论文标题
对长浪的小小的测深量测定，幅度很小
Optimization of Bathymetry for Long Waves with Small Amplitude
论文作者
论文摘要
本文涉及以幅度较小的长波的情况下，面向测深的优化。在这两个假设下，自由表面不可压缩的Navier-Stokes系统可以写为波动方程，在该方程中，在空间操作员中，测深显示作为参数。然后寻找时间谐波的场并写下底部地形作为平坦底部的扰动，我们最终得到了具有阻抗边界条件的异质Helmholtz方程。通过这种方式，我们研究了在异质介质中的Helmholtz方程的一些PDE受限的优化问题，这些介质的系数仅与有界变化界定。我们为一般成本函数提供了必要的条件，即至少具有一个最佳解决方案。我们还证明了解决方案对考虑的Helmholtz方程的有限元近似以及离散最佳倾向的收敛。我们通过一些数值实验结束了本文，以说明理论结果，并表明他们的某些假设实际上可以被删除。
This paper deals with bathymetry-oriented optimization in the case of long waves with small amplitude. Under these two assumptions, the free-surface incompressible Navier-Stokes system can be written as a wave equation where the bathymetry appears as a parameter in the spatial operator. Looking then for time-harmonic fields and writing the bottom topography as a perturbation of a flat bottom, we end up with a heterogeneous Helmholtz equation with impedance boundary condition. In this way, we study some PDE-constrained optimization problem for a Helmholtz equation in heterogeneous media whose coefficients are only bounded with bounded variation. We provide necessary condition for a general cost function to have at least one optimal solution. We also prove the convergence of a finite element approximation of the solution to the considered Helmholtz equation as well as the convergence of discrete optimum toward the continuous ones. We end this paper with some numerical experiments to illustrate the theoretical results and show that some of their assumptions could actually be removed.
