A variational frequency-dependent stabilization for the Helmholtz equation with noisy Cauchy data

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. Unlike the original version of the truncation, our perturbation is driven by the frequencies, rather than the noise level. This deals with real-world circumstances that one only measures data once or even does not know the noise level in advance, but needs to truncate high Fourier frequencies appropriately. The corresponding approximate problem is of a hyperbolic equation, which is also a crucial aspect of this approach. Error estimate between the approximate and true solutions is derived with respect to the noise level and to the frequencies is derived. 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.