Abstract The problem of identification of coefficients and initial conditions for a boundary value problem for parabolic equations that reduces to a minimization problem of a misfit function is investigated. Firstly, the tensor train decomposition approach is presented as a global convergence algorithm. The idea of the proposed method is to extract the tensor structure of the optimized functional and use it for multidimensional optimization problems. Secondly, for the refinement of the unknown parameters, three local optimization approaches are implemented and compared: Nelder–Mead simplex method, gradient method of minimum errors, adaptive gradient method. For gradient methods, the evident formula for the continuous gradient of the misfit function is obtained. The identification problem for the diffusive logistic mathematical model which can be applied to social sciences (online social networks), economy (spatial Solow model) and epidemiology (coronavirus COVID-19, HIV, etc.) is considered. The numerical results for information propagation in online social network are presented and discussed.

中文翻译：

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抛物线系统识别中的全局和局部优化

摘要 研究了简化为失配函数最小化问题的抛物线方程边值问题的系数和初始条件的识别问题。首先，张量训练分解方法作为全局收敛算法被提出。所提出方法的思想是提取优化泛函的张量结构并将其用于多维优化问题。其次，对于未知参数的细化，实现并比较了三种局部优化方法：Nelder-Mead单纯形法、最小误差梯度法、自适应梯度法。对于梯度方法，得到了misfit函数的连续梯度的明显公式。考虑了可应用于社会科学（在线社交网络）、经济（空间索洛模型）和流行病学（冠状病毒 COVID-19、HIV 等）的扩散逻辑数学模型的识别问题。提出并讨论了在线社交网络中信息传播的数值结果。