  In hydrology, the degradation coefficient is one of the key parameters to
describe the water quality change and to determine the water carrying capacity.
This paper is devoted to identify the degradation coefficient in an anomalous
diffusion process by using the average flux data at the accessible part of
boundary. The main challenges in inverse degradation coefficient problems
(IDCP) is the average flux measurement data only provide very limited
information and cause the severe ill-posedness of IDCP. Firstly, we prove the
average flux measurement data can uniquely determine the degradation
coefficient. The existence and uniqueness of weak solution for the direct
problem are established, and the Lipschitz continuity of the corresponding
forward operator is also obtained. Secondly, to overcome the ill-posedness, we
combine the variational regularization method with Laplace approximations (LA)
to solve the IDCP. This hybrid method is essentially the combination of
deterministic regularization method and stochastic method. Thus, it is able to
calculate the minimizer (MAP point) more rapidly and accurately, but also
enables captures the statistics information and quantifying the uncertainty of
the solution. Furthermore, the existence, stability and convergence of the
minimizer of the variational problem are proved. The convergence rate estimate
between the LA posterior distribution and the actual posterior distribution in
the sense of Hellinger distance is given, and the skewness are introduced for
characterizing the symmetry or slope of LA solution, especially the
relationship with the symmetry of the measurement data. Finally, the
one-dimensional and two-dimensional numerical examples are presented to confirm
the efficiency and robustness of the proposed method.
