An Analysis of Infinite Dimensional Bayesian Inverse Shape Acoustic Scattering and Its Numerical Approximation

We present and analyze an infinite dimensional Bayesian inference formulation, and its numerical approximation, for the inverse problem of inferring the shape of an obstacle from scattered acoustic waves. Given a Gaussian prior measure on the shape space, whose covariance operator is the inverse of the Laplacian, the Bayesian solution of the inverse problem is the posterior measure given by the Radon--Nikodym derivative with respect to the Gaussian prior measure. The well-posedness of the Bayesian formulation in infinite dimensions is proved, including the justification of the Radon--Nikodym derivative and the continuous dependence of the posterior measure on the observation data via the Hellinger distance. The proof is made possible by a suitable shape parametrization in a Banach space setting and the regularity of the forward solution with respect to the smoothness of the shape. This also facilitates proving the Lipschitz continuity of the observation operator with respect to the scatterer shape via the shape derivative of the forward solution. Next, a finite dimensional approximation to the Bayesian posterior is proposed and the corresponding approximation error is quantified. The approximation strategy involves a Nyström scheme for approximating a boundary integral formulation of the forward Helmholtz problem and a Karhunen--Loève truncation for approximating the prior measure. Weak convergence of the resulting finite dimensional approximation, as well as convergence in the Hellinger distance, are investigated. In particular, we determine the convergence rate as a function of the number of Nyström quadrature points and the number of truncated terms in the Karhunen--Loève series. Finally, we estimate the error between the exact posterior moments, e.g., posterior mean and variance, and their finite dimensional approximate counterparts in terms of the errors due to forward equation approximation and prior approximation. The main result of this work is that the convergence rate for approximating the Bayesian inverse problem is spectral, and this directly inherits the spectral convergence rates of the approximations of both the prior and the forward problems.