We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale ${\mathbb{H}}^r\left({\mathbb{R}}^n\right)$ and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale ${\mathbb{H}}^r\left({\mathbb{R}}^n\right)$. Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales ${\mathbb{H}}^r\left({\mathbb{R}}^n\right)$ are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.

中文翻译：

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时间分数阶扩散方程反源问题的Tikhonov方法和最优误差界

我们研究在时间分数扩散方程中确定多维空间相关热源的线性但不适定的逆问题。我们证明问题在希尔伯特量表中是不适当的${\mathbb{H}}^{\mathrm{[R}}（{\mathbb{[R}}^{ñ}）$并为最坏情况的误差建立全局阶数最优下限。接下来，我们使用Tikhonov正则化方法在希尔伯特尺度上处理此问题${\mathbb{H}}^{\mathrm{[R}}（{\mathbb{[R}}^{ñ}）$。Hilbert量表中正则化算子族的参数的局部最优选择${\mathbb{H}}^{\mathrm{[R}}（{\mathbb{[R}}^{ñ}）$通过先验和后验方法进行分析。数值实现被提出来说明我们的理论发现。