A note on a quasi-solution for an inverse boundary static Klein–Gordon equation with Gaussian noisy data

In this study, we consider a new quasi-solution to the Cauchy problem for the stationary Klein–Gordon equation associated with a set of Gaussian data. This problem arises in the reconstruction of the meson field in a homogeneous medium, assuming that only certain boundary parts are accessible. Our quasi-solution is constructed using the recently developed variational quasi-reversibility inversion, designed to deal with the exponential instability of this equation, along with the so-called truncated Fourier approximator used to handle the stochasticity in the data. As an interesting aspect of this inversion, a linear second-order differential equation can approximate an elliptic model well. Through our analysis, a Hölder error estimate between our quasi-solution and the true solution is established in the expectation operator. We also investigate the solvability and observability of the inversion scheme. Finally, we present some numerical tests to verify the numerical performance of the inversion.