基于Carleman估计的时间分数阶扩散方程反源、反系数问题的稳定性及数值反演

The studies of fractional diffusion equation, which have a wide range of practical applications in many fields of applied science, arise from modeling a class of process with temporal and spatial non-local dynamic behaviors. Hence, systematic studies on related forward and inverse problems play important roles in both theoretical and practical applications. In this project, we will develop the existing local Carleman estimate for the time fractional diffusion equation to a global result by selecting the suitable weight function for Carleman estimate, as well as establish the global Lipschitz conditional stability for inverse source and inverse coefficient problem of time-fractional diffusion equation. Furthermore, a strategy for the selection of the regularization parameter, which guarantees that the regularized solution has the same convergence rate with the conditional stability result, is to be proposed so as to get the stable inversion of regularized solution. Based on the duality theory for the time-fractional diffusion equation, the Fréchet derivative for the operator of the forward problem is obtained through the dual problem of the corresponding inverse problem, and highly efficient and stable numerical inversion is also achieved for the corresponding inverse problem. It is expected that through the study of this topic, we might further enrich and develop the theoretical and numerical methods for the inverse problem of fractional diffusion equation, and provide new understandings for anomalous diffusion phenomena, as well as provide some guidance on relevant practical issues.

分数阶扩散方程的研究源自对一类具有时空非局部动力学特性过程的建模，在应用科学诸多领域都有广泛应用，因此与之相关正反问题的研究尤为重要且具有广泛的理论和应用前景。本课题拟通过选取合适的Carleman估计权重函数，将已有的时间分数阶扩散方程Carleman估计从局部推广到全局，建立时间分数阶扩散方程反源及反系数问题的全局Lipschtiz条件稳定性，系统研究保证正则化解具有与条件稳定性结果相同收敛阶的正则化参数选取策略，实现正则化解的稳定收敛；并基于时间分数阶扩散方程对偶理论，通过相应反问题的对偶问题求得正问题算子的Fréchet导数，获得数值求解相应反问题的对偶方法，实现反问题稳定高效的数值反演。期望通过本课题的研究，进一步丰富和发展分数阶扩散方程反问题的理论和数值方法，并对反常扩散现象提供新的认识。