  Inverse scattering aims to infer information about a hidden object by using
the received scattered waves and training data collected from forward
mathematical models. Recent advances in computing have led to increasing
attention towards functional inverse inference, which can reveal more detailed
properties of a hidden object. However, rigorous studies on functional inverse,
including the reconstruction of the functional input and quantification of
uncertainty, remain scarce. Motivated by an inverse scattering problem where
the objective is to infer the functional input representing the refractive
index of a bounded scatterer, a new Bayesian framework is proposed. It contains
a surrogate model that takes into account the functional inputs directly
through kernel functions, and a Bayesian procedure that infers functional
inputs through the posterior distribution. Furthermore, the proposed Bayesian
framework is extended to reconstruct functional inverse by integrating
multi-fidelity simulations, including a high-fidelity simulator solved by
finite element methods and a low-fidelity simulator called the Born
approximation. When compared with existing alternatives developed by finite
basis expansion, the proposed method provides more accurate functional
recoveries with smaller prediction variations.
