Identification of Space-Time Distributed Parameters in the Gierer-Meinhardt Reaction-Diffusion System

We consider parameter identification for the classic Gierer--Meinhardt reaction-diffusion system. The original Gierer--Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp. 30--39] was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology. In our paper the parameters are extended in time and space and used as distributed control variables. The methodology employs PDE-constrained optimization in the context of image-driven spatiotemporal pattern formation. We prove the existence of optimal solutions, derive an optimality system, and determine optimal solutions. The results of numerical experiments in two dimensions are presented using the finite element method, which illustrates the convergence of a variable-step gradient algorithm for finding the optimal parameters of the system. A practical target function is constructed for the optimal control algorithm corresponding to the actual image of a marine angel...

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