arXiv subjects

Qingxiang Shi

Publications and source records attributed to Qingxiang Shi.

5 recordsLinked to original sources

arXiv2025

Identifying convex obstacles from backscattering far field data

The recovery of anomalies from backscattering far field data is a long-standing open problem in inverse scattering theory. We make a first step in this direction by establishing the unique identifiability of convex impenetrable obstacles from backscattering far field measurements. Specifically, we prove that both the boundary and the boundary conditions of the convex obstacle are uniquely determined by the far field pattern measured in backscattering directions for all frequencies. The key tool is Majda's asymptotic estimate of the far field patterns in the high-frequency regime. Furthermore, we introduce a fast and stable numerical algorithm for reconstructing the boundary and computing the boundary condition. A key feature of the algorithm is that the boundary condition can be computed even if the boundary is not known, and vice versa. Numerical experiments demonstrate the validity and robustness of the proposed algorithm.

math.NA↗

arXiv2025

The Radon Transform-Based Sampling Methods for Biharmonic Sources from the Scattered Fields

This paper presents three quantitative sampling methods for reconstructing extended sources of the biharmonic wave equation using scattered field data. The first method employs an indicator function that solely relies on scattered fields $ u^s$ measured on a single circle, eliminating the need for Laplacian or derivative data. Its theoretical foundation lies in an explicit formula for the source function, which also serves as a constructive proof of uniqueness. To improve computational efficiency, we introduce a simplified double integral formula for the source function, at the cost of requiring additional measurements $Δu^s$. This advancement motivates the second indicator function, which outperforms the first method in both computational speed and reconstruction accuracy. The third indicator function is proposed to reconstruct the support boundary of extended sources from the scattered fields $ u^s$ at a finite number of sensors. By analyzing singularities induced by the source boundary, we establish the uniqueness of annulus and polygon-shaped sources. A key characteristic of the first and third indicator functions is their link between scattered fields and the Radon transform of the source function. Numerical experiments demonstrate that the proposed sampling methods achieve high-resolution imaging of the source support or the source function itself.

math-ph↗

arXiv2025

The high resolution sampling methods for acoustic sources from multi-frequency far field patterns at sparse observation directions

This work is dedicated to novel uniqueness results and high resolution sampling methods for source support from multi-frequency sparse far field patterns. With a single pair of observation directions $\pm\hat{x}$, we prove that the lines $\{z\in\mathbb R^2|\, \hat{x}\cdot z = \hat{x}\cdot y, \,y\in A_{\hat{x}}\}$ can be determined by multi-frequency far field patterns at the directions $\pm\hat{x}$, where $A_{\hat{x}}$ denotes a set containing the corners of the boundary and points whose normal vector to the boundary is parallel to $\hat{x}$. Furthermore, if the source support is composed of polygons and annuluses, then we prove that the support can be determined by multi-frequency far field patterns at sparse directions. Precisely, the lowest number of the observation directions is given in terms of the number of the corners and the annuluses. Inspired by the uniqueness arguments, we introduce two novel indicators to determine the source support. Numerical examples in two dimensions are presented to show the validity and robustness of the two indicators for reconstructing the boundaries of the source support with a high resolution. The second indicator also shows its powerful ability to determine the unknown source function.

math.NA↗

arXiv2025

Identifying point sources for biharmonic wave equation from the scattered fields at sparse sensors

This work is dedicated to uniqueness and numerical algorithms for determining the point sources of the biharmonic wave equation using scattered fields at sparse sensors. We first show that the point sources in both $\mathbb{R}^2$ and $\mathbb{R}^3$ can be uniquely determined from the multifrequency sparse scattered fields. In particular, to deal with the challenges arising from the fundamental solution of the biharmonic wave equation in $\mathbb{R}^2$, we present an innovative approach that leverages the Fourier transform and Funk-Hecke formula. Such a technique can also be applied for identifying the point sources of the Helmholtz equation. Moreover, we present the uniqueness results for identifying multiple point sources in $\mathbb{R}^3$ from the scattered fields at sparse sensors with finitely many frequencies. Based on the constructive uniqueness proofs, we propose three numerical algorithms for identifying the point sources by using multifrequency sparse scattered fields. The numerical experiments are presented to verify the effectiveness and robustness of the algorithms.

math.AP↗

arXiv2025

A quantitative sampling method for elastic and electromagnetic sources

This work is dedicated to a novel sampling method for accurately reconstructing elastic and electromagnetic sources from the far field patterns. We show that the proposed indicators in the form of integrals with full far field patterns are exactly the source functions. These facts not only give constructive uniqueness proofs of the inverse source problems, but also establish the theoretical basis of the proposed sampling methods. Furthermore, we derive the stability estimates for the corresponding discrete indicators using the far field patterns with finitely many observations and frequencies. We have also proposed the indicators with partial far field patterns and proved their validity for providing the derivative information of the unknown sources. Numerical examples are presented to verify the accuracy and stability of the proposed quantitative sampling method.

math.NA↗