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.
翻译:本条认为Helmholtz方程式的一个棘手问题,众所周知,其解决办法在投入方面极不稳定。在变异准反逆方法的框架内,对根本问题适用了Fourier短路,从而使我们能够找到一个稳定的近似解决办法。相应的近似问题是双曲方程式,这也是这一方法的一个关键方面。在噪音水平方面得出了近似和真实解决办法之间的误差估计。根据这一分析,Lipschitz在噪音水平方面的稳定性如下。提供了一些数字例子,以了解我们的数字算法如何运作。