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Inverse scattering aims to infer information about a hidden object by usingthe received scattered waves and training data collected from forwardmathematical models. Recent advances in computing have led to increasingattention towards functional inverse inference, which can reveal more detailedproperties of a hidden object. However, rigorous studies on functional inverse,including the reconstruction of the functional input and quantification ofuncertainty, remain scarce. Motivated by an inverse scattering problem wherethe objective is to infer the functional input representing the refractiveindex of a bounded scatterer, a new Bayesian framework is proposed. It containsa surrogate model that takes into account the functional inputs directlythrough kernel functions, and a Bayesian procedure that infers functionalinputs through the posterior distribution. Furthermore, the proposed Bayesianframework is extended to reconstruct functional inverse by integratingmulti-fidelity simulations, including a high-fidelity simulator solved byfinite element methods and a low-fidelity simulator called the Bornapproximation. When compared with existing alternatives developed by finitebasis expansion, the proposed method provides more accurate functionalrecoveries with smaller prediction variations.

Advancing inverse scattering with surrogate modeling and Bayesian inference for functional inputs