Diffusion–reaction models are used to describe development processes in the framework of morphogen theory. The images of the concentration fields for the subset of the interacting morphogens are available. In order to interpret this data in terms of the model parameters, the inverse source problem is stated. The sensitivity operator, composed of the independent adjoint problem solutions ensemble, allows transforming the inverse problem to the family of nonlinear ill-posed operator equations. The equations are solved with the Newton–Kantorovich-type algorithm. The approach is applied to the morphogen synthesis region identification problem for the model of regulation of the renewing zone size in biological tissue.

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A numerical algorithm for the solution of an inverse coefficient problem for nonstationary, nonlinear production-destruction type model is proposed and tested on an example of the Lorenz’63 system. With an ensemble of adjoint problem solutions, the inverse problem is transformed into a quasi-linear matrix problem and solved with Newton-type algorithm. Two different ways of the adjoint ensemble construction are compared. In the first one, a trigonometric basis is used. In the second one in situ measurements are taken into account. Local convergence properties of the algorithm are studied numerically to find out when the use of more data can lead to the degradation of the reconstruction results.

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