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

TL;DR

This paper introduces a variational stabilization method for the Helmholtz equation with noisy Cauchy data, using Fourier truncation and hyperbolic reformulation to achieve stable solutions with error bounds, supported by numerical tests.

Contribution

It proposes a novel variational quasi-reversibility approach with Fourier truncation to stabilize the Helmholtz problem under noisy conditions, providing error estimates and numerical validation.

Findings

Stable approximate solutions are obtained despite noise.

Error estimates show Lipschitz stability with respect to noise level.

Numerical examples demonstrate the effectiveness of the method.

Abstract

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.