A Bayesian Scheme for Reconstructing Obstacles in Acoustic Waveguides

Abstract

In this paper, we investigate inverse obstacle scattering problems in acoustic waveguides with low-frequency data. A Bayesian inference scheme, combining a multi-fidelity strategy and surrogate model with guided modes and a deep neural network is proposed to reconstruct the shapes of unknown scattering objects. First, the inverse problem is reformulated as a statistical inference problem using Bayes’ formula, which provides statistical characteristics of the posterior distribution and quantification of the uncertainties. The well-posedness of the posterior distribution is proved by using the f-divergence. Subsequently, a Markov Chain Monte Carlo algorithm is used to explore the posterior density. We propose a new multi-fidelity surrogate model to accelerate the sampling procedure while maintaining high accuracy. Our numerical simulations demonstrate that this method not only produces high-quality reconstructions but also substantially reduces computational costs.

Appendices

Appendix A

Theorem 6.1

(Fernique Theorem, theorem 6.9 in [18]) If $\mu = N(0, C)$ is a Gaussian measure on some Banach space Z, so that $\mu (Z) = 1$, then there is a $\varepsilon$> 0 such that

$$\int _Z\exp (\varepsilon \Vert \varvec{z}\Vert _Z^2)d\mu (\varvec{z})<\infty .$$

Table 15 Example of f-divergence functions

Appendix B

For the sake of convenience, let $\displaystyle \pi _1(\varvec{z})=\frac{d\mu _1}{d\mu _0}$ and $\displaystyle \pi _2(\varvec{z})=\frac{d\mu _2}{d\mu _0}$ be the Radon–Nikodym derivatives of $\mu _1$ and $\mu _2$ with respect to $\mu _0$, respectively.

Appendix C

Algorithm 2

Data-driven method by neural network