Consider the recovery of the weight function in distributed time-fractional diffusion system using the interior measurement, which arises in some ultra-slow diffusion phenomena. Due to the nonlinear and nonlocal dependance of the measurement data on the weight function, such an inverse problem is novel and important. Based on the regularities of the direct problems for general diffusion equations with distributed time fractional derivative shown in this paper, we establish the theoretical framework for the optimization version of the inverse problem, including existence of the minimizer, the differentiability of the cost functional, as well as the gradient of the cost functional, in suitable functional spaces. Then an iteration process of descent type is realized efficiently for minimizing the regularizing cost functional in terms of the gradient type iteration, with numerical examples showing the validity of the proposed scheme.

考虑使用内部测量在分布式时间分数扩散系统中恢复权函数，这在一些超慢扩散现象中出现。由于测量数据对权重函数的非线性和非局部依赖性，这样的逆问题是新颖而重要的。基于本文给出的具有分布时间分数阶导数的一般扩散方程直接问题的规律，我们建立了反问题优化版本的理论框架，包括极小值的存在性、代价函数的可微性，如以及在合适的功能空间中成本泛函的梯度。