论文标题
用嘈杂的cauchy数据分析和模拟Helmholtz方程的变分稳定化
Analysis and simulation of a variational stabilization for the Helmholtz equation with noisy Cauchy data
论文作者
论文摘要
本文考虑了Helmholtz方程的Cauchy问题，其解决方案众所周知，相对于输入而言是指数不稳定的。在变异准可逆性方法的框架中，应用傅立叶截断来适当扰动潜在问题，这使我们能够获得稳定的近似解决方案。相应的近似问题是双曲线方程，这也是该方法的关键方面。相对于噪声水平得出了近似解决方案之间的错误估计。通过此分析，Lipschitz相对于噪声水平的稳定性随之而来。提供了一些数值示例，以查看我们的数值算法如何效果很好。
This article considers a Cauchy problem of Helmholtz equations whose solution is well known to be exponentially unstable with respect to the inputs. In the framework of variational quasi-reversibility method, a Fourier truncation is applied to appropriately perturb the underlying problem, which allows us to obtain a stable approximate solution. The corresponding approximate problem is of a hyperbolic equation, which is also a crucial aspect of this approach. Error estimates between the approximate and true solutions are derived with respect to the noise level. From this analysis, the Lipschitz stability with respect to the noise level follows. Some numerical examples are provided to see how our numerical algorithm works well.
