On an inverse problem in the parabolic equation arising from groundwater pollution problem
For researchers working on inverse problems in groundwater pollution, this work provides a comparison of parameter choice rules, but the results are incremental and limited to specific cases.
This paper addresses an inverse problem of determining a source term in a parabolic equation from groundwater pollution data. The Tikhonov regularization method is used, and numerical experiments show that the a posteriori parameter choice rule yields faster convergence than the a priori rule in some applications.
In this paper, we consider an inverse problem to determine a source term in a parabolic equation, where the data are obtained at a certain time. In general, this problem is ill-posed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solutions is obtained. We also propose both methods, a priori and a posteriori parameter choice rules. In addition, the proposed methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions. Eventually, from the numerical results it shows that the a posteriori parameter choice rule method gives a better the convergence speed in comparison with the a priori parameter choice rule method in some specific applications.