# Identification of the degradation coefficient for an anomalous diffusion process in hydrology

## TL;DR

This paper develops a hybrid variational regularization and Laplace approximation method to accurately identify the degradation coefficient in anomalous diffusion processes in hydrology, addressing ill-posed inverse problems.

## Contribution

It introduces a novel combined deterministic-stochastic approach for inverse degradation coefficient problems, ensuring uniqueness, stability, and quantifying uncertainty.

## Key findings

- Proved unique determination of the degradation coefficient from average flux data.
- Established existence, stability, and convergence of the solution.
- Numerical examples demonstrate the method's efficiency and robustness.

## Abstract

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.