Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem

A discrete assimilation system for a one-dimensional variable coefficient convection-diffusion equation is constructed. The variational adjoint method combined with the regularization technique is employed to retrieve the initial condition and diffusion coefficient with the aid of a set of simulated observations. Several numerical experiments are performed: (a) retrieving both the initial condition and diffusion coefficient jointly (Experiment JR), (b) retrieving either of them separately (Experiment SR), (c) retrieving only the diffusion coefficient with the iteration count increased to 800 (Experiment NoR-SR), and (d) retrieving only the diffusion coefficient with the consideration of a regularization term based on the Experiment NoR-SR (Experiment AdR-SR). The results indicate that within the limit of 100 iterations, the retrieval quality of the Experiment SR is better than those from the Experiment JR. Compared with the initial condition, the diffusion coefficient is a little difficult to retrieve, whereas we still achieve the desired result by increasing the iterations or integrating the regularization term into the cost functional for the improvement with respect to the diffusion coefficient. Further comparisons between the Experiment NoR-SR and AdR-SR show that the regularization term can really help not only improve the precision of retrieval to a large extent, but also speed up the convergence of solution, even if some perturbations are imposed on those observations.