Infinite-dimensional Bayesian approach for inverse scattering problems of a fractional Helmholtz equation

This paper focuses on a fractional Helmholtz equation describing wave propagation in the attenuating medium. According to physical interpretations, the fractional Helmholtz equation can be divided into loss- and dispersion-dominated fractional Helmholtz equations. In the first part of this work, we establish the well-posedness of the loss-dominated fractional Helmholtz equation (an integer- and fractional-order mixed elliptic equation) for a general wavenumber and prove the Lipschitz continuity of the scattering field with respect to the scatterer. Meanwhile, we only prove the well-posedness of the dispersion-dominated fractional Helmholtz equation (a high-order fractional elliptic equation) for a sufficiently small wavenumber due to its complexity. In the second part, we generalize infinite-dimensional Bayesian inverse theory to allow a part of the noise depends on the target function (the function that needs to be estimated). We also prove that the estimated function tends to be the true function if both the model reduction error and the white noise vanish. We eventually apply our theory to the loss-dominated model with an absorbing boundary condition.