. Math. 26 143-73) in the case of two and three parameters