Numerical method for the inverse scattering by random periodic structures

TL;DR

This work tackles inverse scattering by random periodic acoustic-elastic interfaces and introduces a two-step Monte Carlo continuation (TS-MCC) framework to recover the statistics of the random surface from acoustic measurements. By combining Monte Carlo sampling across realizations with a wavenumber continuation and a two-step surface update (Tikhonov-regularized below-interface field followed by Landweber updates), the method reconstructs the mean profile, covariance, and, for non-Gaussian processes, the full distribution via kernel density estimation. The approach leverages Helmholtz decomposition and layer-potential representations to formulate a tractable nonlinear inverse problem, while avoiding reliance on prior stochastic models. Numerical experiments validate robustness to Gaussian and non-Gaussian randomness, demonstrate improved accuracy with higher frequency content, and highlight practical implications for uncertainty quantification in manufactured components.

Abstract

Due to manufacturing defects or wear and tear, industrial components may have uncertainties. In order to evaluate the performance of machined components, it is crucial to quantify the uncertainty of the scattering surface. This brings up an important class of inverse scattering problems for random interface reconstruction. In this paper, we present an efficient numerical algorithm for the inverse scattering problem of acoustic-elastic interaction with random periodic interfaces. The proposed algorithm combines the Monte Carlo technique and the continuation method with respect to the wavenumber, which can accurately reconstruct the key statistics of random periodic interfaces from the measured data of the acoustic scattered field. In the implementation of our algorithm, a key two-step strategy is employed: Firstly, the elastic displacement field below the interface is determined by Tikhonov regularization based on the dynamic interface condition; Secondly, the profile function is iteratively updated and optimised using the Landweber method according to the kinematic interface condition. Such a algorithm does not require a priori information about the stochastic structures and performs well for both stationary Gaussian and non-Gaussian stochastic processes. Numerical experiments demonstrate the reliability and effectiveness of our proposed method.

Figures (15)

• Figure 1: Problem geometry of acoustic-elastic interaction.
• Figure 2: Realisations of random surfaces in one period with different parameters. Solid lines indicate given deterministic contour functions and dashed lines denote random samples. Left column: $l=2$, rows 1-3 correspond to $\sigma=1/3, 1/6, 1/12$, respectively. Right column: $\sigma=1/5$, rows 1-3 correspond to $l=0.5,1,2$, respectively.
• Figure 3: The reconstruction results of Example \ref{['Example1']}. Left column: the solid line indicates the given deterministic profile function and the dashed line denotes the reconstructed mean profile function. Middle column: exact covariance matrix. Right column: reconstructed covariance matrix. Rows 1-3 correspond to $l=2$ and $\sigma=1/6, 1/9, 1/12$, respectively.
• Figure 4: The reconstruction results of Example \ref{['Example1']}. Left column: the solid line indicates the given deterministic profile function and the dashed line denotes the reconstructed mean profile function. Middle column: exact covariance matrix. Right column: reconstructed covariance matrix. Rows 1-3 correspond to $\sigma=1/12$ and $\l=0.5,1,2$, respectively.
• Figure 5: The reconstruction results of Example \ref{['Example1']}. Left column: the solid line indicates the given deterministic profile function (DPF) and the dashed lines denote the reconstructed mean profile functions (RMPFs). Middle column: the reconstructed covariance matrix for $\kappa_Q=3$. Right column: the reconstructed covariance matrix for $\kappa_Q=4$.

Theorems & Definitions (5)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5