SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2981-A3008, January 2021.
We develop an adjoint approach for recovering the topographical function included in the source term of one-dimensional hyperbolic balance laws. We focus on a specific system, namely, the shallow water equations, in an effort to recover the riverbed topography. The novelty of this work is the ability to robustly recover the bottom topography using only noisy boundary data from one measurement event and the inclusion of two regularization terms in the iterative update scheme. The adjoint scheme is determined from a linearization of the forward system and is used to compute the gradient of a cost function. The bottom topography function is recovered through an iterative process given by a three-operator splitting method which allows the feasibility of including two regularization terms. Numerous numerical tests demonstrate the robustness of the method regardless of the choice of initial guess and in the presence of discontinuities in the solution of the forward problem.

中文翻译：

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通过伴随方法从浅水方程恢复瞬态底部地形函数

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2981-A3008 页，2021 年 1 月。
我们开发了一种伴随方法来恢复包含在一维双曲平衡定律源项中的地形函数。我们专注于一个特定的系统，即浅水方程，以努力恢复河床地形。这项工作的新颖之处在于能够仅使用来自一个测量事件的噪声边界数据以及在迭代更新方案中包含两个正则化项来稳健地恢复底部地形。伴随方案由前向系统的线性化确定，用于计算成本函数的梯度。底部地形函数是通过迭代过程恢复的，该过程由三运算符拆分方法给出，该方法允许包括两个正则化项的可行性。