A Bayesian scheme for reconstructing obstacles in acoustic waveguides. (English) Zbl 07761541
Summary: 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.
MSC:
| 62F15 | Bayesian inference |
| 45Q05 | Inverse problems for integral equations |
| 81Q37 | Quantum dots, waveguides, ratchets, etc. |
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