  This paper considers the temporal discretization of an inverse problem
subject to a time fractional diffusion equation. Firstly, the convergence of
the L1 scheme is established with an arbitrary sectorial operator of spectral
angle $< \pi/2 $, that is the resolvent set of this operator contains $
\{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< \theta\}$ for some $
\pi/2 < \theta < \pi $. The relationship between the time fractional order
$\alpha \in (0, 1)$ and the constants in the error estimates is precisely
characterized, revealing that the L1 scheme is robust as $ \alpha $ approaches
$ 1 $. Then an inverse problem of a fractional diffusion equation is analyzed,
and the convergence analysis of a temporal discretization of this inverse
problem is given. Finally, numerical results are provided to confirm the
theoretical results.
