On the recovery of initial status for linearized shallow-water wave equation by data assimilation with error analysis

We recover the initial status of an evolution system governed by linearized shallow-water wave equations in a 2-dimensional bounded domain by data assimilation technique, with the aim of determining the initial wave height from the measurement of wave distribution in an interior domain. Since we specify only one component of the solution to the governed system and the observation is only measured in part of the interior domain, taking into consideration of the engineering restriction on the measurement process, this problem is ill-posed. Based on the known well-posedness result of the forward problem, this inverse problem is reformulated as an optimizing problem with data-fit term and the penalty term involving the background of the wave amplitude as a-prior information. We establish the Euler-Lagrange equation for the optimal solution in terms of its adjoint system. The unique solvability of this Euler-Lagrange equation is rigorously proven. Then the optimal approximation error of the regularizing solution to the exact solution is established in terms of the noise level of measurement data and the a-prior background distribution, based on the Lax-Milgram theorem. Finally, we propose an iterative algorithm to realize this process, with several numerical examples to validate the efficacy of our proposed method.