  In this work, we consider the numerical recovery of a spatially dependent
diffusion coefficient in a subdiffusion model from distributed observations.
The subdiffusion model involves a Caputo fractional derivative of order
$\alpha\in(0,1)$ in time. The numerical estimation is based on the regularized
output least-squares formulation, with an $H^1(\Omega)$ penalty. We prove the
well-posedness of the continuous formulation, e.g., existence and stability.
Next, we develop a fully discrete scheme based on the Galerkin finite element
method in space and backward Euler convolution quadrature in time. We prove the
subsequential convergence of the sequence of discrete solutions to a solution
of the continuous problem as the discretization parameters (mesh size and time
step size) tend to zero. Further, under an additional regularity condition on
the exact coefficient, we derive convergence rates in a weighted $L^2(\Omega)$
norm for the discrete approximations to the exact coefficient {in the one- and
two-dimensional cases}. The analysis relies heavily on suitable nonstandard
nonsmooth data error estimates for the direct problem. We provide illustrative
numerical results to support the theoretical study.
