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.

我们通过数据同化技术恢复了二维有界域中线性化浅水波动方程支配的演化系统的初始状态，目的是通过测量内部域中的波分布来确定初始波高。由于我们只指定了受治理系统的解的一个分量，并且观测仅在内部域的一部分中测量，因此考虑到对测量过程的工程限制，这个问题是病态的。基于前向问题的已知适定性结果，该逆问题被重新表述为一个优化问题，其中数据拟合项和涉及波幅背景的惩罚项作为先验信息。我们建立了 Euler-Lagrange 方程，用于根据其伴随系统进行的最优解。这个欧拉-拉格朗日方程的独特可解性得到了严格的证明。然后，根据测量数据的噪声水平和 a 先验背景分布，基于 Lax-Milgram 定理，建立正则化解对精确解的最优近似误差。最后，我们提出了一种迭代算法来实现这一过程，并通过几个数值示例来验证我们提出的方法的有效性。