arXiv subjects

Guanqiu Ma

Publications and source records attributed to Guanqiu Ma.

10 recordsLinked to original sources

arXiv2026

A novel sampling method for reconstruction of a moving point acoustic source in $\mathbb{R}^3$

In this paper, we introduce a novel sampling method to recovering the trajectory of a moving point source in R^3, where both the spatial location and emission moment of the moving point source are unknown. Combining algebraic theory with geometric knowledge, we prove the uniqueness of the source location by using measured data from five observation points. Our sampling method constructs an indicator function based on the property that the residual of the time difference of arrival constraint formula vanishes at the true source location. It achieves the reconstruction of the spatial positions and emission moments of a moving point source only using data from five observation points and their corresponding arrival times. The algorithm not only reduces the required number of observation points, but also improves computational efficiency, stability, and noise resistance. Numerical experiments verify the effectiveness of the method.

math.NA↗

arXiv2026

DDC-PINNs: A Predictor-Corrector Approach Based on Neural Network-Driven Domain Decomposition and Classical ODE Solvers for Time-Dependent PDEs

When solving time-dependent partial differential equations (PDEs), traditional physics-informed neural networks (PINNs) may encounter several challenges. In particular, standard PINNs do not explicitly account for the temporal evolution order of time-dependent problems during training, which may affect the quality of temporal evolution in time-dependent PDEs. In addition, a single neural network may face difficulties in simultaneously representing different physical behaviors across multiple regions of the computational domain.To address these issues, we propose a domain-decomposition-based causal PINNs (DDC-PINNs) framework. The term causal refers to the fact that the temporal evolution is performed sequentially through classical ordinary differential equation (ODE) integration, thereby respecting the natural temporal ordering of time-dependent PDEs. The proposed framework enhances spatial approximation through domain decomposition and employs a sequential temporal-evolution strategy for time-dependent problems.Within this framework, an approximate solution is first obtained using domain-decomposition PINNs. Subsequently, the time-derivative term in the original PDE is retained, while the remaining solution-dependent terms are replaced by the obtained approximation, thereby transforming the original PDE into an auxiliary ODE system. Classical numerical methods for ODEs are then employed to perform temporal evolution without repeated neural-network optimization. As a result, DDC-PINNs decouples spatial approximation from temporal evolution while preserving the temporal evolution order through sequential ODE integration.Numerical experiments on several benchmark problems demonstrate the effectiveness of the proposed framework and provide proof-of-concept validation of the DDC-PINNs methodology.

math.NA↗

arXiv2026

A frequency-domain method to inverse moving source problem with unknown radiating moment

This paper introduces a multi-frequency factorization method for imaging a time-dependent source, specifically to recover its spatial support and the associated excitation instants. Using far-field data from two opposite directions, we establish a computational criterion that characterizes both the unknown pulse moments and the narrowest strip (perpendicular to the direction) enclosing the source support. Central to our inversion scheme is the construction of indicator functions, defined pointwise over the spatial and temporal sampling variables. The proposed inversion scheme permits the recovery of the $Θ$-convex support domain from far-field data at sparse observation directions. Uniqueness in determining the convex hull of the support and the excitation instants-using all observation directions-is also established as a direct consequence of the factorization method. The effectiveness and feasibility of the approach are examined through comprehensive numerical simulations in two and three dimensions.

math.NA↗

arXiv2025

Imaging a moving point source in R^3 from the time of arrival at sparse observation points

In this paper, we introduce a novel numerical method for reconstructing the trajectory within three-dimensional space, where both the emission moment and spatial location of the point source are unknown. Our approach relies solely on measuring the time of arrival at five or seven properly chosen observation points. By utilizing the distinctive geometric configuration of these five or seven observation points, we establish the uniqueness of the trajectory and emission moment of the point source through rigorous mathematical proofs. Moreover, we analyze the stability of our proposed method. The effectiveness of the method is also verified by numerical experiments.

math.NA↗

arXiv2024

Detection of a piecewise linear crack with one incident wave

This paper is concerned with inverse crack scattering problems for time-harmonic acoustic waves. We prove that a piecewise linear crack with the sound-soft boundary condition in two dimensions can be uniquely determined by the far-field data corresponding to a single incident plane wave or point source. We propose two non-iterative methods for imaging the location and shape of a crack. The first one is a contrast sampling method, while the second one is a variant of the classical factorization method but only with one incoming wave. Newton's iteration method is then employed for getting a more precise reconstruction result. Numerical examples are presented to show the effectiveness of the proposed hybrid method.

