Bathymetry reconstruction via optimal control in well-balanced finite element methods for the shallow water equations
Accurate prediction of shallow water flows relies on precise bottom topography data, yet direct bathymetric surveys are expensive and time-consuming. In contrast, remote sensing platforms such as radar or satellite altimetry provide accurate free surface observations. This disparity motivates a data-driven reconstruction strategy: invert the shallow water equations to estimate the bathymetry that yields the best fit to the governing dynamics. We introduce a new direct reconstruction technique that extracts bathymetric features from widely available free surface measurements. The underlying inverse problem of determining an unknown bathymetry profile from observed wave elevations is inherently ill-posed. Small perturbations in the data may lead to large deviations in the reconstructed topography, and discontinuities or sharp gradients further exacerbate instability. To stabilize the inversion, we formulate an optimal-control problem, wherein a cost functional penalizes deviations between simulated and measured free surface elevation while enforcing a state equation for the flow dynamics. To suppress noise and preserve sharp depth variations, the framework is augmented with $L^1$ regularization and total variation denoising. These sparsity-promoting terms encourage piecewise-smooth solutions, allowing changes in the bathymetry to be captured without excessive smoothing. Numerical experiments on synthetic noisy data and discontinuous bathymetry demonstrate robust performance in reconstructing unknown bathymetry.