arXiv subjects

Zhiqi Sun

Publications and source records attributed to Zhiqi Sun.

3 recordsLinked to original sources

arXiv2026

Inverse scattering for three-dimensional random obstacles with multi-frequency data

In many practical scenarios the shapes of scatterers exhibit uncertain geometric variations arising from diverse physical or environmental factors. For inverse scattering problems which are inherently ill-posed, the presence of such geometric uncertainties may have a non-negligible impact on the recovery process. With the aim of recovering both obstacle geometry and statistics of the shape uncertainties, in this paper we study an inverse acoustic scattering problem for three-dimensional smooth star-shaped obstacles with random isotropic fluctuations. We propose an efficient Monte Carlo-based multi-frequency recursive linearization algorithm in which the far-field operator is linearized with respect to the geometry parameters and frequency continuation is employed to recover the unknown geometry from coarse to fine scales. Based on the reconstructed samples, we further estimate the reference geometry and key statistics of the shape fluctuation field including Karhunen--Lo\`eve eigenvalues, covariance hyper-parameters for Gaussian perturbations and covariance structure, representative marginal distributions for non-Gaussian perturbations. We also prove that the probability law of the far-field data uniquely determines the radial function in distribution which implies uniqueness of the reference shape and related statistics. Numerical experiments demonstrate the effectiveness of the proposed method in recovering both the scatterer shapes and the associated statistical information under Gaussian and non-Gaussian random variations.

math.NA↗

arXiv2026

Inverse acoustic scattering for random obstacles with multi-frequency data

We study an inverse random obstacle scattering problems in $\mathbb{R}^2$ where the scatterer is formulated by a Gaussian process defined on the angular parameter domain. Equipped with a modified covariance function which is mathematically well-defined and physically consistent, the Gaussian process admits a parameterization via Karhunen--Lo\`eve (KL) expansion. Based on observed multi-frequency data, we develop a two-stage inversion method: the first stage reconstructs the baseline shape of the random scatterer and the second stage estimates the statistical characteristics of the boundary fluctuations, including KL eigenvalues and covariance hyperparameters. We further provide theoretical justifications for the modeling and inversion pipeline, covering well-definedness of the Gaussian-process model, convergence for the two-stage procedure and a brief discussion on uniqueness. Numerical experiments demonstrate stable recovery of both geometric and statistical information for obstacles with simple and more complex shapes.

math.NA↗

arXiv2025

An inverse random diffraction grating problem for the Helmholtz equation

This paper investigates the inverse scattering problem of time-harmonic plane waves incident on a perfectly reflecting random periodic structure. To simulate random perturbations arising from manufacturing defects and surface wear in real-world grating profiles, we propose a stochastic surface modeling framework motivated by the discretization of the Wiener process. Our approach introduces randomness at discrete nodes and then applies linear interpolation to construct the surface, marking a novel attempt to incorporate the concepts of the Wiener process into random surface representation. Under this framework, each realization of the random surface generates a Lipschitz-continuous diffraction grating, mathematically represented as a sum of a baseline profile and a weighted linear combination of local `tent' basis functions, meanwhile preserving key statistics of the random surface. Building on this representation, we introduce the Recursive Parametric Smoothing Strategy (RPSS) to invert the key statistics of our random surfaces. Combined with Monte Carlo sampling and a wavenumber continuation strategy, our reconstruction scheme demonstrates effectiveness across multiple benchmark scenarios. Several numerical results are presented along with some discussions in the end on reconstruction mechanisms and future extensions.

math.NA↗