IDENTIFICATION OF SPACE-TIME DISTRIBUTED PARAMETERS IN THE GIERER-MEINHARDT REACTION-DIFFUSION SYSTEM
IDENTIFICATION OF SPACE-TIME DISTRIBUTED PARAMETERS IN THE GIERER-MEINHARDT REACTION-DIFFUSION SYSTEM
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DOI：
10.1137/120885784
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发表时间：
2014-01-01
影响因子：
1.9
通讯作者：
Trenchea, Catalin
中科院分区：
文献类型：
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作者：
Garvie, Marcus R.;Trenchea, Catalin
考虑经典Gierer-Meinhardt反应扩散系统的参数辨识问题。原Gierer-Meinhardt模型[A]。Gierer和H. Meinhardt, Kybernetik, 12 (1972), pp. 30-39]是用恒定参数制定的，并已被用作研究发育生物学模式形成的原型系统。本文将这些参数在时间和空间上进行扩展，并作为分布式控制变量。该方法在图像驱动的时空模式形成的背景下采用pde约束优化。证明了最优解的存在性，导出了一个最优系统，并确定了最优解。用有限元方法给出了二维数值实验结果，说明了变步长梯度算法求解系统最优参数的收敛性。针对海仙鱼的实际图像，构造了一个实用的目标函数作为最优控制算法。
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 angelfish.