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This article considers a Cauchy problem of Helmholtz equations whose solutionis well known to be exponentially unstable with respect to the inputs. In theframework of variational quasi-reversibility method, a Fourier truncation isapplied to appropriately perturb the underlying problem, which allows us toobtain a stable approximate solution. The corresponding approximate problem isof a hyperbolic equation, which is also a crucial aspect of this approach.Error estimates between the approximate and true solutions are derived withrespect to the noise level. From this analysis, the Lipschitz stability withrespect to the noise level follows. Some numerical examples are provided to seehow our numerical algorithm works well.

Analysis and simulation of a variational stabilization for the Helmholtz equation with noisy Cauchy data