math.NA↗

arXiv2024

Inverse wave-number-dependent source problems for the Helmholtz equation with partial information on radiating period

This paper addresses a factorization method for imaging the support of a wave-number-dependent source function from multi-frequency data measured at a finite pair of symmetric receivers in opposite directions. The source function is given by the inverse Fourier transform of a compactly supported time-dependent source whose initial moment or terminal moment for radiating is unknown. Using the multi-frequency far-field data at two opposite observation directions, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the directions. A new parameter is incorporated into the design of test functions for indicating the unknown moment. The data from a finite pair of opposite directions can be used to recover the $Θ$-convex polygon of the support. Uniqueness in recovering the convex hull of the support is obtained as a by-product of our analysis using all observation directions. Similar results are also discussed with the multi-frequency near-field data from a finite pair of observation positions in three dimensions. We further comment on possible extensions to source functions with two disconnected supports. Extensive numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. The theoretical framework explored here should be seen as the frequency-domain analysis for inverse source problems in the time domain.

math.NA↗

arXiv2023

Imaging a moving point source from multi-frequency data measured at one and sparse observation points (part II): near-field case in 3D

In this paper, we introduce a frequency-domain approach to extract information on the trajectory of a moving point source. The method hinges on the analysis of multi-frequency near-field data recorded at one and sparse observation points in three dimensions. The radiating period of the moving point source is supposed to be supported on the real axis and a priori known. In contrast to inverse stationary source problems, one needs to classify observable and non-observable measurement positions. The analogue of these concepts in the far-field regime were firstly proposed in the authors' previous paper (SIAM J. Imag. Sci., 16 (2023): 1535-1571). In this paper we shall derive the observable and non-observable measurement positions for straight and circular motions in $\R^3$. In the near-field case, we verify that the smallest annular region centered at an observable position that contains the trajectory can be imaged for an admissible class of orbit functions. Using the data from sparse observable positions, it is possible to reconstruct the $Θ$-convex domain of the trajectory. Intensive 3D numerical tests with synthetic data are performed to show effectiveness and feasibility of this new algorithm.

math.NA↗

arXiv2022

Imaging a moving point source from multi-frequency data measured at one and sparse observation directions (part I): far-field case

We propose a multi-frequency algorithm for imaging the trajectory of a moving point source from one and sparse far-field observation directions in the frequency domain. The starting and terminal time points of the moving source are both supposed to be known. We introduce the concept of observable directions (angles) in the far-field region and derive all observable directions (angles) for straight and circular motions. At an observable direction, it is verified that the smallest trip containing the trajectory and perpendicular to the direction can be imaged, provided the orbit function possesses a certain monotonical property. Without the monotonicity one can only expect to recover a thinner strip. The far-field data measured at sparse observable directions can be used to recover the $Θ$-convex domain of the trajectory. Both two- and three-dimensional numerical examples are implemented to show effectiveness and feasibility of the approach.

math.NA↗

arXiv2021

Factorization method for inverse time-harmonic elastic scattering with a single plane wave

This paper is concerned with the factorization method with a single far-field pattern to recover an arbitrary convex polygonal scatterer/source in linear elasticity. The approach also applies to the compressional (resp. shear) part of the far-field pattern excited by a single compressional (resp. shear) plane wave. The one-wave factorization is based on the scattering data for a priori given testing scatterers. It can be regarded as a domain-defined sampling method and does not require forward solvers. We derive the spectral system of the far-field operator for rigid disks and show that, using testing disks, the one-wave factorization method can be justified independently of the classical factorization method.

math.NA↗

arXiv2021

Factorization method with one plane wave: from model-driven and data-driven perspectives

The factorization method by Kirsch (1998) provides a necessary and sufficient condition for characterizing the shape and position of an unknown scatterer by using far-field patterns of infinitely many time-harmonic plane waves at a fixed frequency. This paper is concerned with the factorization method with a single far-field pattern to recover a convex polygonal scatterer/source. Its one-wave version relies on the absence of analytical continuation of the scattered/radiated wave-fields in corner domains. It can be regarded as a domain-defined sampling method and does not require forward solvers. In this paper we provide a rigorous mathematical justification of the one-wave factorization method and present some preliminary numerical examples. In particular, the proposed scheme can be interpreted as a model-driven and data-driven method, because it essentially depends on the scattering model and a priori given \emph{sample data}.

math.NA↗